[ { "id": "Trigonometry_0000", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "The Pythagorean theorem & its converse", "level": "5", "keywords": [ "pythagorean theorem", "distance" ], "problem_v1": "A $35 ft$ ladder leans up agains the side of a house, with the base of the ladder a distance $6 ft$ from the wall. If the ladder is moved out by $5 ft$, how far down the wall will the top of the ladder move? distance moved down the wall [ANS] $ft$.", "answer_v1": [ "1.25538384746103" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A $40 ft$ ladder leans up agains the side of a house, with the base of the ladder a distance $3 ft$ from the wall. If the ladder is moved out by $3 ft$, how far down the wall will the top of the ladder move? distance moved down the wall [ANS] $ft$.", "answer_v2": [ "0.339901483787884" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A $35 ft$ ladder leans up agains the side of a house, with the base of the ladder a distance $4 ft$ from the wall. If the ladder is moved out by $4 ft$, how far down the wall will the top of the ladder move? distance moved down the wall [ANS] $ft$.", "answer_v3": [ "0.697227226625778" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0001", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "The Pythagorean theorem & its converse", "level": "2", "keywords": [ "triangle", "Pythagorean" ], "problem_v1": "Two perpendicular sides of a triangle are 5.1 mm and 6.1 mm long respectively. Determine the length of the remaining side of the triangle. [ANS] mm", "answer_v1": [ "7.95110055275369" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two perpendicular sides of a triangle are 2.3 mm and 7.9 mm long respectively. Determine the length of the remaining side of the triangle. [ANS] mm", "answer_v2": [ "8.22800097228968" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two perpendicular sides of a triangle are 3.3 mm and 6.1 mm long respectively. Determine the length of the remaining side of the triangle. [ANS] mm", "answer_v3": [ "6.93541635375988" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0002", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "The Pythagorean theorem & its converse", "level": "2", "keywords": [ "algebra", "coordinate geometry", "distance", "midpoint" ], "problem_v1": "The distance between $(4,-3)$ and $(-6,-4)$ is [ANS]\nThe midpoint of the line segment that joins the given points is given by: $($ [ANS], [ANS] $)$.", "answer_v1": [ "10.0498756211209", "-1", "-3.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The distance between $(1,-1)$ and $(-9,-7)$ is [ANS]\nThe midpoint of the line segment that joins the given points is given by: $($ [ANS], [ANS] $)$.", "answer_v2": [ "11.6619037896906", "-4", "-4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The distance between $(2,-2)$ and $(-8,-5)$ is [ANS]\nThe midpoint of the line segment that joins the given points is given by: $($ [ANS], [ANS] $)$.", "answer_v3": [ "10.4403065089106", "-3", "-3.5" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0003", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "calculus", "trigonometry", "angles", "radians" ], "problem_v1": "Convert from radians to degrees.\n(a) $\\frac{5}{2}$ [ANS]\n(b) $\\frac{4 \\pi}{3}$ [ANS]\n(c) $\\frac{11}{12}$ [ANS]\n(d) $-\\frac{3 \\pi}{4}$ [ANS]", "answer_v1": [ "143.239", "240", "52.5211", "-135" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Convert from radians to degrees.\n(a) $\\frac{5}{2}$ [ANS]\n(b) $\\frac{2 \\pi}{3}$ [ANS]\n(c) $\\frac{7}{12}$ [ANS]\n(d) $-\\frac{5 \\pi}{4}$ [ANS]", "answer_v2": [ "143.239", "120", "33.4226", "-225" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Convert from radians to degrees.\n(a) $\\frac{5}{2}$ [ANS]\n(b) $\\frac{2 \\pi}{3}$ [ANS]\n(c) $\\frac{11}{12}$ [ANS]\n(d) $-\\frac{3 \\pi}{4}$ [ANS]", "answer_v3": [ "143.239", "120", "52.5211", "-135" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0004", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "Trig", "Radian", "Degree", "trigonometry", "angle measure", "radian/degree conversion", "angles" ], "problem_v1": "(a) Convert $\\frac{16}{12}\\pi$ from radians to degrees. $\\frac{16}{12}\\pi$=[ANS] degrees. (b) Convert $218 ^{\\circ}$ from degrees to radians. $218 ^{\\circ}$=[ANS] radians.", "answer_v1": [ "240", "3.80482" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Convert $\\frac{2}{19}\\pi$ from radians to degrees. $\\frac{2}{19}\\pi$=[ANS] degrees. (b) Convert $-631 ^{\\circ}$ from degrees to radians. $-631 ^{\\circ}$=[ANS] radians.", "answer_v2": [ "18.9474", "-11.013" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Convert $\\frac{7}{13}\\pi$ from radians to degrees. $\\frac{7}{13}\\pi$=[ANS] degrees. (b) Convert $-399 ^{\\circ}$ from degrees to radians. $-399 ^{\\circ}$=[ANS] radians.", "answer_v3": [ "96.9231", "-6.96386" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0005", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "Trig", "Degree", "Radian", "calculus", "trigonometry", "angle measure", "radian/degree conversion", "angles" ], "problem_v1": "Convert the following radian measures to degree measures:\n$\\begin{array}{cccc}\\hline 1. & \\frac{4\\pi}{6} &=& [ANS]degrees \\\\\\hline 2. & \\frac{3\\pi}{4} &=& [ANS]degrees \\\\\\hline 3. & \\frac{4\\pi}{3} &=& [ANS]degrees \\\\\\hline 4. & \\frac{5\\pi}{2} &=& [ANS]degrees \\\\\\hline 5. & 2\\pi &=& [ANS]degrees \\\\\\hline\\end{array}$", "answer_v1": [ "120", "135", "240", "450", "360" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Convert the following radian measures to degree measures:\n$\\begin{array}{cccc}\\hline 1. & \\frac{\\pi}{6} &=& [ANS]degrees \\\\\\hline 2. & \\frac{5\\pi}{4} &=& [ANS]degrees \\\\\\hline 3. & \\frac{\\pi}{3} &=& [ANS]degrees \\\\\\hline 4. & \\frac{3\\pi}{2} &=& [ANS]degrees \\\\\\hline 5. & 5\\pi &=& [ANS]degrees \\\\\\hline\\end{array}$", "answer_v2": [ "30", "225", "60", "270", "900" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Convert the following radian measures to degree measures:\n$\\begin{array}{cccc}\\hline 1. & \\frac{2\\pi}{6} &=& [ANS]degrees \\\\\\hline 2. & \\frac{2\\pi}{4} &=& [ANS]degrees \\\\\\hline 3. & \\frac{2\\pi}{3} &=& [ANS]degrees \\\\\\hline 4. & \\frac{3\\pi}{2} &=& [ANS]degrees \\\\\\hline 5. & 5\\pi &=& [ANS]degrees \\\\\\hline\\end{array}$", "answer_v3": [ "60", "90", "120", "270", "900" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0006", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "sine", "cosine", "radians", "degrees" ], "problem_v1": "What angle (in degrees) corresponds to 17.4 rotations around the unit circle? 17.4 rotations is an angle of [ANS] degrees.", "answer_v1": [ "6264" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What angle (in degrees) corresponds to 11.4 rotations around the unit circle? 11.4 rotations is an angle of [ANS] degrees.", "answer_v2": [ "4104" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What angle (in degrees) corresponds to 13.4 rotations around the unit circle? 13.4 rotations is an angle of [ANS] degrees.", "answer_v3": [ "4824" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0007", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "radians", "degrees", "tangent", "arc length" ], "problem_v1": "Determine the exact radian measure for the angle $325^{\\circ}$. Do not give a decimal approximation, and recall in order to enter $\\pi$ you must type pi. $325^{\\circ}=$ [ANS] radians", "answer_v1": [ "325*pi/180" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the exact radian measure for the angle $285^{\\circ}$. Do not give a decimal approximation, and recall in order to enter $\\pi$ you must type pi. $285^{\\circ}=$ [ANS] radians", "answer_v2": [ "285*pi/180" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the exact radian measure for the angle $300^{\\circ}$. Do not give a decimal approximation, and recall in order to enter $\\pi$ you must type pi. $300^{\\circ}=$ [ANS] radians", "answer_v3": [ "300*pi/180" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0008", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "radians", "degrees", "tangent", "arc length" ], "problem_v1": "Determine the degree measure for the angle $240$ radians. $240$ radians=[ANS] degrees", "answer_v1": [ "240*180/pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the degree measure for the angle $30$ radians. $30$ radians=[ANS] degrees", "answer_v2": [ "30*180/pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the degree measure for the angle $120$ radians. $120$ radians=[ANS] degrees", "answer_v3": [ "120*180/pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0009", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "3", "keywords": [ "sine", "cosine", "radians", "degrees" ], "problem_v1": "What is the exact radian angle measure for $135^{\\circ}$ as a fraction of $\\pi$? [ANS] radians What is a decimal approximation for the radian angle measure for $135^{\\circ}$ accurate to three decimal places? [ANS] radians", "answer_v1": [ "3*pi/4", "2.35619" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "What is the exact radian angle measure for $30^{\\circ}$ as a fraction of $\\pi$? [ANS] radians What is a decimal approximation for the radian angle measure for $30^{\\circ}$ accurate to three decimal places? [ANS] radians", "answer_v2": [ "pi/6", "0.523599" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "What is the exact radian angle measure for $45^{\\circ}$ as a fraction of $\\pi$? [ANS] radians What is a decimal approximation for the radian angle measure for $45^{\\circ}$ accurate to three decimal places? [ANS] radians", "answer_v3": [ "pi/4", "0.785398" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0010", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "sine", "cosine", "radians", "degrees" ], "problem_v1": "An angle of $9 \\pi$ radians can be converted to an angle of [ANS] degrees.", "answer_v1": [ "1620" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An angle of $3 \\pi$ radians can be converted to an angle of [ANS] degrees.", "answer_v2": [ "540" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An angle of $5 \\pi$ radians can be converted to an angle of [ANS] degrees.", "answer_v3": [ "900" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0011", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "algebra", "trigonometric function", "angle measure", "trigonometry", "angle measure", "radian/degree conversion" ], "problem_v1": "The radian measure of an angle of 266 degrees is [ANS].", "answer_v1": [ "4.64257581030492" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The radian measure of an angle of 38 degrees is [ANS].", "answer_v2": [ "0.663225115757845" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The radian measure of an angle of 116 degrees is [ANS].", "answer_v3": [ "2.02458193231342" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0012", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "algebra", "trigonometric function", "angle measure", "trigonometry", "angle measure", "radian/degree conversion" ], "problem_v1": "Convert 1.5 in radians to degrees: [ANS].", "answer_v1": [ "85.9436692696235" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Convert-2.5 in radians to degrees: [ANS].", "answer_v2": [ "-143.239448782706" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Convert-1.1 in radians to degrees: [ANS].", "answer_v3": [ "-63.0253574643906" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0013", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "algebra", "trigonometric function", "angle measure", "trigonometry", "angle measure", "radian/degree conversion" ], "problem_v1": "Convert $\\frac{7}{8}\\pi$ in radians to degrees: [ANS].", "answer_v1": [ "157.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Convert $\\frac{1}{2}\\pi$ in radians to degrees: [ANS].", "answer_v2": [ "90" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Convert $\\frac{3}{4}\\pi$ in radians to degrees: [ANS].", "answer_v3": [ "135" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0014", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "algebra" ], "problem_v1": "Fill in the following equations: $45^\\circ=$ [ANS] rad $270^\\circ=$ [ANS] rad $-570^\\circ=$ [ANS] rad Hint: Everything follows from the fact that 360^\\circ=2\\pi \\hbox{~rad}.", "answer_v1": [ "0.785398163397448", "4.71238898038469", "-9.94837673636768" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Fill in the following equations: $45^\\circ=$ [ANS] rad $30^\\circ=$ [ANS] rad $-694^\\circ=$ [ANS] rad Hint: Everything follows from the fact that 360^\\circ=2\\pi \\hbox{~rad}.", "answer_v2": [ "0.785398163397448", "0.523598775598299", "-12.1125850088406" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Fill in the following equations: $45^\\circ=$ [ANS] rad $114^\\circ=$ [ANS] rad $-578^\\circ=$ [ANS] rad Hint: Everything follows from the fact that 360^\\circ=2\\pi \\hbox{~rad}.", "answer_v3": [ "0.785398163397448", "1.98967534727354", "-10.0880030765272" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0015", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "Minutes", "Seconds", "Degrees", "Angles" ], "problem_v1": "Convert each angle to decimal degrees. The angle $101^\\circ$ 35' equals [ANS] degrees. The angle $73^\\circ$ 43' equals [ANS] degrees. The angle $28^\\circ$ 20' 34\" equals [ANS] degrees.", "answer_v1": [ "101.583333333333", "73.7166666666667", "28.3427777777778" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Convert each angle to decimal degrees. The angle $-34^\\circ$ 55' equals [ANS] degrees. The angle $-211^\\circ$ 20' equals [ANS] degrees. The angle $86^\\circ$ 19' 11\" equals [ANS] degrees.", "answer_v2": [ "-34.9166666666667", "-211.333333333333", "86.3197222222222" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Convert each angle to decimal degrees. The angle $13^\\circ$ 36' equals [ANS] degrees. The angle $-133^\\circ$ 33' equals [ANS] degrees. The angle $19^\\circ$ 21' 48\" equals [ANS] degrees.", "answer_v3": [ "13.6", "-133.55", "19.3633333333333" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0016", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "angle measure", "quadrant", "degrees" ], "problem_v1": "For each angle (in degrees) below, determine the quadrant in which the terminal side of the angle is found. [NOTE: Enter '1' for quadrant I, '2' for quadrant II, '3' for quadrant III, and '4' for quadrant IV.]\n(a) $364^\\circ$ is found in quadrant [ANS]\n(b) $118^\\circ$ is found in quadrant [ANS]\n(c) $174^\\circ$ is found in quadrant [ANS]\n(d) $321^\\circ$ is found in quadrant [ANS]", "answer_v1": [ "1", "2", "2", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For each angle (in degrees) below, determine the quadrant in which the terminal side of the angle is found. [NOTE: Enter '1' for quadrant I, '2' for quadrant II, '3' for quadrant III, and '4' for quadrant IV.]\n(a) $-601^\\circ$ is found in quadrant [ANS]\n(b) $622^\\circ$ is found in quadrant [ANS]\n(c) $-505^\\circ$ is found in quadrant [ANS]\n(d) $-239^\\circ$ is found in quadrant [ANS]", "answer_v2": [ "2", "3", "3", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For each angle (in degrees) below, determine the quadrant in which the terminal side of the angle is found. [NOTE: Enter '1' for quadrant I, '2' for quadrant II, '3' for quadrant III, and '4' for quadrant IV.]\n(a) $-269^\\circ$ is found in quadrant [ANS]\n(b) $152^\\circ$ is found in quadrant [ANS]\n(c) $-319^\\circ$ is found in quadrant [ANS]\n(d) $70^\\circ$ is found in quadrant [ANS]", "answer_v3": [ "2", "2", "1", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0017", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Radians, converting radians & degrees", "level": "2", "keywords": [ "trigonometry", "decimal degrees", "minutes", "seconds", "DD", "DMS" ], "problem_v1": "For each of the following, use at least 5 decimal places. $43^{\\circ} 33' 21''$=[ANS] $^{\\circ}$ $\\sin(43^{\\circ} 33' 21'')$=[ANS]\n$77.2608333333333^{\\circ}$=[ANS] $^{\\circ}$ [ANS] $'$ [ANS] $''$", "answer_v1": [ "43.5558333333333", "sin(43.5558*pi/180)", "77", "15", "39" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For each of the following, use at least 5 decimal places. $23^{\\circ} 17' 45''$=[ANS] $^{\\circ}$ $\\sin(23^{\\circ} 17' 45'')$=[ANS]\n$87.1275^{\\circ}$=[ANS] $^{\\circ}$ [ANS] $'$ [ANS] $''$", "answer_v2": [ "23.2958333333333", "sin(23.2958*pi/180)", "87", "7", "39" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For each of the following, use at least 5 decimal places. $29^{\\circ} 21' 17''$=[ANS] $^{\\circ}$ $\\sin(29^{\\circ} 21' 17'')$=[ANS]\n$79.1947222222222^{\\circ}$=[ANS] $^{\\circ}$ [ANS] $'$ [ANS] $''$", "answer_v3": [ "29.3547222222222", "sin(29.3547*pi/180)", "79", "11", "41" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0018", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "3", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "radians", "degrees", "tangent", "arc length" ], "problem_v1": "Find the arc length corresponding to the given angle (in degrees) on a circle of radius 7.7. An angle of $31^{\\circ}$ has an arc length of [ANS] units.", "answer_v1": [ "7.7*31*pi/180" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the arc length corresponding to the given angle (in degrees) on a circle of radius 2.7. An angle of $41^{\\circ}$ has an arc length of [ANS] units.", "answer_v2": [ "2.7*41*pi/180" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the arc length corresponding to the given angle (in degrees) on a circle of radius 4.5. An angle of $33^{\\circ}$ has an arc length of [ANS] units.", "answer_v3": [ "4.5*33*pi/180" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0019", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "trigonometry", "angle measure", "arc length" ], "problem_v1": "A circular arc of length 14 feet subtends a central angle of 55 degrees. Find the radius of the circle in feet. (Note: You can enter $\\pi$ as 'pi' in your answer.) [ANS] feet", "answer_v1": [ "14.5843461351483" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A circular arc of length 19 feet subtends a central angle of 15 degrees. Find the radius of the circle in feet. (Note: You can enter $\\pi$ as 'pi' in your answer.) [ANS] feet", "answer_v2": [ "72.5744843391902" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A circular arc of length 14 feet subtends a central angle of 25 degrees. Find the radius of the circle in feet. (Note: You can enter $\\pi$ as 'pi' in your answer.) [ANS] feet", "answer_v3": [ "32.0855614973262" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0020", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "algebra", "trigonometric function", "angle measure" ], "problem_v1": "In a circle of radius 8, the length of the arc that subtends a central angle of 196 degrees is [ANS].", "answer_v1": [ "27.3667626712711" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In a circle of radius 2, the length of the arc that subtends a central angle of 309 degrees is [ANS].", "answer_v2": [ "10.786134777325" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In a circle of radius 4, the length of the arc that subtends a central angle of 204 degrees is [ANS].", "answer_v3": [ "14.2418866962737" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0021", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "algebra", "trigonometric function", "angle measure", "trigonometry", "angle measure", "arc length" ], "problem_v1": "Find the distance along an arc on the surface of the earth that subtends a central angle of 8 minutes (1 minute=1/60 degree). The radius of the earth is 3960 miles. Your answer is [ANS] miles.", "answer_v1": [ "9.21533845053006" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the distance along an arc on the surface of the earth that subtends a central angle of 1 minutes (1 minute=1/60 degree). The radius of the earth is 3960 miles. Your answer is [ANS] miles.", "answer_v2": [ "1.15191730631626" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the distance along an arc on the surface of the earth that subtends a central angle of 4 minutes (1 minute=1/60 degree). The radius of the earth is 3960 miles. Your answer is [ANS] miles.", "answer_v3": [ "4.60766922526503" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0022", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "5", "keywords": [ "algebra" ], "problem_v1": "You travel again in your trusty car with the 25 inch wheels. You installed a nifty gadget that measures how often your wheels turn each hour. It shows that right now they turn 55000 times per hour. You figure you are going at a speed of [ANS] miles per hour. (Round your answer to the nearest tenth of a mile per hour.) Hint: Multiply the circumference of the wheel with the number of rotations per hour. Convert inches to miles.", "answer_v1": [ "68.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You travel again in your trusty car with the 25 inch wheels. You installed a nifty gadget that measures how often your wheels turn each hour. It shows that right now they turn 41000 times per hour. You figure you are going at a speed of [ANS] miles per hour. (Round your answer to the nearest tenth of a mile per hour.) Hint: Multiply the circumference of the wheel with the number of rotations per hour. Convert inches to miles.", "answer_v2": [ "50.8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You travel again in your trusty car with the 25 inch wheels. You installed a nifty gadget that measures how often your wheels turn each hour. It shows that right now they turn 46000 times per hour. You figure you are going at a speed of [ANS] miles per hour. (Round your answer to the nearest tenth of a mile per hour.) Hint: Multiply the circumference of the wheel with the number of rotations per hour. Convert inches to miles.", "answer_v3": [ "57" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0023", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "5", "keywords": [ "Word Problem", "Arc Length" ], "problem_v1": "Paul Ernster kicked many filed goals during his time on the NAU football team. Assume that the distance between the goal posts, 18.5 ft, is the length of an arc on a circle of radius 43 yards (the length of the fild goal to be kicked). Ernster aims to kick the ball midway between the uprights. To score a field goal, what is the maximum number of degrees that the actual trajectory can deviate from the intended trajectory? [ANS] degrees.", "answer_v1": [ "4.10841829841869" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Paul Ernster kicked many filed goals during his time on the NAU football team. Assume that the distance between the goal posts, 18.5 ft, is the length of an arc on a circle of radius 22 yards (the length of the fild goal to be kicked). Ernster aims to kick the ball midway between the uprights. To score a field goal, what is the maximum number of degrees that the actual trajectory can deviate from the intended trajectory? [ANS] degrees.", "answer_v2": [ "8.03009031054563" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Paul Ernster kicked many filed goals during his time on the NAU football team. Assume that the distance between the goal posts, 18.5 ft, is the length of an arc on a circle of radius 29 yards (the length of the fild goal to be kicked). Ernster aims to kick the ball midway between the uprights. To score a field goal, what is the maximum number of degrees that the actual trajectory can deviate from the intended trajectory? [ANS] degrees.", "answer_v3": [ "6.09179264937944" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0024", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "", "keywords": [ "Angular Velocity" ], "problem_v1": "Suppose a car runs over a nail while driving at a speed of 62 miles per hour, and the nail is lodged in the tire tread 16 inches from the center of the wheel. What is the angular velocity of the nail in radians per hour? [ANS] radians per hour", "answer_v1": [ "245520" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose a car runs over a nail while driving at a speed of 51 miles per hour, and the nail is lodged in the tire tread 18 inches from the center of the wheel. What is the angular velocity of the nail in radians per hour? [ANS] radians per hour", "answer_v2": [ "179520" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose a car runs over a nail while driving at a speed of 55 miles per hour, and the nail is lodged in the tire tread 16 inches from the center of the wheel. What is the angular velocity of the nail in radians per hour? [ANS] radians per hour", "answer_v3": [ "217800" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0025", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "Linear Velocity" ], "problem_v1": "What is the linear velocity in MILES PER HOUR of the tip of a lawnmower blade spinning at 2800 revolutions per minute in a lawnmower that cuts a path that is 24 inches wide? [ANS] miles per hour", "answer_v1": [ "199.919532501169" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the linear velocity in MILES PER HOUR of the tip of a lawnmower blade spinning at 2000 revolutions per minute in a lawnmower that cuts a path that is 30 inches wide? [ANS] miles per hour", "answer_v2": [ "178.499582590329" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the linear velocity in MILES PER HOUR of the tip of a lawnmower blade spinning at 2300 revolutions per minute in a lawnmower that cuts a path that is 24 inches wide? [ANS] miles per hour", "answer_v3": [ "164.219615983103" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0026", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "angle measure", "arc length", "radius" ], "problem_v1": "The radius of the circle with a central angle of 5 radians that intercepts an arc with length 81 cm is [ANS] cm. The radius of the circle with a central angle of $196^\\circ$ that intercepts an arc with length 16 miles is [ANS] miles.", "answer_v1": [ "16.2", "4.67720649086386" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The radius of the circle with a central angle of 3 radians that intercepts an arc with length 57 cm is [ANS] cm. The radius of the circle with a central angle of $309^\\circ$ that intercepts an arc with length 3 miles is [ANS] miles.", "answer_v2": [ "19", "0.556269704010508" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The radius of the circle with a central angle of 4 radians that intercepts an arc with length 64 cm is [ANS] cm. The radius of the circle with a central angle of $204^\\circ$ that intercepts an arc with length 7 miles is [ANS] miles.", "answer_v3": [ "16", "1.96603164995871" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0027", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "3", "keywords": [ "trigonometry", "angle", "degree" ], "problem_v1": "A searchlight rotates through one complete revolution every 8 seconds. Determine: the angle covered in one second=[ANS]\nthe time it takes the light to rotate through $180^{\\circ}$=[ANS]", "answer_v1": [ "360/8", "8/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A searchlight rotates through one complete revolution every 3 seconds. Determine: the angle covered in one second=[ANS]\nthe time it takes the light to rotate through $180^{\\circ}$=[ANS]", "answer_v2": [ "360/3", "3/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A searchlight rotates through one complete revolution every 5 seconds. Determine: the angle covered in one second=[ANS]\nthe time it takes the light to rotate through $180^{\\circ}$=[ANS]", "answer_v3": [ "360/5", "5/2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0028", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "radian", "unit circle", "area", "sector" ], "problem_v1": "You instructor is considering the installation of a circular dart board with radius $5.1$ on the office wall. instead of actually reading your papers, it has been suggested that grades be assigned by chance by dividing the board into three grade sectors and tossing a dart. The area of an \"A\" sector with central angle $2.266$ is [ANS]. The area of a \"B\" sector with central angle $1.64819$ is [ANS]. The area of a \"C\" sector with central angle $135.734 ^o$ is [ANS]. Assume a dart is randomly thrown at the board and the grade is noted on your paper. List your grade in order of increasing likelihood: [ANS] [ANS] [ANS]. If the chance of your receiving a given grade is based upon the size of the sector divided by the size of the whole circle, determine the likelihood that you will receive an \"A\". [ANS]", "answer_v1": [ "29.46933", "21.4346", "30.808845", "C", "A", "B", "0.360645" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS", "MCS", "MCS", "NV" ], "options_v1": [ [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [] ], "problem_v2": "You instructor is considering the installation of a circular dart board with radius $3.5$ on the office wall. instead of actually reading your papers, it has been suggested that grades be assigned by chance by dividing the board into three grade sectors and tossing a dart. The area of an \"A\" sector with central angle $2.987$ is [ANS]. The area of a \"B\" sector with central angle $2.06019$ is [ANS]. The area of a \"C\" sector with central angle $70.8176 ^o$ is [ANS]. Assume a dart is randomly thrown at the board and the grade is noted on your paper. List your grade in order of increasing likelihood: [ANS] [ANS] [ANS]. If the chance of your receiving a given grade is based upon the size of the sector divided by the size of the whole circle, determine the likelihood that you will receive an \"A\". [ANS]", "answer_v2": [ "18.295375", "12.6186", "7.5705", "A", "B", "C", "0.475396" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS", "MCS", "MCS", "NV" ], "options_v2": [ [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [] ], "problem_v3": "You instructor is considering the installation of a circular dart board with radius $4.1$ on the office wall. instead of actually reading your papers, it has been suggested that grades be assigned by chance by dividing the board into three grade sectors and tossing a dart. The area of an \"A\" sector with central angle $2.266$ is [ANS]. The area of a \"B\" sector with central angle $2.36919$ is [ANS]. The area of a \"C\" sector with central angle $94.4234 ^o$ is [ANS]. Assume a dart is randomly thrown at the board and the grade is noted on your paper. List your grade in order of increasing likelihood: [ANS] [ANS] [ANS]. If the chance of your receiving a given grade is based upon the size of the sector divided by the size of the whole circle, determine the likelihood that you will receive an \"A\". [ANS]", "answer_v3": [ "19.04573", "19.913", "13.85144", "B", "A", "C", "0.360645" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS", "MCS", "MCS", "NV" ], "options_v3": [ [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [] ] }, { "id": "Trigonometry_0029", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "radian", "unit circle", "arclength", "area" ], "problem_v1": "A circular sector has radius $r=5.1$ and central angle $\\theta=100^\\circ$. Determine: Arclength=[ANS]. Area=[ANS].", "answer_v1": [ "8.90117918517108", "22.6980069221863" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A circular sector has radius $r=3.5$ and central angle $\\theta=155^\\circ$. Determine: Arclength=[ANS]. Area=[ANS].", "answer_v2": [ "9.46841119206924", "16.5697195861212" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A circular sector has radius $r=4.1$ and central angle $\\theta=105^\\circ$. Determine: Arclength=[ANS]. Area=[ANS].", "answer_v3": [ "7.51364242983559", "15.402966981163" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0030", "subject": "Trigonometry", "topic": "Geometric and algebraic foundations for trigonometry", "subtopic": "Arc length, sector area, angular and linear velocity", "level": "2", "keywords": [ "radian", "unit circle", "velocity", "speed", "linear", "angular" ], "problem_v1": "An jet is waiting to land at a busy airport and so is flying around the airport in a perfectly circular manner with radius $5.1$ miles. They are going pretty fast and discover that they are completing $0.6$ revolutions per minute. The angle covered per minute is [ANS] radians. The airspeed of the plane is [ANS] miles per hour. A camera in the control tower is taping the plane's flight. The speed that the camera must turn to keep up is [ANS] radians per second. The total angle (in radians) that the camera turns in $11$ minutes is [ANS].", "answer_v1": [ "3.76991118430775", "1153.59282239817", "0.0628318530717959", "41.4690230273853" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "An jet is waiting to land at a busy airport and so is flying around the airport in a perfectly circular manner with radius $3.5$ miles. They are going pretty fast and discover that they are completing $0.85$ revolutions per minute. The angle covered per minute is [ANS] radians. The airspeed of the plane is [ANS] miles per hour. A camera in the control tower is taping the plane's flight. The speed that the camera must turn to keep up is [ANS] radians per second. The total angle (in radians) that the camera turns in $6$ minutes is [ANS].", "answer_v2": [ "5.34070751110265", "1121.54857733156", "0.0890117918517108", "32.0442450666159" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "An jet is waiting to land at a busy airport and so is flying around the airport in a perfectly circular manner with radius $4.1$ miles. They are going pretty fast and discover that they are completing $0.6$ revolutions per minute. The angle covered per minute is [ANS] radians. The airspeed of the plane is [ANS] miles per hour. A camera in the control tower is taping the plane's flight. The speed that the camera must turn to keep up is [ANS] radians per second. The total angle (in radians) that the camera turns in $8$ minutes is [ANS].", "answer_v3": [ "3.76991118430775", "927.398151339707", "0.0628318530717959", "30.159289474462" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0031", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Unit circle", "level": "2", "keywords": [ "trigonometry", "unit circle", "precalculus" ], "problem_v1": "If $P(t)=(\\cos t, \\sin t)$ has coordinates (0.814,0.581), find the coordinates of a. $P(t+\\pi)$ $x$=[ANS] $y$=[ANS]\nb. $P(-t)$ $x$=[ANS] $y$=[ANS]\nc. $P(t-\\pi)$ $x$=[ANS] $y$=[ANS]\nc. $P(-t-\\pi)$ $x$=[ANS] $y$=[ANS]", "answer_v1": [ "-0.814", "-0.581", "0.814", "-0.581", "-0.814", "-0.581", "-0.814", "0.581" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "If $P(t)=(\\cos t, \\sin t)$ has coordinates (0.11,0.994), find the coordinates of a. $P(t+\\pi)$ $x$=[ANS] $y$=[ANS]\nb. $P(-t)$ $x$=[ANS] $y$=[ANS]\nc. $P(t-\\pi)$ $x$=[ANS] $y$=[ANS]\nc. $P(-t-\\pi)$ $x$=[ANS] $y$=[ANS]", "answer_v2": [ "-0.11", "-0.994", "0.11", "-0.994", "-0.11", "-0.994", "-0.11", "0.994" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "If $P(t)=(\\cos t, \\sin t)$ has coordinates (0.394,0.919), find the coordinates of a. $P(t+\\pi)$ $x$=[ANS] $y$=[ANS]\nb. $P(-t)$ $x$=[ANS] $y$=[ANS]\nc. $P(t-\\pi)$ $x$=[ANS] $y$=[ANS]\nc. $P(-t-\\pi)$ $x$=[ANS] $y$=[ANS]", "answer_v3": [ "-0.394", "-0.919", "0.394", "-0.919", "-0.394", "-0.919", "-0.394", "0.919" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Trigonometry_0032", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Unit circle", "level": "2", "keywords": [ "algebra", "unit circle" ], "problem_v1": "If the point $P(\\frac{13}{15},y)$ is on the unit circle in quadrant IV, then $y=$ [ANS].", "answer_v1": [ "-0.498887651569859" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If the point $P(\\frac{4}{7},y)$ is on the unit circle in quadrant IV, then $y=$ [ANS].", "answer_v2": [ "-0.82065180664829" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If the point $P(\\frac{7}{9},y)$ is on the unit circle in quadrant IV, then $y=$ [ANS].", "answer_v3": [ "-0.628539361054709" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0033", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "1", "keywords": [ "calculus", "functions", "algebraic functions", "polynomial functions", "transcendental functions", "rational functions" ], "problem_v1": "Identify the following function as polynomial, rational, algebraic, or transcendental. $f(x)=\\sin\\!\\left(8x^{2}+6\\right)$ [ANS] A. algebraic B. rational C. transcendental D. polynomial", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Identify the following function as polynomial, rational, algebraic, or transcendental. $f(x)=\\sin\\!\\left(x^{2}+10\\right)$ [ANS] A. algebraic B. rational C. polynomial D. transcendental", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Identify the following function as polynomial, rational, algebraic, or transcendental. $f(x)=\\sin\\!\\left(4x^{2}+6\\right)$ [ANS] A. algebraic B. transcendental C. polynomial D. rational", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Trigonometry_0034", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "3", "keywords": [ "Trig", "Transform", "trigonometry", "transformation", "precalculus", "trigonometric graphs" ], "problem_v1": "Find the equation of a sine wave that is obtained by shifting the graph of $y=\\sin(x)$ to the right 7 units and downward 6 units and is vertically stretched by a factor of 6 when compared to $y=\\sin(x)$. $y=$ [ANS]", "answer_v1": [ "6*sin(x-7)-6" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of a sine wave that is obtained by shifting the graph of $y=\\sin(x)$ to the right 1 units and downward 9 units and is vertically stretched by a factor of 3 when compared to $y=\\sin(x)$. $y=$ [ANS]", "answer_v2": [ "3*sin(x-1)-9" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of a sine wave that is obtained by shifting the graph of $y=\\sin(x)$ to the right 3 units and downward 6 units and is vertically stretched by a factor of 4 when compared to $y=\\sin(x)$. $y=$ [ANS]", "answer_v3": [ "4*sin(x-3)-6" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0035", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "3", "keywords": [ "sine", "cosine", "radians", "degrees" ], "problem_v1": "(a) Find another angle $\\phi$ between $0^\\circ$ and $360^\\circ$ that has the same cosine as $75^{\\circ}$. (That is, find $\\phi$ satisfying $\\cos(\\phi)=\\cos(75^\\circ)$.) $\\phi=$ [ANS] degrees. (b) Find another angle $\\phi$ between $0^\\circ$ and $360^\\circ$ that has the same sine as $75^{\\circ}$. (That is, find $\\phi$ satisfying $\\sin(\\phi)=\\sin(75^\\circ)$.) $\\phi=$ [ANS] degrees.", "answer_v1": [ "285", "105" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find another angle $\\phi$ between $0^\\circ$ and $360^\\circ$ that has the same cosine as $53^{\\circ}$. (That is, find $\\phi$ satisfying $\\cos(\\phi)=\\cos(53^\\circ)$.) $\\phi=$ [ANS] degrees. (b) Find another angle $\\phi$ between $0^\\circ$ and $360^\\circ$ that has the same sine as $53^{\\circ}$. (That is, find $\\phi$ satisfying $\\sin(\\phi)=\\sin(53^\\circ)$.) $\\phi=$ [ANS] degrees.", "answer_v2": [ "307", "127" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find another angle $\\phi$ between $0^\\circ$ and $360^\\circ$ that has the same cosine as $61^{\\circ}$. (That is, find $\\phi$ satisfying $\\cos(\\phi)=\\cos(61^\\circ)$.) $\\phi=$ [ANS] degrees. (b) Find another angle $\\phi$ between $0^\\circ$ and $360^\\circ$ that has the same sine as $61^{\\circ}$. (That is, find $\\phi$ satisfying $\\sin(\\phi)=\\sin(61^\\circ)$.) $\\phi=$ [ANS] degrees.", "answer_v3": [ "299", "119" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0036", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "phase shift" ], "problem_v1": "Suppose $y=11 \\sin{\\big(7(t+13) \\big)}+8$. In your answers, enter pi for $\\pi$.\n(a) The midline of the graph is the line with equation [ANS]\n(b) The amplitude of the graph is [ANS]\n(c) The period of the graph is [ANS]", "answer_v1": [ "y = 8", "11", "2*pi/7" ], "answer_type_v1": [ "EQ", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $y=5 \\sin{\\big(2(t+15) \\big)}-6$. In your answers, enter pi for $\\pi$.\n(a) The midline of the graph is the line with equation [ANS]\n(b) The amplitude of the graph is [ANS]\n(c) The period of the graph is [ANS]", "answer_v2": [ "y = -6", "5", "2*pi/2" ], "answer_type_v2": [ "EQ", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $y=5 \\sin{\\big(4(t+13) \\big)}+2$. In your answers, enter pi for $\\pi$.\n(a) The midline of the graph is the line with equation [ANS]\n(b) The amplitude of the graph is [ANS]\n(c) The period of the graph is [ANS]", "answer_v3": [ "y = 2", "5", "2*pi/4" ], "answer_type_v3": [ "EQ", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0037", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "phase shift" ], "problem_v1": "Suppose $y=2 \\cos{(8 t+19)}+3$. In your answers, enter pi for $\\pi$.\n(a) What is the phase shift? [ANS]\n(b) What is the horizontal shift? [ANS]", "answer_v1": [ "-19", "-19/8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $y=-5 \\cos{(3 t+27)}-2$. In your answers, enter pi for $\\pi$.\n(a) What is the phase shift? [ANS]\n(b) What is the horizontal shift? [ANS]", "answer_v2": [ "-27", "-27/3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $y=-3 \\cos{(5 t+19)}+1$. In your answers, enter pi for $\\pi$.\n(a) What is the phase shift? [ANS]\n(b) What is the horizontal shift? [ANS]", "answer_v3": [ "-19", "-19/5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0039", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "sin", "cos" ], "problem_v1": "Find amplitude and midline of the function $y=10 \\cos{(5x)}-5$\n(a) The midline is the line with equation [ANS]\n(b) The amplitude is [ANS]", "answer_v1": [ "y = -5", "10" ], "answer_type_v1": [ "EQ", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find amplitude and midline of the function $y=-16 \\sin{(2x)}+9$\n(a) The midline is the line with equation [ANS]\n(b) The amplitude is [ANS]", "answer_v2": [ "y = 9", "16" ], "answer_type_v2": [ "EQ", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find amplitude and midline of the function $y=-8 \\cos{(3x)}-7$\n(a) The midline is the line with equation [ANS]\n(b) The amplitude is [ANS]", "answer_v3": [ "y = -7", "8" ], "answer_type_v3": [ "EQ", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0040", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "periodic", "period", "amplitude", "midline" ], "problem_v1": "Let $y=f(x)$ be a periodic function whose values are given below. Find the period, amplitude, and midline.\n$\\begin{array}{cccccccccc}\\hline x & 5 & 15 & 25 & 35 & 45 & 55 & 65 & 75 & 85 \\\\\\hlinef(x) & 13 & 11 & 2 & 13 & 11 & 2 & 13 & 11 & 2 \\\\\\hline\\end{array}$\n(a) The period of the graph is [ANS]\n(b) The midline of the graph is [ANS]\n(c) The amplitude of the graph is [ANS]", "answer_v1": [ "30", "7.5", "5.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $y=f(x)$ be a periodic function whose values are given below. Find the period, amplitude, and midline.\n$\\begin{array}{cccccccccc}\\hline x &-8 & 12 & 32 & 52 & 72 & 92 & 112 & 132 & 152 \\\\\\hlinef(x) & 17 & 15 &-2 & 17 & 15 &-2 & 17 & 15 &-2 \\\\\\hline\\end{array}$\n(a) The period of the graph is [ANS]\n(b) The midline of the graph is [ANS]\n(c) The amplitude of the graph is [ANS]", "answer_v2": [ "60", "7.5", "9.5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $y=f(x)$ be a periodic function whose values are given below. Find the period, amplitude, and midline.\n$\\begin{array}{cccccccccc}\\hline x &-4 & 6 & 16 & 26 & 36 & 46 & 56 & 66 & 76 \\\\\\hlinef(x) & 16 & 14 &-3 & 16 & 14 &-3 & 16 & 14 &-3 \\\\\\hline\\end{array}$\n(a) The period of the graph is [ANS]\n(b) The midline of the graph is [ANS]\n(c) The amplitude of the graph is [ANS]", "answer_v3": [ "30", "6.5", "9.5" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0041", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "periodic", "period", "amplitude", "midline" ], "problem_v1": "Decide whether the following table appears to represent a periodic function. If so enter the value of its period in the blank. If the table does not appear to be periodic enter NONE.\n$\\begin{array}{ccccccccc}\\hline x & 0 & 4 & 8 & 12 & 16 & 20 & 24 & 28 \\\\\\hline g(x) & 0.6 & 7.9 &-0.9 & 0.6 & 7.9 &-0.9 & 0.6 & 7.9 \\\\\\hline\\end{array}$\nThe period is [ANS] (Enter NONE if not periodic.)", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Decide whether the following table appears to represent a periodic function. If so enter the value of its period in the blank. If the table does not appear to be periodic enter NONE.\n$\\begin{array}{ccccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\\hline g(x) & 0.9 & 6.9 &-1.4 & 0.9 & 6.9 &-1.4 & 0.9 & 6.9 \\\\\\hline\\end{array}$\nThe period is [ANS] (Enter NONE if not periodic.)", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Decide whether the following table appears to represent a periodic function. If so enter the value of its period in the blank. If the table does not appear to be periodic enter NONE.\n$\\begin{array}{ccccccccc}\\hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\\\\\hline g(x) & 0.6 & 7.4 &-1.2 & 0.6 & 7.4 &-1.2 & 0.6 & 7.4 \\\\\\hline\\end{array}$\nThe period is [ANS] (Enter NONE if not periodic.)", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0042", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "periodic", "period", "amplitude", "midline" ], "problem_v1": "The table below gives the height $h=f(t)$ in feet of a weight on a spring where $t$ is time in seconds.\n$\\begin{array}{ccccccccccccccccc}\\hline t(sec) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\\\\\hline h(feet) & 4.9 & 6.9 & 7.9 & 8.2 & 7.9 & 6.9 & 4.9 & 2.9 & 1.9 & 1.6 & 1.9 & 2.9 & 4.9 & 6.9 & 7.9 & 8.2 \\\\\\hline\\end{array}$\n(a) What is the period of $\\ f(t)$? [ANS]s (include) (b) What is the midline of $\\ f(t)$? [ANS]ft (include) (c) What is the amplitude of $\\ f(t)$? [ANS]ft (include)", "answer_v1": [ "12", "4.9", "3.3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The table below gives the height $h=f(t)$ in feet of a weight on a spring where $t$ is time in seconds.\n$\\begin{array}{ccccccccccccccccc}\\hline t(sec) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\\\\\hline h(feet) & 3.7 & 2.5 & 1.5 & 1.3 & 1.5 & 2.5 & 3.7 & 4.9 & 5.9 & 6.1 & 5.9 & 4.9 & 3.7 & 2.5 & 1.5 & 1.3 \\\\\\hline\\end{array}$\n(a) What is the period of $\\ f(t)$? [ANS]s (include) (b) What is the midline of $\\ f(t)$? [ANS]ft (include) (c) What is the amplitude of $\\ f(t)$? [ANS]ft (include)", "answer_v2": [ "12", "3.7", "2.4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The table below gives the height $h=f(t)$ in feet of a weight on a spring where $t$ is time in seconds.\n$\\begin{array}{ccccccccccccccccc}\\hline t(sec) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\\\\\hline h(feet) & 4.1 & 5.5 & 6.5 & 6.7 & 6.5 & 5.5 & 4.1 & 2.7 & 1.7 & 1.5 & 1.7 & 2.7 & 4.1 & 5.5 & 6.5 & 6.7 \\\\\\hline\\end{array}$\n(a) What is the period of $\\ f(t)$? [ANS]s (include) (b) What is the midline of $\\ f(t)$? [ANS]ft (include) (c) What is the amplitude of $\\ f(t)$? [ANS]ft (include)", "answer_v3": [ "12", "4.1", "2.6" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0043", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "algebra", "trigonometric function graph" ], "problem_v1": "For $y=7 \\sin (7x-\\frac{\\pi}{8})$, its amplitude is [ANS] ; its period is [ANS] ; its phase shift is [ANS] ;", "answer_v1": [ "7", "0.897597901025655", "0.0560998688141034" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "For $y=3 \\sin (10x-\\frac{\\pi}{2})$, its amplitude is [ANS] ; its period is [ANS] ; its phase shift is [ANS] ;", "answer_v2": [ "3", "0.628318530717959", "0.15707963267949" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "For $y=4 \\sin (7x-\\frac{\\pi}{4})$, its amplitude is [ANS] ; its period is [ANS] ; its phase shift is [ANS] ;", "answer_v3": [ "4", "0.897597901025655", "0.112199737628207" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0045", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "functions", "domain", "graph", "maximum/minimum" ], "problem_v1": "Determine which of the following statements are true and which are false. Enter the T or F in front of each statement. Remember that $x\\in(-1,1)$ is the same as $-1 < x < 1$ and $x\\in[-1,1]$ means $-1 \\le x \\le 1$. [ANS] 1. The function $\\sin(x)$ on the domain $x\\in[-\\pi/2,\\pi/2]$ has at least one input which produces a largest output value. [ANS] 2. The function $\\sin(x)$ on the domain $x\\in(-\\pi/2,\\pi/2)$ has at least one input which produces a largest output value. [ANS] 3. The function $f(x)=x^2$ with domain $x\\in[-3,3]$ has at least one input which produces a smallest output value. [ANS] 4. The function $\\sin(x)$ on the domain $x\\in(-\\pi/2,\\pi/2)$ has at least one input which produces a smallest output value. [ANS] 5. The function $f(x)=x^2$ with domain $x\\in(-3,3)$ has at least one input which produces a smallest output value.", "answer_v1": [ "T", "F", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Determine which of the following statements are true and which are false. Enter the T or F in front of each statement. Remember that $x\\in(-1,1)$ is the same as $-1 < x < 1$ and $x\\in[-1,1]$ means $-1 \\le x \\le 1$. [ANS] 1. The function $f(x)=x^2$ with domain $x\\in(-3,3)$ has at least one input which produces a largest output value. [ANS] 2. The function $\\sin(x)$ on the domain $x\\in[-\\pi/2,\\pi/2]$ has at least one input which produces a smallest output value. [ANS] 3. The function $f(x)=x^2$ with domain $x\\in(-3,3)$ has at least one input which produces a smallest output value. [ANS] 4. The function $f(x)=x^2$ with domain $x\\in[-3,3]$ has at least one input which produces a smallest output value. [ANS] 5. The function $\\sin(x)$ on the domain $x\\in[-\\pi/2,\\pi/2]$ has at least one input which produces a largest output value.", "answer_v2": [ "F", "T", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Determine which of the following statements are true and which are false. Enter the T or F in front of each statement. Remember that $x\\in(-1,1)$ is the same as $-1 < x < 1$ and $x\\in[-1,1]$ means $-1 \\le x \\le 1$. [ANS] 1. The function $f(x)=x^2$ with domain $x\\in[-3,3]$ has at least one input which produces a largest output value. [ANS] 2. The function $\\sin(x)$ on the domain $x\\in(-\\pi/2,\\pi/2)$ has at least one input which produces a smallest output value. [ANS] 3. The function $f(x)=x^2$ with domain $x\\in(-3,3)$ has at least one input which produces a smallest output value. [ANS] 4. The function $\\sin(x)$ on the domain $x\\in(-\\pi/2,\\pi/2)$ has at least one input which produces a largest output value. [ANS] 5. The function $f(x)=x^2$ with domain $x\\in(-3,3)$ has at least one input which produces a largest output value.", "answer_v3": [ "T", "F", "T", "F", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0046", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "calculus", "function", "trigonometric functions", "inverse trigonometric functions", "transformation of functions", "translations" ], "problem_v1": "Consider the function $y=7+6 \\cos(6x)$.\n(a) What is its amplitude? [ANS]. (b) What is its period? [ANS]. (c) Sketch its graph and use your sketch to determine the largest value the graph takes. Largest value=[ANS].", "answer_v1": [ "6", "1.0472", "13" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the function $y=2+8 \\cos(3x)$.\n(a) What is its amplitude? [ANS]. (b) What is its period? [ANS]. (c) Sketch its graph and use your sketch to determine the largest value the graph takes. Largest value=[ANS].", "answer_v2": [ "8", "2.0944", "10" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the function $y=4+6 \\cos(3x)$.\n(a) What is its amplitude? [ANS]. (b) What is its period? [ANS]. (c) Sketch its graph and use your sketch to determine the largest value the graph takes. Largest value=[ANS].", "answer_v3": [ "6", "2.0944", "10" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0047", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "4", "keywords": [ "calculus", "function", "trigonometric functions", "inverse trigonometric functions", "transformation of functions", "translations" ], "problem_v1": "On the graph of $f(x)=7 \\sin(6 \\pi x)$, points $P$ and $Q$ are at consecutive lowest and highest points with $P$ occuring before $Q$. Find the slope of the line which passes through $P$ and $Q$. Slope=[ANS].", "answer_v1": [ "84" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "On the graph of $f(x)=2 \\sin(8 \\pi x)$, points $P$ and $Q$ are at consecutive lowest and highest points with $P$ occuring before $Q$. Find the slope of the line which passes through $P$ and $Q$. Slope=[ANS].", "answer_v2": [ "32" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "On the graph of $f(x)=4 \\sin(6 \\pi x)$, points $P$ and $Q$ are at consecutive lowest and highest points with $P$ occuring before $Q$. Find the slope of the line which passes through $P$ and $Q$. Slope=[ANS].", "answer_v3": [ "48" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0048", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "calculus", "function", "trigonometric functions", "inverse trigonometric functions", "transformation of functions", "translations" ], "problem_v1": "Find the period and amplitude of $r=0.7 \\sin(6 t)+6$ period=[ANS]\namplitude=[ANS]", "answer_v1": [ "2*pi/6", "0.7" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the period and amplitude of $r=0.1 \\sin(9 t)+2$ period=[ANS]\namplitude=[ANS]", "answer_v2": [ "2*pi/9", "0.1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the period and amplitude of $r=0.3 \\sin(6 t)+3$ period=[ANS]\namplitude=[ANS]", "answer_v3": [ "2*pi/6", "0.3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0049", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trigonometry", "sine", "cosine", "frequency" ], "problem_v1": "Determine the frequency of the curve determined by $y=\\cos (117 \\pi x)$, where $x$ is time in seconds. Frequency [ANS]", "answer_v1": [ "58.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the frequency of the curve determined by $y=\\sin (187 \\pi x)$, where $x$ is time in seconds. Frequency [ANS]", "answer_v2": [ "93.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the frequency of the curve determined by $y=\\sin (122 \\pi x)$, where $x$ is time in seconds. Frequency [ANS]", "answer_v3": [ "61" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0050", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trigonometry", "identity", "odd", "even" ], "problem_v1": "Which of these functions are neither odd nor even? [ANS] A. $f(x)=x+\\sin(x)$ B. $f(\\beta)=1+\\csc(\\beta)$ C. $f(t)=2+\\tan(t)$ D. $f(x)=\\cos(x)+\\sin(x)$ E. $f(x)=\\cos(2x)$ F. ${f(x)=\\frac{\\sin(x)}{x}}$", "answer_v1": [ "BCD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of these functions are neither odd nor even? [ANS] A. $f(x)=\\cos(2x)$ B. $f(\\beta)=1+\\csc(\\beta)$ C. ${f(x)=\\frac{\\cos(2x)}{x}}$ D. ${f(x)=\\frac{\\sin(x)}{x}}$ E. $f(\\alpha)=1+\\sec(\\alpha)$ F. $f(x)=\\cos(x)+\\sin(x)$", "answer_v2": [ "BF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of these functions are neither odd nor even? [ANS] A. $f(x)=\\cos(2x)$ B. $f(x)=x\\cos(x)$ C. $f(\\alpha)=1+\\sec(\\alpha)$ D. $f(x)=\\sin(2x)$ E. $f(t)=2+\\tan(t)$ F. $f(x)=\\cos(x)+\\sin(x)$", "answer_v3": [ "EF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Trigonometry_0051", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trigonometry", "amplitude", "period", "phase shift" ], "problem_v1": "For the curve\n$y=23 \\cos(6 \\pi x-4)$ determine each of the following:\nAmplitude=[ANS]\nPeriod=[ANS]\nPhase shift=[ANS] Note: Use a negative for a shift to the left. Note: Use a negative for a shift to the left.", "answer_v1": [ "23", "0.333333333333333", "0.212207" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "For the curve\n$y=17 \\cos(3 \\pi x-9)$ determine each of the following:\nAmplitude=[ANS]\nPeriod=[ANS]\nPhase shift=[ANS] Note: Use a negative for a shift to the left. Note: Use a negative for a shift to the left.", "answer_v2": [ "17", "0.666666666666667", "0.95493" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "For the curve\n$y=19 \\cos(4 \\pi x+4)$ determine each of the following:\nAmplitude=[ANS]\nPeriod=[ANS]\nPhase shift=[ANS] Note: Use a negative for a shift to the left.", "answer_v3": [ "19", "0.5", "-0.31831" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0052", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Sine & cosine functions - definitions, graphs, & properties", "level": "2", "keywords": [ "algebra", "analytic trigonometry", "trigonometric identities" ], "problem_v1": "Suppose:\n$\\sin(x)=0.4618$ $\\cos(x)=0.8870$ $\\tan(x)=0.5206$ Then,\n$\\sin(-x)=$ [ANS]\n$\\cos(-x)=$ [ANS]\n$\\tan(-x)=$ [ANS]\nand $\\sec(-x)\\csc(-x)\\cot(-x)\\sin(-x)=$ [ANS]", "answer_v1": [ "-1.0000*0.4618", "0.8870", "-1.0000*0.5206", "-1.0000/0.4618" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose:\n$\\sin(x)=-0.7590$ $\\cos(x)=0.6511$ $\\tan(x)=-1.1657$ Then,\n$\\sin(-x)=$ [ANS]\n$\\cos(-x)=$ [ANS]\n$\\tan(-x)=$ [ANS]\nand $\\sec(-x)\\csc(-x)\\cot(-x)\\sin(-x)=$ [ANS]", "answer_v2": [ "-1.0000*-0.7590", "0.6511", "-1.0000*-1.1657", "-1.0000/-0.7590" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose:\n$\\sin(x)=-0.3390$ $\\cos(x)=0.9408$ $\\tan(x)=-0.3603$ Then,\n$\\sin(-x)=$ [ANS]\n$\\cos(-x)=$ [ANS]\n$\\tan(-x)=$ [ANS]\nand $\\sec(-x)\\csc(-x)\\cot(-x)\\sin(-x)=$ [ANS]", "answer_v3": [ "-1.0000*-0.3390", "0.9408", "-1.0000*-0.3603", "-1.0000/-0.3390" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0053", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Tangent & cotangent functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trigonometry", "precalculus", "trigonometric graphs" ], "problem_v1": "Enter T or F depending on whether the statement is true or false (You must enter T or F--True and False will not work.) [ANS] 1. $y=\\tan (2x)$ has period $\\pi$ [ANS] 2. The domain of $\\tan (x)$ is all real numbers [ANS] 3. The range of $\\tan (x)$ is all real numbers [ANS] 4. $\\tan (x)$ and $\\cot x$ are odd functions", "answer_v1": [ "F", "F", "T", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Enter T or F depending on whether the statement is true or false (You must enter T or F--True and False will not work.) [ANS] 1. $\\sec (x)$ and $\\tan (x)$ are undefined for the same values of $x$ [ANS] 2. $\\tan (x)$ and $\\cot x$ are odd functions [ANS] 3. $y=\\tan (2x)$ has period $\\pi$ [ANS] 4. $\\sin (x)$ is odd and $\\cos (x)$ is even", "answer_v2": [ "T", "T", "F", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Enter T or F depending on whether the statement is true or false (You must enter T or F--True and False will not work.) [ANS] 1. $\\tan (x)=\\tan (-x)$ for all $x$ in the domain of $\\tan (x)$ [ANS] 2. $\\sin (x)$ is odd and $\\cos (x)$ is even [ANS] 3. $\\sec (x)$ and $\\tan (x)$ are undefined for the same values of $x$ [ANS] 4. The domain of $\\tan (x)$ is all real numbers", "answer_v3": [ "F", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0054", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Tangent & cotangent functions - definitions, graphs, & properties", "level": "2", "keywords": [ "tangent", "secant", "cosecant", "cotangent", "period" ], "problem_v1": "Determine the period for the function $y=2 \\tan (\\frac{11\\pi}{8}x-\\frac{9\\pi}{10})$ Type 'pi' for $\\pi$ in your answer(s), if needed. Period=[ANS]", "answer_v1": [ "0.727272727272727" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the period for the function $y=4 \\cot (\\frac{2\\pi}{13}x-\\frac{4\\pi}{5})$ Type 'pi' for $\\pi$ in your answer(s), if needed. Period=[ANS]", "answer_v2": [ "6.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the period for the function $y=6 \\tan (\\frac{5\\pi}{9}x-\\frac{5\\pi}{8})$ Type 'pi' for $\\pi$ in your answer(s), if needed. Period=[ANS]", "answer_v3": [ "1.8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0055", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "calculus", "trigonometric functions", "inverse functions" ], "problem_v1": "Compute $\\tan^{-1}(\\tan \\frac{5\\pi}{8})=$ [ANS]", "answer_v1": [ "-1.1781" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute $\\tan^{-1}(\\tan \\frac{3\\pi}{4})=$ [ANS]", "answer_v2": [ "-0.785398" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute $\\tan^{-1}(\\tan \\frac{5\\pi}{4})=$ [ANS]", "answer_v3": [ "0.785398" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0056", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "Trig", "Inverse" ], "problem_v1": "Evaluate the following expressions. Your answer must be an angle in radians in the interval $[-\\frac{\\pi}{2},\\frac{\\pi}{2}]$.\n$\\begin{array}{cccc}\\hline 1. & \\sin^{-1}(\\frac{1}{2}) &=& [ANS] \\\\\\hline 2. & \\sin^{-1}(\\frac{\\sqrt{2}}{2}) &=& [ANS] \\\\\\hline 3. & \\sin^{-1}(0) &=& [ANS] \\\\\\hline\\end{array}$", "answer_v1": [ "0.523599", "0.785398", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the following expressions. Your answer must be an angle in radians in the interval $[-\\frac{\\pi}{2},\\frac{\\pi}{2}]$.\n$\\begin{array}{cccc}\\hline 1. & \\sin^{-1}(-\\frac{1}{2}) &=& [ANS] \\\\\\hline 2. & \\sin^{-1}(-1) &=& [ANS] \\\\\\hline 3. & \\sin^{-1}(0) &=& [ANS] \\\\\\hline\\end{array}$", "answer_v2": [ "-0.523599", "-1.5708", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the following expressions. Your answer must be an angle in radians in the interval $[-\\frac{\\pi}{2},\\frac{\\pi}{2}]$.\n$\\begin{array}{cccc}\\hline 1. & \\sin^{-1}(-\\frac{\\sqrt{3}}{2}) &=& [ANS] \\\\\\hline 2. & \\sin^{-1}(-\\frac{\\sqrt{2}}{2}) &=& [ANS] \\\\\\hline 3. & \\sin^{-1}(-\\frac{1}{2}) &=& [ANS] \\\\\\hline\\end{array}$", "answer_v3": [ "-1.0472", "-0.785398", "-0.523599" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0057", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "algebra", "analytic trigonometry", "inverse trigonometric function" ], "problem_v1": "Find the value of each expression if defined; otherwise, input undefined.\n(a) $\\tan(\\tan^{-1} 8)=$ [ANS]. (b) $\\sin^{-1}[\\sin (-\\frac{\\pi}{6})]=$ [ANS] radians.", "answer_v1": [ "8", "-0.523598775598299" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the value of each expression if defined; otherwise, input undefined.\n(a) $\\tan(\\tan^{-1} 2)=$ [ANS]. (b) $\\sin^{-1}[\\sin (-\\frac{\\pi}{6})]=$ [ANS] radians.", "answer_v2": [ "2", "-0.523598775598299" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the value of each expression if defined; otherwise, input undefined.\n(a) $\\tan(\\tan^{-1} 4)=$ [ANS]. (b) $\\sin^{-1}[\\sin (-\\frac{\\pi}{6})]=$ [ANS] radians.", "answer_v3": [ "4", "-0.523598775598299" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0058", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "1", "keywords": [ "trig functions in degrees", "arcsin", "arccos", "arctan" ], "problem_v1": "In each of the following equations, determine which variable represents the angle and which variable represents the value of the trigonometric function.\n(a) In the equation $q=\\sin p$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function. (b) In the equation $p=\\tan^{-1}q$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function. (c) In the equation $\\arccos d=f$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function.", "answer_v1": [ "p", "q", "p", "q", "f", "d" ], "answer_type_v1": [ "OE", "OE", "OE", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "In each of the following equations, determine which variable represents the angle and which variable represents the value of the trigonometric function.\n(a) In the equation $a=\\sin w$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function. (b) In the equation $b=\\tan^{-1}f$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function. (c) In the equation $\\arccos s=f$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function.", "answer_v2": [ "w", "a", "b", "f", "f", "s" ], "answer_type_v2": [ "OE", "OE", "OE", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "In each of the following equations, determine which variable represents the angle and which variable represents the value of the trigonometric function.\n(a) In the equation $c=\\sin p$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function. (b) In the equation $d=\\tan^{-1}n$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function. (c) In the equation $\\arccos a=f$, [ANS] represents the angle and [ANS] represents the value of the trigonometric function.", "answer_v3": [ "p", "c", "d", "n", "f", "a" ], "answer_type_v3": [ "OE", "OE", "OE", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Trigonometry_0059", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trig functions in degrees", "arcsin", "arccos", "arctan" ], "problem_v1": "For each of the following, find an angle $\\phi$ satisfying the given equation. (Round your answers to the nearest $0.001^{\\circ}$.)\n(a) If $0^{\\circ} \\leq \\phi \\leq 90^{\\circ}$ and $ \\sin(\\phi)=0.561$, then $\\phi=$ [ANS] degrees. (b) If $0^{\\circ} \\leq \\phi \\leq 90^{\\circ}$ and $ \\cos(\\phi)=0.612$, then $\\phi=$ [ANS] degrees. (c) If $0^{\\circ} \\leq \\phi \\leq 90^{\\circ}$ and $ \\tan(\\phi)=721.863$, then $\\phi=$ [ANS] degrees.", "answer_v1": [ "34.125", "52.266", "89.921" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "For each of the following, find an angle $\\theta$ satisfying the given equation. (Round your answers to the nearest $0.001^{\\circ}$.)\n(a) If $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$ and $ \\sin(\\theta)=0.811$, then $\\theta=$ [ANS] degrees. (b) If $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$ and $ \\cos(\\theta)=0.212$, then $\\theta=$ [ANS] degrees. (c) If $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$ and $ \\tan(\\theta)=333.863$, then $\\theta=$ [ANS] degrees.", "answer_v2": [ "54.194", "77.76", "89.828" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "For each of the following, find an angle $\\theta$ satisfying the given equation. (Round your answers to the nearest $0.001^{\\circ}$.)\n(a) If $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$ and $ \\sin(\\theta)=0.561$, then $\\theta=$ [ANS] degrees. (b) If $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$ and $ \\cos(\\theta)=0.312$, then $\\theta=$ [ANS] degrees. (c) If $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$ and $ \\tan(\\theta)=548.163$, then $\\theta=$ [ANS] degrees.", "answer_v3": [ "34.125", "71.82", "89.895" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0060", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trig functions in degrees", "arcsin", "arccos", "arctan" ], "problem_v1": "(a) Evaluate $\\tan t$ if $t=48$ degrees. $\\tan t=$ [ANS]\n(b) Evaluate $\\arctan z$ if $z=48$. $\\arctan z=$ [ANS] degrees (c) Evaluate $\\tan^{-1} b$ if $b=48$. $\\tan^{-1} b=$ [ANS] degrees (d) Evaluate $(\\tan \\beta)^{-1}$ if $\\beta=48$ degrees. $(\\tan \\beta) ^{-1}=$ [ANS]", "answer_v1": [ "1.11061", "88.8065", "88.8065", "0.900404" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "(a) Evaluate $\\tan v$ if $v=78$ degrees. $\\tan v=$ [ANS]\n(b) Evaluate $\\arctan y$ if $y=78$. $\\arctan y=$ [ANS] degrees (c) Evaluate $\\tan^{-1} w$ if $w=78$. $\\tan^{-1} w=$ [ANS] degrees (d) Evaluate $(\\tan \\phi)^{-1}$ if $\\phi=78$ degrees. $(\\tan \\phi) ^{-1}=$ [ANS]", "answer_v2": [ "4.70463", "89.2655", "89.2655", "0.212557" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "(a) Evaluate $\\tan y$ if $y=53$ degrees. $\\tan y=$ [ANS]\n(b) Evaluate $\\arctan z$ if $z=53$. $\\arctan z=$ [ANS] degrees (c) Evaluate $\\tan^{-1} a$ if $a=53$. $\\tan^{-1} a=$ [ANS] degrees (d) Evaluate $(\\tan \\theta)^{-1}$ if $\\theta=53$ degrees. $(\\tan \\theta) ^{-1}=$ [ANS]", "answer_v3": [ "1.32704", "88.9191", "88.9191", "0.753554" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0061", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "3", "keywords": [ "Functions", "Inverse", "Trigonometry", "Inverse Cosine", "Sine", "Inverse Trig" ], "problem_v1": "Find the exact value. $\\small{\\sin} \\left(\\small{2 \\cos^{-1}} \\large{\\left(\\frac{11}{61}\\right)} \\right)$=[ANS]", "answer_v1": [ "2*0.983607*0.180328" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the exact value. $\\small{\\sin} \\left(\\small{2 \\cos^{-1}} \\large{\\left(\\frac{3}{5}\\right)} \\right)$=[ANS]", "answer_v2": [ "2*0.8*0.6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the exact value. $\\small{\\sin} \\left(\\small{2 \\cos^{-1}} \\large{\\left(\\frac{7}{25}\\right)} \\right)$=[ANS]", "answer_v3": [ "2*0.96*0.28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0062", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "inverse", "trigonometry", "domain" ], "problem_v1": "Find the inverse of the following function and state its domain. $f(x)=16 \\cos (10x)+6$ Type 'arccos' for the inverse cosine function in your answer. $\\ f^{-1} (x)$=[ANS]\nDomain=[[ANS], [ANS]]", "answer_v1": [ "(1/10)*arccos((x-6)/16)", "-10", "22" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the inverse of the following function and state its domain. $f(x)=3 \\cos (15x)+2$ Type 'arccos' for the inverse cosine function in your answer. $\\ f^{-1} (x)$=[ANS]\nDomain=[[ANS], [ANS]]", "answer_v2": [ "(1/15)*arccos((x-2)/3)", "-1", "5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the inverse of the following function and state its domain. $f(x)=7 \\cos (10x)+3$ Type 'arccos' for the inverse cosine function in your answer. $\\ f^{-1} (x)$=[ANS]\nDomain=[[ANS], [ANS]]", "answer_v3": [ "(1/10)*arccos((x-3)/7)", "-4", "10" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0063", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Inverse trigonometric functions - definitions, graphs, & properties", "level": "2", "keywords": [ "trigonometry", "inverse trig functions" ], "problem_v1": "Determine each to 1 decimal place: $\\sin(\\alpha)=0.868632$ implies $\\alpha \\ $ [ANS]\n$\\tan(\\beta)=1.04644$ implies $\\beta \\ $ [ANS]\n$\\sec(\\theta)=1.67329$ implies $\\theta \\ $ [ANS]", "answer_v1": [ "60.3", "46.3", "53.3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Determine each to 1 decimal place: $\\sin(\\alpha)=0.195946$ implies $\\alpha \\ $ [ANS]\n$\\tan(\\beta)=3.55761$ implies $\\beta \\ $ [ANS]\n$\\sec(\\theta)=1.05327$ implies $\\theta \\ $ [ANS]", "answer_v2": [ "11.3", "74.3", "18.3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Determine each to 1 decimal place: $\\sin(\\alpha)=0.534352$ implies $\\alpha \\ $ [ANS]\n$\\tan(\\beta)=1.3416$ implies $\\beta \\ $ [ANS]\n$\\sec(\\theta)=1.10609$ implies $\\theta \\ $ [ANS]", "answer_v3": [ "32.3", "53.3", "25.3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0064", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Combinations and compositions of functions", "level": "2", "keywords": [ "calculus", "functions", "composite functions", "domain" ], "problem_v1": "Calculate the composite functions $f \\circ g$ and $g \\circ f$. $f(x)=\\cos\\!\\left(x\\right), g(x)=8x^{3}+6x^{2}-6$ $f(g(x))=$ [ANS]\n$g(f(x))=$ [ANS]", "answer_v1": [ "cos(8*x^3+6*x^2-6)", "8*[cos(x)]^3+6*[cos(x)]^2-6" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the composite functions $f \\circ g$ and $g \\circ f$. $f(x)=\\cos\\!\\left(x\\right), g(x)=x^{3}+10x^{2}-1$ $f(g(x))=$ [ANS]\n$g(f(x))=$ [ANS]", "answer_v2": [ "cos(x^3+10*x^2-1)", "[cos(x)]^3+10*[cos(x)]^2-1" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the composite functions $f \\circ g$ and $g \\circ f$. $f(x)=\\cos\\!\\left(x\\right), g(x)=4x^{3}+7x^{2}-3$ $f(g(x))=$ [ANS]\n$g(f(x))=$ [ANS]", "answer_v3": [ "cos(4*x^3+7*x^2-3)", "4*[cos(x)]^3+7*[cos(x)]^2-3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0065", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Combinations and compositions of functions", "level": "2", "keywords": [ "algebra", "function", "composition of functions" ], "problem_v1": "Write the function $h(x)=\\frac{1}{8x+1}$ as the composition of two functions $f(g(x))$. $f(x)$=[ANS]\n$g(x)$=[ANS]\nDo not use the identity function as one of your answers. That is, do not use $f(x)=x$ or $g(x)=x$.", "answer_v1": [ "1/x", "8*x+1" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Write the function $h(x)=\\sin\\!\\left(2x+6\\right)$ as the composition of two functions $f(g(x))$. $f(x)$=[ANS]\n$g(x)$=[ANS]\nDo not use the identity function as one of your answers. That is, do not use $f(x)=x$ or $g(x)=x$.", "answer_v2": [ "sin(x)", "2*x+6" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Write the function $h(x)=\\sin\\!\\left(4x^{2}\\right)$ as the composition of two functions $f(g(x))$. $f(x)$=[ANS]\n$g(x)$=[ANS]\nDo not use the identity function as one of your answers. That is, do not use $f(x)=x$ or $g(x)=x$.", "answer_v3": [ "sin(x)", "4*x^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0066", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Combinations and compositions of functions", "level": "2", "keywords": [ "composition", "decomposition", "function" ], "problem_v1": "Let $f(x)=\\sin{(8x)}$ and $g(x)=5+\\sqrt{x}$. Find formulas for:\n(a) $f(g(x))=$ [ANS]\n(b) $g(f(x))=$ [ANS]", "answer_v1": [ "sin(8*sqrt(x)+40)", "5+sqrt(sin(8*x))" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=\\sin{(2x)}$ and $g(x)=7+\\sqrt{x}$. Find formulas for:\n(a) $f(g(x))=$ [ANS]\n(b) $g(f(x))=$ [ANS]", "answer_v2": [ "sin(2*sqrt(x)+14)", "7+sqrt(sin(2*x))" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=\\sin{(4x)}$ and $g(x)=5+\\sqrt{x}$. Find formulas for:\n(a) $f(g(x))=$ [ANS]\n(b) $g(f(x))=$ [ANS]", "answer_v3": [ "sin(4*sqrt(x)+20)", "5+sqrt(sin(4*x))" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0067", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Combinations and compositions of functions", "level": "3", "keywords": [ "decomposition", "inverse", "composition", "combinations", "function" ], "problem_v1": "Suppose that $f(x)=g \\big(h(x) \\big)$. In each part, based on one of the functions provided, find a formula for the other formula such that their composition yields $f(x)=g \\big(h(x) \\big)$. Be sure to play close attention to the order of composition. f(x)=\\tan{\\big(4+\\tan{(x)}\\big)}\n(a) If $g(x)=\\tan{(x)}$, then $h(x)=$ [ANS]\n(b) If $h(x)=\\tan{(x)}$, then $g(x)=$ [ANS]", "answer_v1": [ "4+tan(x)", "tan(4+x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $f(x)=g \\big(h(x) \\big)$. In each part, based on one of the functions provided, find a formula for the other formula such that their composition yields $f(x)=g \\big(h(x) \\big)$. Be sure to play close attention to the order of composition. f(x)=\\sin{\\big(6+\\sin{(x)}\\big)}\n(a) If $g(x)=\\sin{(x)}$, then $h(x)=$ [ANS]\n(b) If $h(x)=\\sin{(x)}$, then $g(x)=$ [ANS]", "answer_v2": [ "6+sin(x)", "sin(6+x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $f(x)=g \\big(h(x) \\big)$. In each part, based on one of the functions provided, find a formula for the other formula such that their composition yields $f(x)=g \\big(h(x) \\big)$. Be sure to play close attention to the order of composition. f(x)=\\cos{\\big(4+\\cos{(x)}\\big)}\n(a) If $g(x)=\\cos{(x)}$, then $h(x)=$ [ANS]\n(b) If $h(x)=\\cos{(x)}$, then $g(x)=$ [ANS]", "answer_v3": [ "4+cos(x)", "cos(4+x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0068", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Combinations and compositions of functions", "level": "2", "keywords": [ "Functions", "Composition" ], "problem_v1": "Express $\\small{f(x)=8\\csc^{4}\\!\\left(x\\right)+6\\csc\\!\\left(x\\right)}$ as a composition of two functions; that is, find $\\small{g}$ and $\\small{h}$ such that $\\small{f=g \\circ h}$. Do not use the identity function, $\\small{x}$, as one of the functions.\n$\\begin{array}{ccc}\\hline \\small{g(x)} &=& [ANS] \\\\\\hline \\small{h(x)} &=& [ANS] \\\\\\hline\\end{array}$", "answer_v1": [ "8*x^4+6*x", "csc(x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Express $\\small{f(x)=\\tan^{6}\\!\\left(x\\right)+2\\tan\\!\\left(x\\right)}$ as a composition of two functions; that is, find $\\small{g}$ and $\\small{h}$ such that $\\small{f=g \\circ h}$. Do not use the identity function, $\\small{x}$, as one of the functions.\n$\\begin{array}{ccc}\\hline \\small{g(x)} &=& [ANS] \\\\\\hline \\small{h(x)} &=& [ANS] \\\\\\hline\\end{array}$", "answer_v2": [ "1*x^6+2*x", "tan(x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Express $\\small{f(x)=4\\sec^{5}\\!\\left(x\\right)+3\\sec\\!\\left(x\\right)}$ as a composition of two functions; that is, find $\\small{g}$ and $\\small{h}$ such that $\\small{f=g \\circ h}$. Do not use the identity function, $\\small{x}$, as one of the functions.\n$\\begin{array}{ccc}\\hline \\small{g(x)} &=& [ANS] \\\\\\hline \\small{h(x)} &=& [ANS] \\\\\\hline\\end{array}$", "answer_v3": [ "4*x^5+3*x", "sec(x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0069", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Combinations and compositions of functions", "level": "2", "keywords": [ "calculus", "functions", "models" ], "problem_v1": "Express the function F(x)=\\tan(7 \\pi x) in the form $f \\circ g$. $f(x)$=[ANS]\n$g(x)$=[ANS]", "answer_v1": [ "tan(x)", "7*pi*x" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Express the function F(x)=\\tan(3 \\pi x) in the form $f \\circ g$. $f(x)$=[ANS]\n$g(x)$=[ANS]", "answer_v2": [ "tan(x)", "3*pi*x" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Express the function F(x)=\\tan(4 \\pi x) in the form $f \\circ g$. $f(x)$=[ANS]\n$g(x)$=[ANS]", "answer_v3": [ "tan(x)", "4*pi*x" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0070", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "2", "keywords": [ "sine", "cosine", "radians", "degrees" ], "problem_v1": "Find the coordinates of the point $P$ at an angle of $90^{\\circ}$ on a circle of radius $4.5$. Round your answers to the three decimal places. Enter a point as $(a,b)$ including parentheses. The point $P$ is [ANS]", "answer_v1": [ "(0,4.5)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the coordinates of the point $P$ at an angle of $180^{\\circ}$ on a circle of radius $2.7$. Round your answers to the three decimal places. Enter a point as $(a,b)$ including parentheses. The point $P$ is [ANS]", "answer_v2": [ "(-2.7,0)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the coordinates of the point $P$ at an angle of $90^{\\circ}$ on a circle of radius $3.3$. Round your answers to the three decimal places. Enter a point as $(a,b)$ including parentheses. The point $P$ is [ANS]", "answer_v3": [ "(0,3.3)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0071", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "2", "keywords": [ "tangent", "cotangent", "cosecant", "secant" ], "problem_v1": "Find the exact value as fraction (not a decimal approximation). $ \\sec\\left(\\frac{-\\pi}{3} \\right)$=[ANS]", "answer_v1": [ "2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the exact value as fraction (not a decimal approximation). $ \\sec\\left(\\frac{2\\pi}{3} \\right)$=[ANS]", "answer_v2": [ "-2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the exact value as fraction (not a decimal approximation). $ \\sec\\left(\\frac{5\\pi}{6} \\right)$=[ANS]", "answer_v3": [ "-2/[sqrt(3)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0072", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "2", "keywords": [ "tangent", "cotangent", "cosecant", "secant" ], "problem_v1": "Find the exact value of each:\na) $\\tan{\\left(\\frac{5 \\pi}{3} \\right)}$=[ANS]\nb) $\\tan{\\left(\\frac{7 \\pi}{6} \\right)}$=[ANS]\nc) $\\cot{\\left(\\frac{5 \\pi}{4} \\right)}$=[ANS]\nd) $\\sec{\\left(\\frac{4 \\pi}{3} \\right)}$=[ANS]\ne) $\\csc{\\left(\\frac{\\pi}{3} \\right)}$=[ANS]", "answer_v1": [ "-sqrt(3)", "1/sqrt(3)", "1", "-2", "2/sqrt(3)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find the exact value of each:\na) $\\tan{\\left(\\frac{\\pi}{6} \\right)}$=[ANS]\nb) $\\tan{\\left(\\frac{11 \\pi}{6} \\right)}$=[ANS]\nc) $\\cot{\\left(\\frac{\\pi}{3} \\right)}$=[ANS]\nd) $\\sec{\\left(\\frac{5 \\pi}{6} \\right)}$=[ANS]\ne) $\\csc{\\left(\\frac{7 \\pi}{4} \\right)}$=[ANS]", "answer_v2": [ "1/sqrt(3)", "- 1/sqrt(3)", "1/sqrt(3)", "-2/sqrt(3)", "- sqrt(2)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find the exact value of each:\na) $\\tan{\\left(\\frac{2 \\pi}{3} \\right)}$=[ANS]\nb) $\\tan{\\left(\\frac{5 \\pi}{4} \\right)}$=[ANS]\nc) $\\cot{\\left(\\frac{\\pi}{3} \\right)}$=[ANS]\nd) $\\sec{\\left(\\frac{7 \\pi}{6} \\right)}$=[ANS]\ne) $\\csc{\\left(\\frac{\\pi}{4} \\right)}$=[ANS]", "answer_v3": [ "-sqrt(3)", "1", "1/sqrt(3)", "-2/sqrt(3)", "sqrt(2)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0073", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "3", "keywords": [ "trigonometry", "unit circle" ], "problem_v1": "Evaluate the following expressions. Note: Your answer must be in EXACT form: it cannot contain decimals. It must be either an integer or a fraction. If the answer involves a square root write it as sqrt. For instance, the square root of 2 should be written as sqrt(2). $\\sin (\\frac{2\\pi}{3})=$ [ANS]\n$\\cos (0)=$ [ANS]\n$\\tan (\\frac{\\pi}{6})=$ [ANS]\n$\\cot (-\\frac{\\pi}{3})=$ [ANS]\n$\\sec (\\frac{\\pi}{4})=$ [ANS]\n$\\csc (\\frac{\\pi}{6})=$ [ANS]", "answer_v1": [ "0.866025403784439", "1", "0.577350269189626", "-0.577350269189626", "1.41421356237309", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Evaluate the following expressions. Note: Your answer must be in EXACT form: it cannot contain decimals. It must be either an integer or a fraction. If the answer involves a square root write it as sqrt. For instance, the square root of 2 should be written as sqrt(2). $\\sin (-\\pi)=$ [ANS]\n$\\cos (\\frac{4\\pi}{3})=$ [ANS]\n$\\tan (-\\frac{2\\pi}{3})=$ [ANS]\n$\\cot (\\frac{3\\pi}{4})=$ [ANS]\n$\\sec (-\\frac{\\pi}{4})=$ [ANS]\n$\\csc (-\\frac{\\pi}{3})=$ [ANS]", "answer_v2": [ "-1.22464679914735E-16", "-0.5", "1.73205080756888", "-1", "1.41421356237309", "-1.15470053837925" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Evaluate the following expressions. Note: Your answer must be in EXACT form: it cannot contain decimals. It must be either an integer or a fraction. If the answer involves a square root write it as sqrt. For instance, the square root of 2 should be written as sqrt(2). $\\sin (-\\frac{\\pi}{3})=$ [ANS]\n$\\cos (0)=$ [ANS]\n$\\tan (-\\frac{\\pi}{4})=$ [ANS]\n$\\cot (-\\frac{\\pi}{2})=$ [ANS]\n$\\sec (\\frac{\\pi}{6})=$ [ANS]\n$\\csc (-\\frac{\\pi}{4})=$ [ANS]", "answer_v3": [ "-0.866025403784439", "1", "-1", "-6.12323399573677E-17", "1.15470053837925", "-1.4142135623731" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Trigonometry_0074", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "2", "keywords": [ "trigonometry", "unit circle" ], "problem_v1": "If $\\theta=\\frac {5 \\pi} {3}$, then\n$\\sin (\\theta)$ equals [ANS]\n$\\cos (\\theta)$ equals [ANS]\n$\\tan (\\theta)$ equals [ANS]\n$\\sec (\\theta)$ equals [ANS]", "answer_v1": [ "-0.866025406775933", "0.49999999481858", "-1.73205083150083", "2.00000002072568" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "If $\\theta=\\frac {-7 \\pi} {3}$, then\n$\\sin (\\theta)$ equals [ANS]\n$\\cos (\\theta)$ equals [ANS]\n$\\tan (\\theta)$ equals [ANS]\n$\\sec (\\theta)$ equals [ANS]", "answer_v2": [ "-0.866025399596347", "0.500000007253988", "-1.73205077406414", "1.99999997098405" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "If $\\theta=\\frac {-4 \\pi} {3}$, then\n$\\sin (\\theta)$ equals [ANS]\n$\\cos (\\theta)$ equals [ANS]\n$\\tan (\\theta)$ equals [ANS]\n$\\sec (\\theta)$ equals [ANS]", "answer_v3": [ "0.866025401391243", "-0.500000004145136", "-1.73205078842332", "-1.99999998341946" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0075", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "3", "keywords": [ "trigonometry", "identity", "quadrants" ], "problem_v1": "Given $\\csc(\\alpha)=-2\\sqrt 3/3$ and $3\\pi/2<\\alpha<2\\pi$, find the exact values of the remaining five trigonometric functions. Note: You are not allowed to use decimals in your answer. $\\sin(\\alpha)$=[ANS]. $\\cos(\\alpha)$=[ANS]. $\\tan(\\alpha)$=[ANS]. $\\sec(\\alpha)$=[ANS]. $\\cot(\\alpha)$=[ANS].", "answer_v1": [ "-0.866025403784439", "0.5", "-1.73205080756888", "2", "-0.577350269189626" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Given $\\sin(\\alpha)=\\sqrt 2/2$ and $0<\\alpha<\\pi/2$, find the exact values of the remaining five trigonometric functions. Note: You are not allowed to use decimals in your answer. $\\cos(\\alpha)$=[ANS]. $\\tan(\\alpha)$=[ANS]. $\\csc(\\alpha)$=[ANS]. $\\sec(\\alpha)$=[ANS]. $\\cot(\\alpha)$=[ANS].", "answer_v2": [ "0.707106781186548", "1", "1.4142135623731", "1.4142135623731", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Given $\\cos(\\alpha)=-\\sqrt 2/2$ and $\\pi/2<\\alpha<\\pi$, find the exact values of the remaining five trigonometric functions. Note: You are not allowed to use decimals in your answer. $\\sin(\\alpha)$=[ANS]. $\\tan(\\alpha)$=[ANS]. $\\csc(\\alpha)$=[ANS]. $\\sec(\\alpha)$=[ANS]. $\\cot(\\alpha)$=[ANS].", "answer_v3": [ "0.707106781186548", "-1", "1.4142135623731", "-1.4142135623731", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0076", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of special angles", "level": "2", "keywords": [ "right triangle", "30-60-90" ], "problem_v1": "Suppose you are given a $30^\\circ-60^\\circ-90^\\circ$ triangle with hypotenuse of length $5.1$. Determine the length of the two sides. (Enter these below in order so that the first one is not larger than the second.) First leg=[ANS] Second leg=[ANS]", "answer_v1": [ "2.55", "4.41672955930064" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose you are given a $30^\\circ-60^\\circ-90^\\circ$ triangle with hypotenuse of length $2.3$. Determine the length of the two sides. (Enter these below in order so that the first one is not larger than the second.) First leg=[ANS] Second leg=[ANS]", "answer_v2": [ "1.15", "1.99185842870421" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose you are given a $30^\\circ-60^\\circ-90^\\circ$ triangle with hypotenuse of length $3.3$. Determine the length of the two sides. (Enter these below in order so that the first one is not larger than the second.) First leg=[ANS] Second leg=[ANS]", "answer_v3": [ "1.65", "2.85788383248865" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0077", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "2", "keywords": [ "calculus", "trigonometry", "trigonometric functions" ], "problem_v1": "Find $\\sin \\theta$ and $\\tan \\theta$ if $\\cos \\theta=\\frac {12}{37}$, assuming that $0\\leq\\theta < \\pi/2$. $\\sin \\theta=$ [ANS]\n$\\tan \\theta=$ [ANS]", "answer_v1": [ "0.945946", "2.91667" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find $\\sin \\theta$ and $\\tan \\theta$ if $\\cos \\theta=\\frac {3}{5}$, assuming that $0\\leq\\theta < \\pi/2$. $\\sin \\theta=$ [ANS]\n$\\tan \\theta=$ [ANS]", "answer_v2": [ "0.8", "1.33333" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find $\\sin \\theta$ and $\\tan \\theta$ if $\\cos \\theta=\\frac {7}{25}$, assuming that $0\\leq\\theta < \\pi/2$. $\\sin \\theta=$ [ANS]\n$\\tan \\theta=$ [ANS]", "answer_v3": [ "0.96", "3.42857" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0078", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "tangent", "cotangent", "cosecant", "secant" ], "problem_v1": "If $\\cos(\\phi)=0.8855$ and $3 \\pi/2 \\leq \\phi \\leq 2 \\pi$, approximate the following to four decimal places.\n(a) $\\sin(\\phi)$=[ANS] (Round to four decimal places.)\n(b) $\\tan(\\phi)$=[ANS] (Round to four decimal places.)", "answer_v1": [ "-0.464639", "-0.52472" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $\\cos(\\phi)=0.1865$ and $3 \\pi/2 \\leq \\phi \\leq 2 \\pi$, approximate the following to four decimal places.\n(a) $\\sin(\\phi)$=[ANS] (Round to four decimal places.)\n(b) $\\tan(\\phi)$=[ANS] (Round to four decimal places.)", "answer_v2": [ "-0.982455", "-5.26786" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $\\cos(\\phi)=0.4685$ and $3 \\pi/2 \\leq \\phi \\leq 2 \\pi$, approximate the following to four decimal places.\n(a) $\\sin(\\phi)$=[ANS] (Round to four decimal places.)\n(b) $\\tan(\\phi)$=[ANS] (Round to four decimal places.)", "answer_v3": [ "-0.883463", "-1.88573" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0079", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "tangent", "cotangent", "cosecant", "secant" ], "problem_v1": "Suppose the angle $\\theta$ is in the second quadrant, $\\pi/2 \\leq \\theta \\leq \\pi$, and $ \\sin(\\theta)=\\frac{1}{9}$. Find exact values (as fractions, not decimal approximations) for the following.\n(a) $\\sec{\\theta}$=[ANS]\n(b) $\\tan{\\theta}$=[ANS]", "answer_v1": [ "-[sqrt(9^2)]/[sqrt(9^2-1)]", "-1/[sqrt(9^2-1)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Suppose the angle $\\theta$ is in the second quadrant, $\\pi/2 \\leq \\theta \\leq \\pi$, and $ \\sin(\\theta)=\\frac{1}{3}$. Find exact values (as fractions, not decimal approximations) for the following.\n(a) $\\sec{\\theta}$=[ANS]\n(b) $\\tan{\\theta}$=[ANS]", "answer_v2": [ "-[sqrt(3^2)]/[sqrt(3^2-1)]", "-1/[sqrt(3^2-1)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Suppose the angle $\\theta$ is in the second quadrant, $\\pi/2 \\leq \\theta \\leq \\pi$, and $ \\sin(\\theta)=\\frac{1}{5}$. Find exact values (as fractions, not decimal approximations) for the following.\n(a) $\\sec{\\theta}$=[ANS]\n(b) $\\tan{\\theta}$=[ANS]", "answer_v3": [ "-[sqrt(5^2)]/[sqrt(5^2-1)]", "-1/[sqrt(5^2-1)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0080", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "trigonometry", "unit circle" ], "problem_v1": "If $\\cos(\\theta)=-\\frac {7} {10}$ and $\\theta$ is in quadrant III, then find\n(a) $\\tan(\\theta) \\cot(\\theta)=$ [ANS]\n(b) $\\csc(\\theta) \\tan(\\theta)=$ [ANS]\n(c) $\\sin ^2 (\\theta)+\\cos ^2 (\\theta)=$ [ANS]", "answer_v1": [ "1", "-1.42857142857143", "1" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If $\\cos(\\theta)=-\\frac {5} {12}$ and $\\theta$ is in quadrant II, then find\n(a) $\\tan(\\theta) \\cot(\\theta)=$ [ANS]\n(b) $\\csc(\\theta) \\tan(\\theta)=$ [ANS]\n(c) $\\sin ^2 (\\theta)+\\cos ^2 (\\theta)=$ [ANS]", "answer_v2": [ "1", "-2.4", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If $\\cos(\\theta)=-\\frac {5} {10}$ and $\\theta$ is in quadrant II, then find\n(a) $\\tan(\\theta) \\cot(\\theta)=$ [ANS]\n(b) $\\csc(\\theta) \\tan(\\theta)=$ [ANS]\n(c) $\\sin ^2 (\\theta)+\\cos ^2 (\\theta)=$ [ANS]", "answer_v3": [ "1", "-2", "1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0081", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "1", "keywords": [ "algebra", "trigonometric functions" ], "problem_v1": "Find the following values: $\\sin (0.6)=$ [ANS] ; $\\cos (0.6)=$ [ANS] ; $\\tan (0.6)=$ [ANS].", "answer_v1": [ "0.564642473395035", "0.825335614909678", "0.684136808341692" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the following values: $\\sin (0.1)=$ [ANS] ; $\\cos (0.1)=$ [ANS] ; $\\tan (0.1)=$ [ANS].", "answer_v2": [ "0.0998334166468282", "0.995004165278026", "0.100334672085451" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the following values: $\\sin (0.25)=$ [ANS] ; $\\cos (0.25)=$ [ANS] ; $\\tan (0.25)=$ [ANS].", "answer_v3": [ "0.247403959254523", "0.968912421710645", "0.255341921221036" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0082", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "trigonometry", "identity", "quadrants" ], "problem_v1": "If $\\cos(\\alpha)$=$\\frac {5} {9}$, and $\\alpha$ is in quadrant IV, then $\\sin(\\alpha)$=[ANS]\nExpress your answer in exact form. Your answer may contain NO decimals. Type \"sqrt\" if you need to use a square root.", "answer_v1": [ "-0.831479419283098" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $\\sin(\\alpha)$=$\\frac {3} {13}$, and $\\alpha$ is in quadrant II, then $\\cos(\\alpha)$=[ANS]\nExpress your answer in exact form. Your answer may contain NO decimals. Type \"sqrt\" if you need to use a square root.", "answer_v2": [ "-0.97300851082104" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $\\sin(\\alpha)$=-$\\frac {3} {11}$, and $\\alpha$ is in quadrant IV, then $\\cos(\\alpha)$=[ANS]\nExpress your answer in exact form. Your answer may contain NO decimals. Type \"sqrt\" if you need to use a square root.", "answer_v3": [ "0.962091385841669" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0083", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "2", "keywords": [ "trigonometry" ], "problem_v1": "Determine whether each of the following expressions is Positive or Negative. $\\sin(271^\\circ)$ [ANS] A. Positive B. Negative\n$\\cos(209^\\circ)$ [ANS] A. Negative B. Positive\n$\\sin(\\frac{10\\pi}{17})$ [ANS] A. Negative B. Positive\n$\\cos(\\frac{13\\pi}{17})$ [ANS] A. Positive B. Negative", "answer_v1": [ "B", "A", "B", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "Determine whether each of the following expressions is Positive or Negative. $\\sin(30^\\circ)$ [ANS] A. Positive B. Negative\n$\\cos(335^\\circ)$ [ANS] A. Positive B. Negative\n$\\sin(\\frac{9\\pi}{8})$ [ANS] A. Positive B. Negative\n$\\cos(\\frac{17\\pi}{19})$ [ANS] A. Negative B. Positive", "answer_v2": [ "A", "A", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "Determine whether each of the following expressions is Positive or Negative. $\\sin(113^\\circ)$ [ANS] A. Positive B. Negative\n$\\cos(218^\\circ)$ [ANS] A. Positive B. Negative\n$\\sin(\\frac{13\\pi}{19})$ [ANS] A. Negative B. Positive\n$\\cos(\\frac{19\\pi}{15})$ [ANS] A. Positive B. Negative", "answer_v3": [ "A", "B", "B", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Trigonometry_0084", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "2", "keywords": [ "standard position" ], "problem_v1": "If the point (10,-7) is on the terminal side of the angle $\\theta$ in standard position, $\\sin (\\theta)=$ [ANS] ; $\\cos (\\theta)=$ [ANS] ; $\\tan (\\theta)=$ [ANS].", "answer_v1": [ "-0.573462344363328", "0.81923192051904", "-0.7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If the point (4,-5) is on the terminal side of the angle $\\theta$ in standard position, $\\sin (\\theta)=$ [ANS] ; $\\cos (\\theta)=$ [ANS] ; $\\tan (\\theta)=$ [ANS].", "answer_v2": [ "-0.78086880944303", "0.624695047554424", "-1.25" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If the point (6,-7) is on the terminal side of the angle $\\theta$ in standard position, $\\sin (\\theta)=$ [ANS] ; $\\cos (\\theta)=$ [ANS] ; $\\tan (\\theta)=$ [ANS].", "answer_v3": [ "-0.759256602365297", "0.650791373455968", "-1.16666666666667" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0085", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "right triangle", "angles" ], "problem_v1": "Assume that $\\theta$ is an angle in quadrant III. Find the values of the six triginometric functions given the following:\n\\sec(\\theta)=\\frac{-45}{34}.\n$\\sin(\\theta)$=[ANS]\n$\\cos(\\theta)$=[ANS]\n$\\tan(\\theta)$=[ANS]\n$\\cot(\\theta)$=[ANS]\n$\\sec(\\theta)$=[ANS]\n$\\csc(\\theta)$=[ANS]", "answer_v1": [ "-0.655084576577052", "-0.755555555555556", "0.867023704293157", "1.15337100363969", "-1.32352941176471", "-1.52652044599371" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Assume that $\\theta$ is an angle in quadrant I. Find the values of the six triginometric functions given the following:\n\\tan(\\theta)=\\frac{5}{56}.\n$\\sin(\\theta)$=[ANS]\n$\\cos(\\theta)$=[ANS]\n$\\tan(\\theta)$=[ANS]\n$\\cot(\\theta)$=[ANS]\n$\\sec(\\theta)$=[ANS]\n$\\csc(\\theta)$=[ANS]", "answer_v2": [ "0.0889319379745055", "0.996037705314462", "0.0892857142857143", "11.2", "1.00397805691933", "11.2445542374965" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Assume that $\\theta$ is an angle in quadrant II. Find the values of the six triginometric functions given the following:\n\\cot(\\theta)=\\frac{-19}{37}.\n$\\sin(\\theta)$=[ANS]\n$\\cos(\\theta)$=[ANS]\n$\\tan(\\theta)$=[ANS]\n$\\cot(\\theta)$=[ANS]\n$\\sec(\\theta)$=[ANS]\n$\\csc(\\theta)$=[ANS]", "answer_v3": [ "0.889567018143538", "-0.456804684992627", "-1.94736842105263", "-0.513513513513513", "-2.18911940453531", "1.12414239692354" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Trigonometry_0086", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "trigonometry", "angle", "degree" ], "problem_v1": "Determine any point in the third quadrant for which $\\tan(\\alpha)=3$ One valid point=[ANS]\nFor this angle, $\\sin(\\alpha)=$ [ANS]", "answer_v1": [ "(-1,-3)", "-0.948683298050514" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine any point in the third quadrant for which $\\tan(\\alpha)=7$ One valid point=[ANS]\nFor this angle, $\\sin(\\alpha)=$ [ANS]", "answer_v2": [ "(-1,-7)", "-0.989949493661166" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine any point in the third quadrant for which $\\tan(\\alpha)=6$ One valid point=[ANS]\nFor this angle, $\\sin(\\alpha)=$ [ANS]", "answer_v3": [ "(-1,-6)", "-0.986393923832144" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0087", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "3", "keywords": [ "trigonometry", "functions" ], "problem_v1": "Suppose $\\tan(\\alpha)=-\\frac{5}{2}$ and $\\alpha$ terminates in Quadrant II. Without simplifying any square roots, $\\sin(\\alpha)=$ [ANS] [ANS]\n$\\cos(\\alpha)=$ [ANS] [ANS]\n$\\cot(\\alpha)=$ [ANS] [ANS]\n$\\sec(\\alpha)=$ [ANS] [ANS]\n$\\csc(\\alpha)=$ [ANS] [ANS]", "answer_v1": [ "+", "|5/[sqrt(29)]|", "-", "|-2|/[sqrt(29)]", "-", "|-2/5|", "-", "[sqrt(29)]/|-2|", "+", "[sqrt(29)]/|5|" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV" ], "options_v1": [ [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [] ], "problem_v2": "Suppose $\\tan(\\alpha)=-\\frac{8}{8}$ and $\\alpha$ terminates in Quadrant IV. Without simplifying any square roots, $\\sin(\\alpha)=$ [ANS] [ANS]\n$\\cos(\\alpha)=$ [ANS] [ANS]\n$\\cot(\\alpha)=$ [ANS] [ANS]\n$\\sec(\\alpha)=$ [ANS] [ANS]\n$\\csc(\\alpha)=$ [ANS] [ANS]", "answer_v2": [ "-", "|-8/[sqrt(128)]|", "+", "|8|/[sqrt(128)]", "-", "|8/-8|", "+", "[sqrt(128)]/|8|", "-", "[sqrt(128)]/|-8|" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV" ], "options_v2": [ [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [] ], "problem_v3": "Suppose $\\tan(\\alpha)=-\\frac{4}{2}$ and $\\alpha$ terminates in Quadrant IV. Without simplifying any square roots, $\\sin(\\alpha)=$ [ANS] [ANS]\n$\\cos(\\alpha)=$ [ANS] [ANS]\n$\\cot(\\alpha)=$ [ANS] [ANS]\n$\\sec(\\alpha)=$ [ANS] [ANS]\n$\\csc(\\alpha)=$ [ANS] [ANS]", "answer_v3": [ "-", "|-4/[sqrt(20)]|", "+", "|2|/[sqrt(20)]", "-", "|2/-4|", "+", "[sqrt(20)]/|2|", "-", "[sqrt(20)]/|-4|" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV" ], "options_v3": [ [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [], [ "+", "-" ], [] ] }, { "id": "Trigonometry_0088", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "2", "keywords": [ "trigonometry", "angle", "degree" ], "problem_v1": "Suppose $\\cos(\\alpha)=\\frac{-6}{8}$ and $\\sin(\\alpha)$ is positive. If the point $(-6,y)$ is on the terminal side of the angle $\\alpha$ then (without simplifying any square roots) $y=$ [ANS]", "answer_v1": [ "5.2915" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $\\cos(\\alpha)=\\frac{-9}{12}$ and $\\sin(\\alpha)$ is positive. If the point $(-9,y)$ is on the terminal side of the angle $\\alpha$ then (without simplifying any square roots) $y=$ [ANS]", "answer_v2": [ "7.93725" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $\\cos(\\alpha)=\\frac{-8}{10}$ and $\\sin(\\alpha)$ is positive. If the point $(-8,y)$ is on the terminal side of the angle $\\alpha$ then (without simplifying any square roots) $y=$ [ANS]", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0089", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Trigonometric functions of non-special angles", "level": "5", "keywords": [ "Radians", "Paper", "Unit Circle" ], "problem_v1": "Writing Assignment Prepare a paper (greater than one page with 10 point font) discussing radian measure and the unit circle. In the paper, be certain to:-use some flexible item as the definition of unit measure. 1-2 feet long is good.-draw a circle by holding one end and rotate a pen (or sidewalk chalk) along the other end-lay the flexible item along the perimeter of the circle. One length along the circle gives an angle of one radian; Two lengths give two radians; 1/2 of a length gives 1/2 of a radian; etc. In this manner, create physical examples for each of the following values. Take a picture of your results and share these in your paper.\n$\\begin{array}{cccc}\\hline 1. & sin(1) rounded to the nearest eighth &=& [ANS] \\\\\\hline 2. & sin(4) rounded to the nearest eighth &=& [ANS] \\\\\\hline\\end{array}$\nPrepare a paper, with appropriate references, and share with your instructor.", "answer_v1": [ "0.875", "-0.75" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Writing Assignment Prepare a paper (greater than one page with 10 point font) discussing radian measure and the unit circle. In the paper, be certain to:-use some flexible item as the definition of unit measure. 1-2 feet long is good.-draw a circle by holding one end and rotate a pen (or sidewalk chalk) along the other end-lay the flexible item along the perimeter of the circle. One length along the circle gives an angle of one radian; Two lengths give two radians; 1/2 of a length gives 1/2 of a radian; etc. In this manner, create physical examples for each of the following values. Take a picture of your results and share these in your paper.\n$\\begin{array}{cccc}\\hline 1. & sin(1) rounded to the nearest eighth &=& [ANS] \\\\\\hline 2. & 1 revolution in radians rounded to nearest eighth &=& [ANS] \\\\\\hline\\end{array}$\nPrepare a paper, with appropriate references, and share with your instructor.", "answer_v2": [ "0.875", "6.125" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Writing Assignment Prepare a paper (greater than one page with 10 point font) discussing radian measure and the unit circle. In the paper, be certain to:-use some flexible item as the definition of unit measure. 1-2 feet long is good.-draw a circle by holding one end and rotate a pen (or sidewalk chalk) along the other end-lay the flexible item along the perimeter of the circle. One length along the circle gives an angle of one radian; Two lengths give two radians; 1/2 of a length gives 1/2 of a radian; etc. In this manner, create physical examples for each of the following values. Take a picture of your results and share these in your paper.\n$\\begin{array}{cccc}\\hline 1. & 60 radians rounded to closest axis angle (in radians) &=& [ANS] \\\\\\hline 2. & 1 revolution in radians rounded to nearest eighth &=& [ANS] \\\\\\hline\\end{array}$\nPrepare a paper, with appropriate references, and share with your instructor.", "answer_v3": [ "59.6903", "6.125" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0090", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "5", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "phase shift" ], "problem_v1": "The pressure $P$ (in pounds per square foot), in a pipe varies over time. Six times an hour, the pressure oscillates from a low of 80 to a high of 260 and then back to a low of 80. The pressure at time $t=0$ is 80. Let the function $P=f(t)$ denote the pressure in pipe at time $t$ minutes. Find a possible formula for the function $P=f(t)$ described above. $f(t)=$ [ANS]", "answer_v1": [ "-90*cos(2/10*pi*t)+170" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The pressure $P$ (in pounds per square foot), in a pipe varies over time. Two times an hour, the pressure oscillates from a low of 120 to a high of 210 and then back to a low of 120. The pressure at time $t=0$ is 120. Let the function $P=f(t)$ denote the pressure in pipe at time $t$ minutes. Find a possible formula for the function $P=f(t)$ described above. $f(t)=$ [ANS]", "answer_v2": [ "-45*cos(2/30*pi*t)+165" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The pressure $P$ (in pounds per square foot), in a pipe varies over time. Three times an hour, the pressure oscillates from a low of 90 to a high of 230 and then back to a low of 90. The pressure at time $t=0$ is 90. Let the function $P=f(t)$ denote the pressure in pipe at time $t$ minutes. Find a possible formula for the function $P=f(t)$ described above. $f(t)=$ [ANS]", "answer_v3": [ "-70*cos(2/20*pi*t)+160" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0091", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "5", "keywords": [ "sine", "cosine", "period", "amplitude", "midline", "phase shift" ], "problem_v1": "A ferris wheel is 35 meters in diameter and boarded at ground level. The wheel makes one full rotation every 6 minutes, and at time $t=0$ you are at the 9 o'clock position and descending. Let $f(t)$ denote your height (in meters) above ground at $t$ minutes. Find a formula for $f(t)$. $f(t)=$ [ANS]", "answer_v1": [ "-17.5*sin(2*pi/6*t)+17.5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A ferris wheel is 20 meters in diameter and boarded at ground level. The wheel makes one full rotation every 8 minutes, and at time $t=0$ you are at the 3 o'clock position and ascending. Let $f(t)$ denote your height (in meters) above ground at $t$ minutes. Find a formula for $f(t)$. $f(t)=$ [ANS]", "answer_v2": [ "10*sin(2*pi/8*t)+10" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A ferris wheel is 25 meters in diameter and boarded at ground level. The wheel makes one full rotation every 7 minutes, and at time $t=0$ you are at the 9 o'clock position and ascending. Let $f(t)$ denote your height (in meters) above ground at $t$ minutes. Find a formula for $f(t)$. $f(t)=$ [ANS]", "answer_v3": [ "12.5*sin(2*pi/7*t)+12.5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0092", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "5", "keywords": [ "calculus", "function", "trigonometric functions", "inverse trigonometric functions", "transformation of functions", "translations" ], "problem_v1": "A population of animals oscillates sinusoidally between a low of 500 on January 1 and a high of 1200 on July 1. Graph the population against time and use your graph to find a formula for the population $P$ as a function of time $t$, in months since the start of the year. Assume that the period of $P$ is one year. $P(t)=$ [ANS]", "answer_v1": [ "-350*cos(0.523599*t)+850" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A population of animals oscillates sinusoidally between a low of 100 on January 1 and a high of 1600 on July 1. Graph the population against time and use your graph to find a formula for the population $P$ as a function of time $t$, in months since the start of the year. Assume that the period of $P$ is one year. $P(t)=$ [ANS]", "answer_v2": [ "-750*cos(0.523599*t)+850" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A population of animals oscillates sinusoidally between a low of 200 on January 1 and a high of 1300 on July 1. Graph the population against time and use your graph to find a formula for the population $P$ as a function of time $t$, in months since the start of the year. Assume that the period of $P$ is one year. $P(t)=$ [ANS]", "answer_v3": [ "-550*cos(0.523599*t)+750" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0093", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "5", "keywords": [ "calculus", "function", "trigonometric functions", "inverse trigonometric functions", "transformation of functions", "translations" ], "problem_v1": "A mass is oscillating on the end of a spring. The distance, $y$, of the mass from its equilibrium point is given by the formula\ny=7 z \\cos (12 \\pi w t) where $y$ is in centimeters, $t$ is time in seconds, and $z$ and $w$ are positive constants.\n(a) What is the furthest distance of the mass from its equilibrium point? distance=[ANS] cm (b) How many oscillations are completed in 1 second? number of oscillations=[ANS]", "answer_v1": [ "7*z", "12*w/2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A mass is oscillating on the end of a spring. The distance, $y$, of the mass from its equilibrium point is given by the formula\ny=2 z \\cos (16 \\pi w t) where $y$ is in centimeters, $t$ is time in seconds, and $z$ and $w$ are positive constants.\n(a) What is the furthest distance of the mass from its equilibrium point? distance=[ANS] cm (b) How many oscillations are completed in 1 second? number of oscillations=[ANS]", "answer_v2": [ "2*z", "16*w/2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A mass is oscillating on the end of a spring. The distance, $y$, of the mass from its equilibrium point is given by the formula\ny=4 z \\cos (12 \\pi w t) where $y$ is in centimeters, $t$ is time in seconds, and $z$ and $w$ are positive constants.\n(a) What is the furthest distance of the mass from its equilibrium point? distance=[ANS] cm (b) How many oscillations are completed in 1 second? number of oscillations=[ANS]", "answer_v3": [ "4*z", "12*w/2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0094", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "4", "keywords": [ "trigonometry", "spring", "word problem" ], "problem_v1": "A weight on a vertical spring is given an initial downward velocity of 5 cm/sec from a point 7 cm above equilibrium. Assume that the contstant $\\omega$ has a value of 0.1. Write the formula for the location of the weight at time t. x=[ANS]\nFind the location of the weight 13 seconds after it is set in motion. [ANS] centimeters", "answer_v1": [ "1*(5/0.1)*sin(0.1*t)+(-1*7)*cos(0.1*t)", "46.3054174704875" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A weight on a vertical spring is given an initial downward velocity of 1 cm/sec from a point 10 cm above equilibrium. Assume that the contstant $\\omega$ has a value of 0.1. Write the formula for the location of the weight at time t. x=[ANS]\nFind the location of the weight 3 seconds after it is set in motion. [ANS] centimeters", "answer_v2": [ "1*(1/0.1)*sin(0.1*t)+(-1*10)*cos(0.1*t)", "-6.59816282464266" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A weight on a vertical spring is given an initial upward velocity of 2 cm/sec from a point 7 cm below equilibrium. Assume that the contstant $\\omega$ has a value of 0.1. Write the formula for the location of the weight at time t. x=[ANS]\nFind the location of the weight 6 seconds after it is set in motion. [ANS] centimeters", "answer_v3": [ "-1*(2/0.1)*sin(0.1*t)+(1*7)*cos(0.1*t)", "-5.51550016353296" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0095", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "5", "keywords": [ "trigonometry", "sine", "cosine", "application" ], "problem_v1": "Over the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about \\$ 52000 in May and a minimum of about \\$ 26000 in November. Suppose the months are numbered 1 through 12, and write a function of the form $f(x)=A\\sin(B \\left [x-C \\right])+D$ that models the boutique's revenue during the year, where $x$ corresponds to the month. If needed, you can enter $\\pi$=3.1416... as 'pi' in your answer. $f(x)=$ [ANS]", "answer_v1": [ "(52000-((52000+26000)/2))*sin((3.14159265358979/6)*x+(3.14159265358979/6)*(3-(4+1)))+((52000+26000)/2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Over the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about \\$ 59000 in January and a minimum of about \\$ 21000 in July. Suppose the months are numbered 1 through 12, and write a function of the form $f(x)=A\\sin(B \\left [x-C \\right])+D$ that models the boutique's revenue during the year, where $x$ corresponds to the month. If needed, you can enter $\\pi$=3.1416... as 'pi' in your answer. $f(x)=$ [ANS]", "answer_v2": [ "(59000-((59000+21000)/2))*sin((3.14159265358979/6)*x+(3.14159265358979/6)*(3-(0+1)))+((59000+21000)/2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Over the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about \\$ 52000 in February and a minimum of about \\$ 23000 in August. Suppose the months are numbered 1 through 12, and write a function of the form $f(x)=A\\sin(B \\left [x-C \\right])+D$ that models the boutique's revenue during the year, where $x$ corresponds to the month. If needed, you can enter $\\pi$=3.1416... as 'pi' in your answer. $f(x)=$ [ANS]", "answer_v3": [ "(52000-((52000+23000)/2))*sin((3.14159265358979/6)*x+(3.14159265358979/6)*(3-(1+1)))+((52000+23000)/2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0096", "subject": "Trigonometry", "topic": "Trigonometric functions", "subtopic": "Modeling with trigonometric functions", "level": "2", "keywords": [ "trigonometry", "sine", "cosine", "application" ], "problem_v1": "The volume of air contained in the lungs of a certain athlete is modeled by the equation $v=451 \\sin (75 \\pi t)+886$, where $t$ is time in minutes, and $v$ is volume in cubic centimeters. What is the maximum possible volume of air in the athlete's lungs? Maximum volume=[ANS] cubic centimeters What is the minimum possible volume of air in the athlete's lungs? Minimum volume=[ANS] cubic centimeters How many breaths does the athlete take per minute? [ANS] breaths per minute", "answer_v1": [ "1337", "435", "37.5" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The volume of air contained in the lungs of a certain athlete is modeled by the equation $v=316 \\sin (96 \\pi t)+745$, where $t$ is time in minutes, and $v$ is volume in cubic centimeters. What is the maximum possible volume of air in the athlete's lungs? Maximum volume=[ANS] cubic centimeters What is the minimum possible volume of air in the athlete's lungs? Minimum volume=[ANS] cubic centimeters How many breaths does the athlete take per minute? [ANS] breaths per minute", "answer_v2": [ "1061", "429", "48" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The volume of air contained in the lungs of a certain athlete is modeled by the equation $v=363 \\sin (76 \\pi t)+783$, where $t$ is time in minutes, and $v$ is volume in cubic centimeters. What is the maximum possible volume of air in the athlete's lungs? Maximum volume=[ANS] cubic centimeters What is the minimum possible volume of air in the athlete's lungs? Minimum volume=[ANS] cubic centimeters How many breaths does the athlete take per minute? [ANS] breaths per minute", "answer_v3": [ "1146", "420", "38" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0097", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Sine, cosine, and tangent of an angle in a right triangle", "level": "3", "keywords": [ "right triangle", "sides", "angles", "hypotenuse" ], "problem_v1": "Suppose that triangle ABC is a right triangle with a right angle at $C$ and hypotenuse $c$. Also note that $a$ is the length of the side opposite angle $A$ and $b$ is the length of the side opposite angle $B$. Given that $c$=18 and $A$=$55^ \\circ$, determine the values indicated below. Round to four decimal places when needed.\n$a$=[ANS]\n$b$=[ANS]\n$B$=[ANS] degrees.", "answer_v1": [ "14.7447367972019", "10.3243758543188", "35" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that triangle ABC is a right triangle with a right angle at $C$ and hypotenuse $c$. Also note that $a$ is the length of the side opposite angle $A$ and $b$ is the length of the side opposite angle $B$. Given that $a$=28 and $A$=$24^ \\circ$, determine the values indicated below. Round to four decimal places when needed.\n$b$=[ANS]\n$c$=[ANS]\n$B$=[ANS] degrees.", "answer_v2": [ "62.8890296693181", "68.8406133960787", "66" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that triangle ABC is a right triangle with a right angle at $C$ and hypotenuse $c$. Also note that $a$ is the length of the side opposite angle $A$ and $b$ is the length of the side opposite angle $B$. Given that $a$=19 and $A$=$33^ \\circ$, determine the values indicated below. Round to four decimal places when needed.\n$b$=[ANS]\n$c$=[ANS]\n$B$=[ANS] degrees.", "answer_v3": [ "29.2574343124771", "34.8854907167566", "57" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0098", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "An airplane is flying at the height of ${10.8\\ {\\rm km}}$. The airplane can see both City 1 and City 2 straight ahead of it. The angle of depression to City 1 is $46$ degrees, and the angle of depression to City 2 is $32$ degrees. Find the distance between those two cities. Round your answer to two decimal places if needed. The distance between those two cities is [ANS]km.", "answer_v1": [ "6.85417" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An airplane is flying at the height of ${7.9\\ {\\rm km}}$. The airplane can see both City 1 and City 2 straight ahead of it. The angle of depression to City 1 is $40$ degrees, and the angle of depression to City 2 is $26$ degrees. Find the distance between those two cities. Round your answer to two decimal places if needed. The distance between those two cities is [ANS]km.", "answer_v2": [ "6.78255" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An airplane is flying at the height of ${9.7\\ {\\rm km}}$. The airplane can see both City 1 and City 2 straight ahead of it. The angle of depression to City 1 is $34$ degrees, and the angle of depression to City 2 is $24$ degrees. Find the distance between those two cities. Round your answer to two decimal places if needed. The distance between those two cities is [ANS]km.", "answer_v3": [ "7.40572" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0099", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A ramp is set up from a truck\u2019s trunk to the ground. The ramp\u2019s end at the edge of the trunk is ${4.5\\ {\\rm ft}}$ high. If the angle between the ramp and the ground is $16$ degrees, find the length of the ramp. Round your answer to two decimal places if needed. The length of the ramp is [ANS]ft.", "answer_v1": [ "16.3258" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A ramp is set up from a truck\u2019s trunk to the ground. The ramp\u2019s end at the edge of the trunk is ${3.1\\ {\\rm ft}}$ high. If the angle between the ramp and the ground is $20$ degrees, find the length of the ramp. Round your answer to two decimal places if needed. The length of the ramp is [ANS]ft.", "answer_v2": [ "9.06379" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A ramp is set up from a truck\u2019s trunk to the ground. The ramp\u2019s end at the edge of the trunk is ${3.6\\ {\\rm ft}}$ high. If the angle between the ramp and the ground is $16$ degrees, find the length of the ramp. Round your answer to two decimal places if needed. The length of the ramp is [ANS]ft.", "answer_v3": [ "13.0606" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0100", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A straight road connects two towns, which are built on a $16$ degree slope. The road is ${7.4\\ {\\rm km}}$ long. Find the difference in elevation between the two towns(vertical distance). Round your answer to two decimal places if needed. The difference in elevation is [ANS]km.", "answer_v1": [ "2.03972" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A straight road connects two towns, which are built on a $24$ degree slope. The road is ${6.7\\ {\\rm km}}$ long. Find the difference in elevation between the two towns(vertical distance). Round your answer to two decimal places if needed. The difference in elevation is [ANS]km.", "answer_v2": [ "2.72514" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A straight road connects two towns, which are built on a $15$ degree slope. The road is ${7.8\\ {\\rm km}}$ long. Find the difference in elevation between the two towns(vertical distance). Round your answer to two decimal places if needed. The difference in elevation is [ANS]km.", "answer_v3": [ "2.01879" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0101", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "4", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A lighthouse has a spotlight on its top, and the spotlight is shining light on a boat on the sea. The lighthouse is ${163\\ {\\rm ft}}$ high, and the angle of depression from the spotlight toward the boat is $49$ degrees. Find the horizontal distance from the boat to the bottom of the lighthouse. Round your answer to two decimal places if needed. The horizontal distance from the boat to the lighthouse is [ANS]ft.", "answer_v1": [ "141.694" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A lighthouse has a spotlight on its top, and the spotlight is shining light on a boat on the sea. The lighthouse is ${62\\ {\\rm ft}}$ high, and the angle of depression from the spotlight toward the boat is $67$ degrees. Find the horizontal distance from the boat to the bottom of the lighthouse. Round your answer to two decimal places if needed. The horizontal distance from the boat to the lighthouse is [ANS]ft.", "answer_v2": [ "26.3174" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A lighthouse has a spotlight on its top, and the spotlight is shining light on a boat on the sea. The lighthouse is ${97\\ {\\rm ft}}$ high, and the angle of depression from the spotlight toward the boat is $50$ degrees. Find the horizontal distance from the boat to the bottom of the lighthouse. Round your answer to two decimal places if needed. The horizontal distance from the boat to the lighthouse is [ANS]ft.", "answer_v3": [ "81.3927" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0102", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A spotlight will be installed on a wall, shining light at a spot ${14\\ {\\rm m}}$ from the bottom of the wall. If the angle between the spotlight\u2019s light beam and the wall is $49$ degrees, how high on the wall should the spotlight be installed? Round your answer to two decimal places if needed. The spotlight should be installed on the wall at the height of [ANS]m.", "answer_v1": [ "12.17" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A spotlight will be installed on a wall, shining light at a spot ${6\\ {\\rm m}}$ from the bottom of the wall. If the angle between the spotlight\u2019s light beam and the wall is $67$ degrees, how high on the wall should the spotlight be installed? Round your answer to two decimal places if needed. The spotlight should be installed on the wall at the height of [ANS]m.", "answer_v2": [ "2.54685" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A spotlight will be installed on a wall, shining light at a spot ${9\\ {\\rm m}}$ from the bottom of the wall. If the angle between the spotlight\u2019s light beam and the wall is $50$ degrees, how high on the wall should the spotlight be installed? Round your answer to two decimal places if needed. The spotlight should be installed on the wall at the height of [ANS]m.", "answer_v3": [ "7.5519" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0103", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A person is flying a kite. The string is fully extended at ${43\\ {\\rm ft}}$. The hand holding the string is very close to his eyes, which are ${5.5\\ {\\rm ft}}$ above the ground. When he looks up at the kite, the angle of elevation is $49$ degrees. Find the height of the kite. Round your answer to two decimal places if needed. The height of the kite is [ANS]ft.", "answer_v1": [ "37.9525" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A person is flying a kite. The string is fully extended at ${22\\ {\\rm ft}}$. The hand holding the string is very close to his eyes, which are ${5\\ {\\rm ft}}$ above the ground. When he looks up at the kite, the angle of elevation is $67$ degrees. Find the height of the kite. Round your answer to two decimal places if needed. The height of the kite is [ANS]ft.", "answer_v2": [ "25.2511" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A person is flying a kite. The string is fully extended at ${29\\ {\\rm ft}}$. The hand holding the string is very close to his eyes, which are ${5\\ {\\rm ft}}$ above the ground. When he looks up at the kite, the angle of elevation is $50$ degrees. Find the height of the kite. Round your answer to two decimal places if needed. The height of the kite is [ANS]ft.", "answer_v3": [ "27.2153" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0104", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A building has two storeys with unequal heights. From an observation point on the ground ${51\\ {\\rm ft}}$ from the bottom of the building, the angle of elevation to the bottom of the second storey is $20$ degrees, and the angle of elevation to the top of the second storey is $31$ degrees. Find the height of the second storey. Round your answer to two decimal places if needed. The height of the second storey is [ANS]ft.", "answer_v1": [ "12.0814" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A building has two storeys with unequal heights. From an observation point on the ground ${48\\ {\\rm ft}}$ from the bottom of the building, the angle of elevation to the bottom of the second storey is $22$ degrees, and the angle of elevation to the top of the second storey is $37$ degrees. Find the height of the second storey. Round your answer to two decimal places if needed. The height of the second storey is [ANS]ft.", "answer_v2": [ "16.7773" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A building has two storeys with unequal heights. From an observation point on the ground ${37\\ {\\rm ft}}$ from the bottom of the building, the angle of elevation to the bottom of the second storey is $20$ degrees, and the angle of elevation to the top of the second storey is $33$ degrees. Find the height of the second storey. Round your answer to two decimal places if needed. The height of the second storey is [ANS]ft.", "answer_v3": [ "10.5612" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0105", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "4", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A guy wire needs to be installed to support a pole. The end of the wire on the pole will be ${14\\ {\\rm ft}}$ from the ground, and the angle formed by the wire and the ground will be $49$ degrees. On the ground, how far away is the guy wire from the pole? Round your answer to two decimal places if needed. On the ground, the distance from the guy wire to the pole is [ANS]ft.", "answer_v1": [ "12.17" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A guy wire needs to be installed to support a pole. The end of the wire on the pole will be ${6\\ {\\rm ft}}$ from the ground, and the angle formed by the wire and the ground will be $67$ degrees. On the ground, how far away is the guy wire from the pole? Round your answer to two decimal places if needed. On the ground, the distance from the guy wire to the pole is [ANS]ft.", "answer_v2": [ "2.54685" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A guy wire needs to be installed to support a pole. The end of the wire on the pole will be ${9\\ {\\rm ft}}$ from the ground, and the angle formed by the wire and the ground will be $50$ degrees. On the ground, how far away is the guy wire from the pole? Round your answer to two decimal places if needed. On the ground, the distance from the guy wire to the pole is [ANS]ft.", "answer_v3": [ "7.5519" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0106", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "4", "keywords": [ "trigonometry", "sine", "cosine", "tangent", "right triangle" ], "problem_v1": "A person is standing straight on the ground, looking up at an airplane which is taking off. His eyes is ${5.5\\ {\\rm ft}}$ from the ground. Horizontally, the person is ${176\\ {\\rm ft}}$ away from the airplane. The angle of elevation from his eyes to the airplane is $39$ degrees. Find the height of the airplane. Round your answer to two decimal places if needed. The height of the airplane is [ANS]ft.", "answer_v1": [ "148.022" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A person is standing straight on the ground, looking up at an airplane which is taking off. His eyes is ${5\\ {\\rm ft}}$ from the ground. Horizontally, the person is ${108\\ {\\rm ft}}$ away from the airplane. The angle of elevation from his eyes to the airplane is $57$ degrees. Find the height of the airplane. Round your answer to two decimal places if needed. The height of the airplane is [ANS]ft.", "answer_v2": [ "171.305" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A person is standing straight on the ground, looking up at an airplane which is taking off. His eyes is ${5\\ {\\rm ft}}$ from the ground. Horizontally, the person is ${131\\ {\\rm ft}}$ away from the airplane. The angle of elevation from his eyes to the airplane is $40$ degrees. Find the height of the airplane. Round your answer to two decimal places if needed. The height of the airplane is [ANS]ft.", "answer_v3": [ "114.922" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0107", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "algebra", "trigonometry", "The law of cosines" ], "problem_v1": "A steep mountain is inclined 74 degree to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 950 ft from the base to the top of the mountain. Find the shortest length of cable needed. Your answer is [ANS] ft;", "answer_v1": [ "3907.09253843542" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A steep mountain is inclined 74 degree to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 810 ft from the base to the top of the mountain. Find the shortest length of cable needed. Your answer is [ANS] ft;", "answer_v2": [ "3840.05084558204" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A steep mountain is inclined 74 degree to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 860 ft from the base to the top of the mountain. Find the shortest length of cable needed. Your answer is [ANS] ft;", "answer_v3": [ "3863.5455125847" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0108", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "trigonometry", "calculus" ], "problem_v1": "A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be $23 ^\\circ$ and $26 ^\\circ$. How high (in feet) is the ballon? [ANS]", "answer_v1": [ "17280.3976316522" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be $15 ^\\circ$ and $19 ^\\circ$. How high (in feet) is the ballon? [ANS]", "answer_v2": [ "6378.0445997577" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be $18 ^\\circ$ and $21 ^\\circ$. How high (in feet) is the ballon? [ANS]", "answer_v3": [ "11172.3693396369" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0109", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "trigonometry" ], "problem_v1": "A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is $30 ^\\circ$. From a point 1500 feet closer to the mountain along the plain, they find that the angle of elevation is $34 ^\\circ$. How high (in feet) is the mountain? [ANS]", "answer_v1": [ "6012.26889910529" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is $24 ^\\circ$. From a point 1000 feet closer to the mountain along the plain, they find that the angle of elevation is $29 ^\\circ$. How high (in feet) is the mountain? [ANS]", "answer_v2": [ "2262.49965008929" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is $26 ^\\circ$. From a point 1000 feet closer to the mountain along the plain, they find that the angle of elevation is $30 ^\\circ$. How high (in feet) is the mountain? [ANS]", "answer_v3": [ "3142.15386228576" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0110", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [], "problem_v1": "The captain of a ship at sea sights a lighthouse which is $220$ feet tall. The captain measures the angle of elevation to the top of the lighthouse to be $23 ^\\circ$. How far is the ship from the base of the lighthouse? [ANS] feet", "answer_v1": [ "518.28752114221" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The captain of a ship at sea sights a lighthouse which is $300$ feet tall. The captain measures the angle of elevation to the top of the lighthouse to be $15 ^\\circ$. How far is the ship from the base of the lighthouse? [ANS] feet", "answer_v2": [ "1119.61524361039" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The captain of a ship at sea sights a lighthouse which is $220$ feet tall. The captain measures the angle of elevation to the top of the lighthouse to be $18 ^\\circ$. How far is the ship from the base of the lighthouse? [ANS] feet", "answer_v3": [ "677.090379005597" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0111", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "4", "keywords": [ "Line", "Slope", "Rise", "Run" ], "problem_v1": "If the distance from the town of Bree to Weathertop is 16 miles on a 45 degree upward slope, what is the elevation gain (omit units)? [ANS]", "answer_v1": [ "11.3137084989848" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If the distance from the town of Bree to Weathertop is 3 miles on a 45 degree upward slope, what is the elevation gain (omit units)? [ANS]", "answer_v2": [ "2.12132034355964" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If the distance from the town of Bree to Weathertop is 7 miles on a 45 degree upward slope, what is the elevation gain (omit units)? [ANS]", "answer_v3": [ "4.94974746830583" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0112", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "3", "keywords": [ "algebra" ], "problem_v1": "You are on a pleasure cruise through the universe and you crash in the ocean of an unknown planet. Your spaceship floats on the water and its top is 32 feet above the surface of the water. You swim away from the spaceship until you see its top on the horizon. Your laser range meter tells you that your eyes are 2.8 miles away from the top of the space ship. (You are a capable--if reckless and curious--swimmer.) The radius of the planet is [ANS] miles.", "answer_v1": [ "646.79696969697" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You are on a pleasure cruise through the universe and you crash in the ocean of an unknown planet. Your spaceship floats on the water and its top is 39 feet above the surface of the water. You swim away from the spaceship until you see its top on the horizon. Your laser range meter tells you that your eyes are 2 miles away from the top of the space ship. (You are a capable--if reckless and curious--swimmer.) The radius of the planet is [ANS] miles.", "answer_v2": [ "270.765537587413" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You are on a pleasure cruise through the universe and you crash in the ocean of an unknown planet. Your spaceship floats on the water and its top is 32 feet above the surface of the water. You swim away from the spaceship until you see its top on the horizon. Your laser range meter tells you that your eyes are 2.3 miles away from the top of the space ship. (You are a capable--if reckless and curious--swimmer.) The radius of the planet is [ANS] miles.", "answer_v3": [ "436.42196969697" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0113", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "algebra" ], "problem_v1": "You are about to land your Cessna airplane in Salt Lake City. You are approaching the runway at a ground speed of 76 miles per hour and you are sinking at 350 feet per minute. (The ground speed is the speed of the point on the ground directly underneath your plane. You can also think of it as the horizontal component of your current velocity.) You are going to hit the runway at an angle of [ANS] degrees. (Enter your answer as a mathematical expression, or with at least three digits beyond the decimal point.) Hint: Draw a triangle showing your descent and the horizontal distance you cover in one minute.", "answer_v1": [ "2.99570066237461" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You are about to land your Cessna airplane in Salt Lake City. You are approaching the runway at a ground speed of 80 miles per hour and you are sinking at 210 feet per minute. (The ground speed is the speed of the point on the ground directly underneath your plane. You can also think of it as the horizontal component of your current velocity.) You are going to hit the runway at an angle of [ANS] degrees. (Enter your answer as a mathematical expression, or with at least three digits beyond the decimal point.) Hint: Draw a triangle showing your descent and the horizontal distance you cover in one minute.", "answer_v2": [ "1.70860040763434" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You are about to land your Cessna airplane in Salt Lake City. You are approaching the runway at a ground speed of 76 miles per hour and you are sinking at 260 feet per minute. (The ground speed is the speed of the point on the ground directly underneath your plane. You can also think of it as the horizontal component of your current velocity.) You are going to hit the runway at an angle of [ANS] degrees. (Enter your answer as a mathematical expression, or with at least three digits beyond the decimal point.) Hint: Draw a triangle showing your descent and the horizontal distance you cover in one minute.", "answer_v3": [ "2.22628661003906" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0114", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "bearings", "directions" ], "problem_v1": "Directions (or bearings) on Earth are measured in degrees, running from zero to $360^\\circ$, clockwise, starting with $0^\\circ$ being due North. So due East for example, is $90^\\circ$, due South $180^\\circ$, and Northwest is $315^\\circ$. You are swinging a rock clockwise (looking from above) around your head and you are trying to hit a broom stick $18$ feet due east of you. The rock moves in a circle of a radius of $3$ feet around your head. When you release your sling the rock will continue to move along the tangent to the circle though its position at the time of the release. When you release the rock the sling is pointing in a direction of [ANS] degrees. Ignore the vertical motion of the rock.\nIt's unrealistic, but remember that unless otherwise stated WeBWorK expects your answer to be within one tenth of one percent of the true answer.\nHint: You can solve this problem using calculus and computing the tangent to a circle. However, you can also solve it using plain trigonometry. The moral is that you want to use whatever requires the least amount of fuss for the problem at hand.", "answer_v1": [ "9.59407" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Directions (or bearings) on Earth are measured in degrees, running from zero to $360^\\circ$, clockwise, starting with $0^\\circ$ being due North. So due East for example, is $90^\\circ$, due South $180^\\circ$, and Northwest is $315^\\circ$. You are swinging a rock clockwise (looking from above) around your head and you are trying to hit a broom stick $10$ feet due east of you. The rock moves in a circle of a radius of $4$ feet around your head. When you release your sling the rock will continue to move along the tangent to the circle though its position at the time of the release. When you release the rock the sling is pointing in a direction of [ANS] degrees. Ignore the vertical motion of the rock.\nIt's unrealistic, but remember that unless otherwise stated WeBWorK expects your answer to be within one tenth of one percent of the true answer.\nHint: You can solve this problem using calculus and computing the tangent to a circle. However, you can also solve it using plain trigonometry. The moral is that you want to use whatever requires the least amount of fuss for the problem at hand.", "answer_v2": [ "23.5782" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Directions (or bearings) on Earth are measured in degrees, running from zero to $360^\\circ$, clockwise, starting with $0^\\circ$ being due North. So due East for example, is $90^\\circ$, due South $180^\\circ$, and Northwest is $315^\\circ$. You are swinging a rock clockwise (looking from above) around your head and you are trying to hit a broom stick $13$ feet due east of you. The rock moves in a circle of a radius of $3.5$ feet around your head. When you release your sling the rock will continue to move along the tangent to the circle though its position at the time of the release. When you release the rock the sling is pointing in a direction of [ANS] degrees. Ignore the vertical motion of the rock.\nIt's unrealistic, but remember that unless otherwise stated WeBWorK expects your answer to be within one tenth of one percent of the true answer.\nHint: You can solve this problem using calculus and computing the tangent to a circle. However, you can also solve it using plain trigonometry. The moral is that you want to use whatever requires the least amount of fuss for the problem at hand.", "answer_v3": [ "15.6185" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0115", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "trigonometry", "chord", "word problem" ], "problem_v1": "Suppose an arc is intercepted by a central angle of $\\theta$ radians in a circle of radius r. The chord associated with that arc is the straight line segment joining the endpoints of the arc. The length of the chord, c, is given by $c=r\\sqrt{2-2 \\cos(\\theta)}$ A manufacturer of stop signs packs the signs into a circular drum, so that the signs fit snugly. Determine the radius of the drum given that length of each edge (chord) of the stop signs is 14 inches. (Recall that stop signs are regular octagons, 8-sided polygons with all sides and angles congruent.) [ANS] inches", "answer_v1": [ "18.2918815082693" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose an arc is intercepted by a central angle of $\\theta$ radians in a circle of radius r. The chord associated with that arc is the straight line segment joining the endpoints of the arc. The length of the chord, c, is given by $c=r\\sqrt{2-2 \\cos(\\theta)}$ A manufacturer of stop signs packs the signs into a circular drum, so that the signs fit snugly. Determine the radius of the drum given that length of each edge (chord) of the stop signs is 8 inches. (Recall that stop signs are regular octagons, 8-sided polygons with all sides and angles congruent.) [ANS] inches", "answer_v2": [ "10.452503719011" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose an arc is intercepted by a central angle of $\\theta$ radians in a circle of radius r. The chord associated with that arc is the straight line segment joining the endpoints of the arc. The length of the chord, c, is given by $c=r\\sqrt{2-2 \\cos(\\theta)}$ A manufacturer of stop signs packs the signs into a circular drum, so that the signs fit snugly. Determine the radius of the drum given that length of each edge (chord) of the stop signs is 10 inches. (Recall that stop signs are regular octagons, 8-sided polygons with all sides and angles congruent.) [ANS] inches", "answer_v3": [ "13.0656296487638" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0116", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "right triangle", "application" ], "problem_v1": "An aerial photograph from a U-2 spy plane is taken of a building suspected of housing nuclear warheads. When the photograph is taken, the angle of elevation of the sun is $40 ^ \\circ$. By comparing the shadow cast by the building in question to the shadows of other objects of known size in the photograph, scientists determine that the shadow of the building in question is 83 feet long. How tall is the bulding? (Round your answer to two decimal places.) [ANS] feet", "answer_v1": [ "69.6452693877142" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An aerial photograph from a U-2 spy plane is taken of a building suspected of housing nuclear warheads. When the photograph is taken, the angle of elevation of the sun is $26 ^ \\circ$. By comparing the shadow cast by the building in question to the shadows of other objects of known size in the photograph, scientists determine that the shadow of the building in question is 98 feet long. How tall is the bulding? (Round your answer to two decimal places.) [ANS] feet", "answer_v2": [ "47.7977936794544" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An aerial photograph from a U-2 spy plane is taken of a building suspected of housing nuclear warheads. When the photograph is taken, the angle of elevation of the sun is $31 ^ \\circ$. By comparing the shadow cast by the building in question to the shadows of other objects of known size in the photograph, scientists determine that the shadow of the building in question is 84 feet long. How tall is the bulding? (Round your answer to two decimal places.) [ANS] feet", "answer_v3": [ "50.4722919983151" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0117", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "right triangle", "application" ], "problem_v1": "A forest ranger is watching the progress of a forest fire spreading towards her from the top of a 3753-foot mesa. In 6 minutes, the angle of depression to the leading edge of the fire changes from $11.16 ^ \\circ$ to $12.66 ^ \\circ$. How many feet does the fire advance during the 6 minutes that the ranger is observing it? (Round your answer to two decimal places.) [ANS] feet\nAt what speed (in MILES PER HOUR) is the fire spreading towards the ranger? (Round your answer to two decimal places.) [ANS] miles per hour", "answer_v1": [ "2315.97678239937", "4.38631966363517" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A forest ranger is watching the progress of a forest fire spreading towards her from the top of a 3083-foot mesa. In 4 minutes, the angle of depression to the leading edge of the fire changes from $11.87 ^ \\circ$ to $12.23 ^ \\circ$. How many feet does the fire advance during the 4 minutes that the ranger is observing it? (Round your answer to two decimal places.) [ANS] feet\nAt what speed (in MILES PER HOUR) is the fire spreading towards the ranger? (Round your answer to two decimal places.) [ANS] miles per hour", "answer_v2": [ "444.562326623823", "1.26296115518131" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A forest ranger is watching the progress of a forest fire spreading towards her from the top of a 3313-foot mesa. In 4 minutes, the angle of depression to the leading edge of the fire changes from $11.21 ^ \\circ$ to $12.35 ^ \\circ$. How many feet does the fire advance during the 4 minutes that the ranger is observing it? (Round your answer to two decimal places.) [ANS] feet\nAt what speed (in MILES PER HOUR) is the fire spreading towards the ranger? (Round your answer to two decimal places.) [ANS] miles per hour", "answer_v3": [ "1585.22233512337", "4.50347254296411" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0118", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [ "right triangle", "application" ], "problem_v1": "The official Visual Flight Rules require that the distance between the ground and the bottom of the clouds to be more than 1000 feet for a plane to fly without instrumentation. At night, cloud height can be determined by aiming a searchlight straight upward and having an observer find the angle of elevation to the point at which the light hits the clouds. Suppose the observer stands 700 feet away from the searchlight and sights the beam of light on the clouds with an angle of elevation $68 ^ \\circ$. What is the height of the cloud cover? (Round your answer to two decimal places.) [ANS] feet\nWould the Visual Flight Rules allow a plane to fly without instrumentation in these conditions? Answer 'yes' or 'no'. [ANS]", "answer_v1": [ "1732.56079739141", "YES" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "The official Visual Flight Rules require that the distance between the ground and the bottom of the clouds to be more than 1000 feet for a plane to fly without instrumentation. At night, cloud height can be determined by aiming a searchlight straight upward and having an observer find the angle of elevation to the point at which the light hits the clouds. Suppose the observer stands 400 feet away from the searchlight and sights the beam of light on the clouds with an angle of elevation $78 ^ \\circ$. What is the height of the cloud cover? (Round your answer to two decimal places.) [ANS] feet\nWould the Visual Flight Rules allow a plane to fly without instrumentation in these conditions? Answer 'yes' or 'no'. [ANS]", "answer_v2": [ "1881.85204379138", "YES" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "The official Visual Flight Rules require that the distance between the ground and the bottom of the clouds to be more than 1000 feet for a plane to fly without instrumentation. At night, cloud height can be determined by aiming a searchlight straight upward and having an observer find the angle of elevation to the point at which the light hits the clouds. Suppose the observer stands 500 feet away from the searchlight and sights the beam of light on the clouds with an angle of elevation $68 ^ \\circ$. What is the height of the cloud cover? (Round your answer to two decimal places.) [ANS] feet\nWould the Visual Flight Rules allow a plane to fly without instrumentation in these conditions? Answer 'yes' or 'no'. [ANS]", "answer_v3": [ "1237.54342670815", "YES" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0119", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [], "problem_v1": "A boat is traveling between two ports by first going S 67 $^o$ E for 51 miles. The boat then stops and turns at a 90 $^o$ angle and travels N 23 $^o$ E for 56 miles to reach the ending port. The straight distance between starting and ending ports is [ANS]\nThe bearing from the first port to the second port is [ANS] [ANS] $^o$ [ANS].\n(Express your answers using at least one decimal place.)", "answer_v1": [ "sqrt(51^2+56^2)", "N", "65.3246", "E" ], "answer_type_v1": [ "EX", "MCS", "NV", "MCS" ], "options_v1": [ [], [ "N", "S", "E", "W" ], [], [ "N", "S", "E", "W" ] ], "problem_v2": "A boat is traveling between two ports by first going S 17 $^o$ E for 58 miles. The boat then stops and turns at a 90 $^o$ angle and travels N 73 $^o$ E for 43 miles to reach the ending port. The straight distance between starting and ending ports is [ANS]\nThe bearing from the first port to the second port is [ANS] [ANS] $^o$ [ANS].\n(Express your answers using at least one decimal place.)", "answer_v2": [ "sqrt(58^2+43^2)", "S", "53.5525", "E" ], "answer_type_v2": [ "EX", "MCS", "NV", "MCS" ], "options_v2": [ [], [ "N", "S", "E", "W" ], [], [ "N", "S", "E", "W" ] ], "problem_v3": "A boat is traveling between two ports by first going S 32 $^o$ E for 52 miles. The boat then stops and turns at a 90 $^o$ angle and travels N 58 $^o$ E for 42 miles to reach the ending port. The straight distance between starting and ending ports is [ANS]\nThe bearing from the first port to the second port is [ANS] [ANS] $^o$ [ANS].\n(Express your answers using at least one decimal place.)", "answer_v3": [ "sqrt(52^2+42^2)", "S", "70.9275", "E" ], "answer_type_v3": [ "EX", "MCS", "NV", "MCS" ], "options_v3": [ [], [ "N", "S", "E", "W" ], [], [ "N", "S", "E", "W" ] ] }, { "id": "Trigonometry_0120", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of special triangles & right triangles", "level": "5", "keywords": [], "problem_v1": "Two guy wires from the top of a 700-foot radio antenna are anchored to the ground by two large concrete posts. The antenna is perpendicular to the ground and between the two posts. If the angles of depression from the top of the antenna to each of the posts is $46^o$ and $48^o$ respectively, then the distance between the two posts is [ANS]. (Compute your answers to at least three decimal places.)", "answer_v1": [ "675.982+630.283" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two guy wires from the top of a 200-foot radio antenna are anchored to the ground by two large concrete posts. The antenna is perpendicular to the ground and between the two posts. If the angles of depression from the top of the antenna to each of the posts is $50^o$ and $46^o$ respectively, then the distance between the two posts is [ANS]. (Compute your answers to at least three decimal places.)", "answer_v2": [ "167.82+193.138" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two guy wires from the top of a 400-foot radio antenna are anchored to the ground by two large concrete posts. The antenna is perpendicular to the ground and between the two posts. If the angles of depression from the top of the antenna to each of the posts is $46^o$ and $42^o$ respectively, then the distance between the two posts is [ANS]. (Compute your answers to at least three decimal places.)", "answer_v3": [ "386.276+444.245" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0121", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of law of sines & law of cosines", "level": "5", "keywords": [ "trigonometry", "law of cosines" ], "problem_v1": "Two ships leave a harbor at the same time, traveling on courses that have an angle of $130 ^\\circ$ between them. If the first ship travels at $28$ miles per hour and the second ship travels at $37$ miles per hour, how far apart are the two ships after $2.6$ hours?\ndistance=[ANS]", "answer_v1": [ "153.484937503751" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Two ships leave a harbor at the same time, traveling on courses that have an angle of $110 ^\\circ$ between them. If the first ship travels at $20$ miles per hour and the second ship travels at $43$ miles per hour, how far apart are the two ships after $1.5$ hours?\ndistance=[ANS]", "answer_v2": [ "79.8991110507024" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Two ships leave a harbor at the same time, traveling on courses that have an angle of $120 ^\\circ$ between them. If the first ship travels at $22$ miles per hour and the second ship travels at $37$ miles per hour, how far apart are the two ships after $1.8$ hours?\ndistance=[ANS]", "answer_v3": [ "92.9574095436603" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0122", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of law of sines & law of cosines", "level": "5", "keywords": [ "algebra", "trigonometry", "The law of cosines" ], "problem_v1": "A pilot flies in a straight path for 1 hour and 30 min. She then makes a course correction, heading 10 degrees to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 675 miles per hour, how far is she from her starting position?\nAnswer: [ANS] miles", "answer_v1": [ "sqrt(1012.5^2+(2*675)^2-2*1012.5*2*675*cos(pi*170/180))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A pilot flies in a straight path for 1 hour and 30 min. She then makes a course correction, heading 10 degrees to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 605 miles per hour, how far is she from her starting position?\nAnswer: [ANS] miles", "answer_v2": [ "sqrt(907.5^2+(2*605)^2-2*907.5*2*605*cos(pi*170/180))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A pilot flies in a straight path for 1 hour and 30 min. She then makes a course correction, heading 10 degrees to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 630 miles per hour, how far is she from her starting position?\nAnswer: [ANS] miles", "answer_v3": [ "sqrt(945^2+(2*630)^2-2*945*2*630*cos(pi*170/180))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0123", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of law of sines & law of cosines", "level": "5", "keywords": [ "trigonometry", "law of sines" ], "problem_v1": "A ship is sailing due north. At a certain point the bearing of a lighthouse is N $43.4 ^\\circ$ E and the distance is 15.5. After a while, the captain notices that the bearing of the lighthouse is now S $52.5 ^\\circ$ E. How far did the ship travel between the two observations of the lighthouse.\ndistance=[ANS]", "answer_v1": [ "19.4338296595196" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A ship is sailing due north. At a certain point the bearing of a lighthouse is N $48.8 ^\\circ$ E and the distance is 10.5. After a while, the captain notices that the bearing of the lighthouse is now S $43.2 ^\\circ$ E. How far did the ship travel between the two observations of the lighthouse.\ndistance=[ANS]", "answer_v2": [ "15.3292645630993" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A ship is sailing due north. At a certain point the bearing of a lighthouse is N $43.8 ^\\circ$ E and the distance is 12.5. After a while, the captain notices that the bearing of the lighthouse is now S $45.9 ^\\circ$ E. How far did the ship travel between the two observations of the lighthouse.\ndistance=[ANS]", "answer_v3": [ "17.4061703397207" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0124", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Applications of law of sines & law of cosines", "level": "5", "keywords": [ "application" ], "problem_v1": "Frank and Marie set sail from the same point. Frank is sailing in the direction $S 7^\\circ E$. Marie is sailing in the direction $S 13^\\circ W$. After 8 hours, Marie was 13 miles due west of Frank. How far had Marie sailed? Round your answer to four decimal places. Distance Marie sailed=[ANS] miles.", "answer_v1": [ "37.7261404719399" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Frank and Marie set sail from the same point. Frank is sailing in the direction $S 2^\\circ E$. Marie is sailing in the direction $S 15^\\circ W$. After 2 hours, Marie was 7 miles due west of Frank. How far had Marie sailed? Round your answer to four decimal places. Distance Marie sailed=[ANS] miles.", "answer_v2": [ "23.927540442971" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Frank and Marie set sail from the same point. Frank is sailing in the direction $S 4^\\circ E$. Marie is sailing in the direction $S 13^\\circ W$. After 4 hours, Marie was 10 miles due west of Frank. How far had Marie sailed? Round your answer to four decimal places. Distance Marie sailed=[ANS] miles.", "answer_v3": [ "34.1197193211921" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0125", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Area of a triangle", "level": "2", "keywords": [ "algebra", "trigonometric functions of angles", "area of a triangle" ], "problem_v1": "Find the area of a triangle with sides of length 13 and 26 and included angle 62 degrees. Your answer is [ANS] ;", "answer_v1": [ "149.218143193159" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of a triangle with sides of length 18 and 21 and included angle 11 degrees. Your answer is [ANS] ;", "answer_v2": [ "36.062900126167" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of a triangle with sides of length 14 and 23 and included angle 28 degrees. Your answer is [ANS] ;", "answer_v3": [ "75.5849216085284" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0126", "subject": "Trigonometry", "topic": "Triangle trigonometry", "subtopic": "Area of a triangle", "level": "3", "keywords": [ "trigonometry", "law of sines", "law of cosines" ], "problem_v1": "A triangular parcel of land has sides of lengths 780 feet, 620 feet and 906 feet. a) What is the area of the parcel of land? Area=[ANS]\nb) If land is valued at 2200 per acre (1 acre is 43,560 feet $^2$), what is the value of the parcel of land? value=[ANS]", "answer_v1": [ "237947.502443291", "12017.550628449" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A triangular parcel of land has sides of lengths 170 feet, 940 feet and 891 feet. a) What is the area of the parcel of land? Area=[ANS]\nb) If land is valued at 1800 per acre (1 acre is 43,560 feet $^2$), what is the value of the parcel of land? value=[ANS]", "answer_v2": [ "74193.0306864297", "3065.82771431528" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A triangular parcel of land has sides of lengths 380 feet, 650 feet and 526 feet. a) What is the area of the parcel of land? Area=[ANS]\nb) If land is valued at 2100 per acre (1 acre is 43,560 feet $^2$), what is the value of the parcel of land? value=[ANS]", "answer_v3": [ "99939.3659375524", "4818.01350938613" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0127", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "3", "keywords": [ "trigonometry", "half-angle", "precalculus", "identities" ], "problem_v1": "Use half angle formulas to fill in the blanks in the identity below: $(\\sin(7x))^4$=[ANS] $-\\frac{1}{2}\\cos($ [ANS] $x)+\\frac{1}{8}\\cos ($ [ANS] $x)$", "answer_v1": [ "0.375", "14", "28" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use half angle formulas to fill in the blanks in the identity below: $(\\sin(2x))^4$=[ANS] $-\\frac{1}{2}\\cos($ [ANS] $x)+\\frac{1}{8}\\cos ($ [ANS] $x)$", "answer_v2": [ "0.375", "4", "8" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use half angle formulas to fill in the blanks in the identity below: $(\\sin(4x))^4$=[ANS] $-\\frac{1}{2}\\cos($ [ANS] $x)+\\frac{1}{8}\\cos ($ [ANS] $x)$", "answer_v3": [ "0.375", "8", "16" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0128", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "4", "keywords": [ "algebra", "analytic trigonometry", "half-angle" ], "problem_v1": "Using a double-angle or half-angle formula to simplify the given expressions.\n(a) If $\\cos^2 36^\\circ-\\sin^2 36^\\circ=\\cos A^\\circ,$ then $A=$ [ANS] degrees; (b) If $\\cos^2 6x-\\sin^2 6x=\\cos B,$ then $B=$ [ANS].", "answer_v1": [ "72", "2*6*x" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Using a double-angle or half-angle formula to simplify the given expressions.\n(a) If $\\cos^2 22^\\circ-\\sin^2 22^\\circ=\\cos A^\\circ,$ then $A=$ [ANS] degrees; (b) If $\\cos^2 9x-\\sin^2 9x=\\cos B,$ then $B=$ [ANS].", "answer_v2": [ "44", "2*9*x" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Using a double-angle or half-angle formula to simplify the given expressions.\n(a) If $\\cos^2 27^\\circ-\\sin^2 27^\\circ=\\cos A^\\circ,$ then $A=$ [ANS] degrees; (b) If $\\cos^2 6x-\\sin^2 6x=\\cos B,$ then $B=$ [ANS].", "answer_v3": [ "54", "2*6*x" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0129", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "3", "keywords": [ "trigonometry", "inverse trigonometric functions", "trig functions", "inverse trig functions" ], "problem_v1": "Simplify the expression \\tan\\left(2\\cos^{-1}(x/6)\\right) answer=", "answer_v1": [ "2*x*sqrt(6^2-x^2)/(2*x^2-6^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Simplify the expression \\tan\\left(2\\cos^{-1}(x/3)\\right) answer=", "answer_v2": [ "2*x*sqrt(3^2-x^2)/(2*x^2-3^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Simplify the expression \\tan\\left(2\\cos^{-1}(x/4)\\right) answer=", "answer_v3": [ "2*x*sqrt(4^2-x^2)/(2*x^2-4^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0130", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "2", "keywords": [ "precalculus", "trigonometry", "identities", "algebra", "analytic trigonometry", "formulas for lowering powers" ], "problem_v1": "Use the formula for lowering the powers to simplify the expression:\n(a) If $\\cos^2 8x-\\sin^2 8x=\\cos (f(x))$, then $f(x)=$ [ANS] ; (b) If $\\cos^4 8x-\\sin^4 8x=\\cos (g(x))$, then $g(x)=$ [ANS]. Hint: For part (b), use $a^4-b^4=(a^2-b^2)(a^2+b^2)$!", "answer_v1": [ "2*8*x", "2*8*x" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Use the formula for lowering the powers to simplify the expression:\n(a) If $\\cos^2 2x-\\sin^2 2x=\\cos (f(x))$, then $f(x)=$ [ANS] ; (b) If $\\cos^4 2x-\\sin^4 2x=\\cos (g(x))$, then $g(x)=$ [ANS]. Hint: For part (b), use $a^4-b^4=(a^2-b^2)(a^2+b^2)$!", "answer_v2": [ "2*2*x", "2*2*x" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Use the formula for lowering the powers to simplify the expression:\n(a) If $\\cos^2 4x-\\sin^2 4x=\\cos (f(x))$, then $f(x)=$ [ANS] ; (b) If $\\cos^4 4x-\\sin^4 4x=\\cos (g(x))$, then $g(x)=$ [ANS]. Hint: For part (b), use $a^4-b^4=(a^2-b^2)(a^2+b^2)$!", "answer_v3": [ "2*4*x", "2*4*x" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0131", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "3", "keywords": [ "trigonometry", "quadrants", "identity", "half-angle" ], "problem_v1": "Given $\\tan(\\alpha)=\\frac{7}{\\sqrt{32}}$ and $\\pi<\\alpha<3\\pi/2$, find the exact value of $\\hbox{} \\tan(\\alpha/2)$. Note: You are not allowed to use decimals in your answer. $\\tan(\\alpha/2)$=[ANS].", "answer_v1": [ "(1--1*[sqrt(32)]/9)/(-1*7/9)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given $\\tan(\\alpha)=\\frac{2}{\\sqrt{77}}$ and $\\alpha$ is in quadrant I, find the exact value of $\\hbox{} \\tan(\\alpha/2)$. Note: You are not allowed to use decimals in your answer. $\\tan(\\alpha/2)$=[ANS].", "answer_v2": [ "(1-1*[sqrt(77)]/9)/(1*2/9)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given $\\tan(\\alpha)=\\frac{3}{\\sqrt{40}}$ and $\\alpha$ is in quadrant I, find the exact value of $\\hbox{} \\tan(\\alpha/2)$. Note: You are not allowed to use decimals in your answer. $\\tan(\\alpha/2)$=[ANS].", "answer_v3": [ "(1-1*[sqrt(40)]/7)/(1*3/7)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0132", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "2", "keywords": [ "trigonometry", "identity", "sum" ], "problem_v1": "Use an identity to find the exact value of the expression: Note: You are not allowed to use decimals in your answer. ${\\sin\\left(\\frac{9\\pi}{8}\\right))}$=[ANS]", "answer_v1": [ "-0.38268343236509" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use an identity to find the exact value of the expression: Note: You are not allowed to use decimals in your answer. ${\\sin\\left(\\frac{\\pi}{8}\\right))}$=[ANS]", "answer_v2": [ "0.38268343236509" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use an identity to find the exact value of the expression: Note: You are not allowed to use decimals in your answer. ${\\sin\\left(\\frac{3\\pi}{8}\\right))}$=[ANS]", "answer_v3": [ "0.923879532511287" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0133", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Double-angle & half-angle formulas", "level": "3", "keywords": [ "trigonometry", "identity", "double angle" ], "problem_v1": "If $\\sin(A)=0.55$ and $\\cos(A) < 0$, determine:\n$\\cos(A)=$ [ANS]\n$\\sin(2A)=$ [ANS]\n$\\cos(2A)=$ [ANS]\n$\\tan(2A)=$ [ANS]\nThe quadrant for $2A$ is [ANS]. Be certain to express all answers to at least four decimal places. Be certain to express all answers to at least four decimal places.", "answer_v1": [ "-0.835165", "-0.918681", "0.395", "-2.32577", "IV" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [ "I", "II", "III", "IV" ] ], "problem_v2": "If $\\sin(A)=0.23$ and $\\cos(A) < 0$, determine:\n$\\cos(A)=$ [ANS]\n$\\sin(2A)=$ [ANS]\n$\\cos(2A)=$ [ANS]\n$\\tan(2A)=$ [ANS]\nThe quadrant for $2A$ is [ANS]. Be certain to express all answers to at least four decimal places. Be certain to express all answers to at least four decimal places.", "answer_v2": [ "-0.973191", "-0.447668", "0.8942", "-0.500635", "IV" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [ "I", "II", "III", "IV" ] ], "problem_v3": "If $\\sin(A)=0.35$ and $\\cos(A) < 0$, determine:\n$\\cos(A)=$ [ANS]\n$\\sin(2A)=$ [ANS]\n$\\cos(2A)=$ [ANS]\n$\\tan(2A)=$ [ANS]\nThe quadrant for $2A$ is [ANS]. Be certain to express all answers to at least four decimal places. Be certain to express all answers to at least four decimal places.", "answer_v3": [ "-0.93675", "-0.655725", "0.755", "-0.86851", "IV" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [ "I", "II", "III", "IV" ] ] }, { "id": "Trigonometry_0134", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Addition & subtraction formulas", "level": "4", "keywords": [ "trigonometry" ], "problem_v1": "Here is a little more review concerning trig functions. Using the formula for sin() and cos() of the sum of two angles. $2\\cos(6x+1)=$ [ANS] $\\cos(6x)$-[ANS] $\\sin(6x)$ $2\\sin(5x+2)=$ [ANS] $\\cos(5x)$+[ANS] $\\sin(5x)$ Now reverse this formula and given the expanded version find the version with just one term. This involves solving a pair of equations--in order to get $A\\cos(x)+B\\sin(x)=R\\sin(x+b)=R\\sin(b)\\cos(x)+R\\cos(b)\\sin(x)$ what values must you choose for $R$ and $b$? (Match coefficients.)\nBy convention we'll assume that the amplitude (the first coefficient on the left hand side) is positive. [ANS] $\\cos(6x+$ [ANS] $)=$ $3\\cos(6x)$+$1 \\sin(6x)$ [ANS] $\\sin(5x+$ [ANS] $)=$ $4\\cos(5x)$+$1 \\sin(5x)$ The upshot of this exercise is that we can always rewrite the sum of multiples of sin() and cos() as a single sin() function with a given amplitude and phase shift. We could also write it as a single cos(), but it would have a different phase in that case. We'll use this many times in interpreting results.", "answer_v1": [ "1.08060461173628", "1.68294196961579", "1.81859485365136", "-0.832293673094285", "3.16227766016838", "-0.321750554396642", "4.12310562561766", "1.32581766366803" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Here is a little more review concerning trig functions. Using the formula for sin() and cos() of the sum of two angles. $4\\cos(2x+4)=$ [ANS] $\\cos(2x)$-[ANS] $\\sin(2x)$ $4\\sin(2x-1)=$ [ANS] $\\cos(2x)$+[ANS] $\\sin(2x)$ Now reverse this formula and given the expanded version find the version with just one term. This involves solving a pair of equations--in order to get $A\\cos(x)+B\\sin(x)=R\\sin(x+b)=R\\sin(b)\\cos(x)+R\\cos(b)\\sin(x)$ what values must you choose for $R$ and $b$? (Match coefficients.)\nBy convention we'll assume that the amplitude (the first coefficient on the left hand side) is positive. [ANS] $\\cos(2x+$ [ANS] $)=$ $3\\cos(2x)$+$-2 \\sin(2x)$ [ANS] $\\sin(2x+$ [ANS] $)=$ $3\\cos(2x)$+$-3 \\sin(2x)$ The upshot of this exercise is that we can always rewrite the sum of multiples of sin() and cos() as a single sin() function with a given amplitude and phase shift. We could also write it as a single cos(), but it would have a different phase in that case. We'll use this many times in interpreting results.", "answer_v2": [ "-2.61457448345445", "-3.02720998123171", "-3.36588393923159", "2.16120922347256", "3.60555127546399", "0.588002603547568", "4.24264068711928", "2.35619449019234" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Here is a little more review concerning trig functions. Using the formula for sin() and cos() of the sum of two angles. $3\\cos(3x+1)=$ [ANS] $\\cos(3x)$-[ANS] $\\sin(3x)$ $3\\sin(3x-3)=$ [ANS] $\\cos(3x)$+[ANS] $\\sin(3x)$ Now reverse this formula and given the expanded version find the version with just one term. This involves solving a pair of equations--in order to get $A\\cos(x)+B\\sin(x)=R\\sin(x+b)=R\\sin(b)\\cos(x)+R\\cos(b)\\sin(x)$ what values must you choose for $R$ and $b$? (Match coefficients.)\nBy convention we'll assume that the amplitude (the first coefficient on the left hand side) is positive. [ANS] $\\cos(3x+$ [ANS] $)=$ $6\\cos(3x)$+$3 \\sin(3x)$ [ANS] $\\sin(3x+$ [ANS] $)=$ $7\\cos(3x)$+$-2 \\sin(3x)$ The upshot of this exercise is that we can always rewrite the sum of multiples of sin() and cos() as a single sin() function with a given amplitude and phase shift. We could also write it as a single cos(), but it would have a different phase in that case. We'll use this many times in interpreting results.", "answer_v3": [ "1.62090691760442", "2.52441295442369", "-0.423360024179602", "-2.96997748980134", "6.70820393249937", "-0.463647609000806", "7.28010988928052", "1.84909598580001" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Trigonometry_0135", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Addition & subtraction formulas", "level": "4", "keywords": [ "calculus" ], "problem_v1": "a) Find $\\tan\\left(\\sin^{-1}(\\frac {4}{7})+\\cos^{-1}(\\frac {4}{8})\\right)$=[ANS]. (Make sure your answer is an algebraic expression with square roots but without trigonometric or inverse trignometric functions.) b) Express in terms of $x$: $\\sin\\left(2 \\tan^{-1}(x)\\right)$=[ANS].", "answer_v1": [ "-11.7856", "2*x/(1+x^2)" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "a) Find $\\tan\\left(\\sin^{-1}(\\frac {1}{6})+\\cos^{-1}(\\frac {1}{3})\\right)$=[ANS]. (Make sure your answer is an algebraic expression with square roots but without trigonometric or inverse trignometric functions.) b) Express in terms of $x$: $\\sin\\left(2 \\tan^{-1}(x)\\right)$=[ANS].", "answer_v2": [ "5.74326", "2*x/(1+x^2)" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "a) Find $\\tan\\left(\\sin^{-1}(\\frac {2}{6})+\\cos^{-1}(\\frac {2}{5})\\right)$=[ANS]. (Make sure your answer is an algebraic expression with square roots but without trigonometric or inverse trignometric functions.) b) Express in terms of $x$: $\\sin\\left(2 \\tan^{-1}(x)\\right)$=[ANS].", "answer_v3": [ "13.927", "2*x/(1+x^2)" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0136", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Addition & subtraction formulas", "level": "3", "keywords": [ "trigonometry", "quadrants", "sum", "Pythagorean", "identity" ], "problem_v1": "Given $\\cos(\\alpha)=-\\frac{\\sqrt{32}}{9}$ and $\\alpha$ is in quadrant II and $\\sin(\\beta)=-\\frac{5}{8}$ and $\\beta$ is in quadrant III. Use sum and difference formulas to find the following: Note: You are not allowed to use decimals in your answer. $\\cos(\\alpha-\\beta)=$ [ANS]", "answer_v1": [ "0.00454227035154714" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given $\\cos(\\alpha)=\\frac{\\sqrt{77}}{9}$ and $\\alpha$ is in quadrant I and $\\sin(\\beta)=-\\frac{8}{9}$ and $\\beta$ is in quadrant III. Use sum and difference formulas to find the following: Note: You are not allowed to use decimals in your answer. $\\cos(\\alpha-\\beta)=$ [ANS]", "answer_v2": [ "-0.644198827533964" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given $\\sin(\\alpha)=-\\frac{3}{7}$ and $\\alpha$ is in quadrant IV and $\\cos(\\beta)=-\\frac{\\sqrt{32}}{9}$ and $\\beta$ is in quadrant II. Use sum and difference formulas to find the following: Note: You are not allowed to use decimals in your answer. $\\cos(\\alpha-\\beta)=$ [ANS]", "answer_v3": [ "-0.90122361333328" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0137", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Addition & subtraction formulas", "level": "3", "keywords": [ "trigonometry", "identity", "sum" ], "problem_v1": "Use a sum or difference identity to find the exact value of each expression: Note: You are not allowed to use decimals in your answer. $\\tan(285 ^ \\circ)$=[ANS]", "answer_v1": [ "-3.73205080756888" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use a sum or difference identity to find the exact value of each expression: Note: You are not allowed to use decimals in your answer. $\\tan(255 ^ \\circ)$=[ANS]", "answer_v2": [ "3.73205080756888" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use a sum or difference identity to find the exact value of each expression: Note: You are not allowed to use decimals in your answer. $\\tan(195 ^ \\circ)$=[ANS]", "answer_v3": [ "0.267949192431123" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0138", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Addition & subtraction formulas", "level": "2", "keywords": [ "trigonometry", "identity", "sum" ], "problem_v1": "Use an identity to find the exact value of each expression: Note: You are not allowed to use decimals in your answer. $\\sin(187^\\circ) \\cos(113^\\circ)+\\cos(187^\\circ) \\sin(113^\\circ)$=[ANS]\n$\\sin(388^\\circ) \\cos(163^\\circ)-\\cos(388^\\circ) \\sin(163^\\circ)$=[ANS]", "answer_v1": [ "-0.866025403784439", "-0.707106781186548" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use an identity to find the exact value of each expression: Note: You are not allowed to use decimals in your answer. $\\sin(8^\\circ) \\cos(37^\\circ)+\\cos(8^\\circ) \\sin(37^\\circ)$=[ANS]\n$\\sin(440^\\circ) \\cos(110^\\circ)-\\cos(440^\\circ) \\sin(110^\\circ)$=[ANS]", "answer_v2": [ "0.707106781186548", "-0.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use an identity to find the exact value of each expression: Note: You are not allowed to use decimals in your answer. $\\sin(39^\\circ) \\cos(96^\\circ)+\\cos(39^\\circ) \\sin(96^\\circ)$=[ANS]\n$\\sin(348^\\circ) \\cos(123^\\circ)-\\cos(348^\\circ) \\sin(123^\\circ)$=[ANS]", "answer_v3": [ "0.707106781186548", "-0.707106781186548" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0139", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Addition & subtraction formulas", "level": "4", "keywords": [ "double-angle", "half-angle", "trigonometry", "analytic\"" ], "problem_v1": "Complete the following:\n$\\cos(163-327)=$ [ANS] $($ [ANS] $)$ $\\cdot \\cos($ [ANS] $)$ [ANS] [ANS] $($ [ANS] $)$ $\\cdot \\sin($ [ANS] $)$ Note that the given angles are in radians. Note that the given angles are in radians.", "answer_v1": [ "cos", "163", "327", "+", "sin", "163", "327" ], "answer_type_v1": [ "MCS", "NV", "NV", "MCS", "MCS", "NV", "NV" ], "options_v1": [ [ "sin", "cos", "tan" ], [], [], [ "+", "-", "*", "/" ], [ "sin", "cos", "tan" ], [], [] ], "problem_v2": "Complete the following:\n$\\cos(117-375)=$ [ANS] $($ [ANS] $)$ $\\cdot \\cos($ [ANS] $)$ [ANS] [ANS] $($ [ANS] $)$ $\\cdot \\sin($ [ANS] $)$ Note that the given angles are in radians. Note that the given angles are in radians.", "answer_v2": [ "cos", "117", "375", "+", "sin", "117", "375" ], "answer_type_v2": [ "MCS", "NV", "NV", "MCS", "MCS", "NV", "NV" ], "options_v2": [ [ "sin", "cos", "tan" ], [], [], [ "+", "-", "*", "/" ], [ "sin", "cos", "tan" ], [], [] ], "problem_v3": "Complete the following:\n$\\cos(133-329)=$ [ANS] $($ [ANS] $)$ $\\cdot \\cos($ [ANS] $)$ [ANS] [ANS] $($ [ANS] $)$ $\\cdot \\sin($ [ANS] $)$ Note that the given angles are in radians. Note that the given angles are in radians.", "answer_v3": [ "cos", "133", "329", "+", "sin", "133", "329" ], "answer_type_v3": [ "MCS", "NV", "NV", "MCS", "MCS", "NV", "NV" ], "options_v3": [ [ "sin", "cos", "tan" ], [], [], [ "+", "-", "*", "/" ], [ "sin", "cos", "tan" ], [], [] ] }, { "id": "Trigonometry_0140", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving basic identities", "level": "1", "keywords": [ "Trig", "Identity", "Identities", "Matching", "precalculus", "trigonometry", "identities", "trigonometric identities" ], "problem_v1": "For each trigonometric expression in the lefthand column, choose the expression from the righthand column that completes a fundamental identity. Enter the appropriate letter (A,B,C,D, or E) in each blank.\n$\\begin{array}{ccccccccccccccccccc}\\hline & [ANS] 1.1-\\cos^2 (x) 1-\\cos^2 (x) [ANS] 2.\\cot(x) \\cot(x) [ANS] 3.\\sec^2 (x) \\sec^2 (x) [ANS] 4.1 1 [ANS] 5.\\tan(x) \\tan(x) & [ANS] & 1. & 1-\\cos^2 (x) & [ANS] & 2. & \\cot(x) & [ANS] & 3. & \\sec^2 (x) & [ANS] & 4. & 1 & [ANS] & 5. & \\tan(x) & & A. \\sin^2 (x) B. \\tan^2 (x)+1 C. \\frac{\\sin (x)}{\\cos (x)} D. \\sin^2 (x)+\\cos^2 (x) E. \\frac{\\cos (x)}{\\sin(x)} \\\\\\hline [ANS] & 1. & 1-\\cos^2 (x) \\\\\\hline [ANS] & 2. & \\cot(x) \\\\\\hline [ANS] & 3. & \\sec^2 (x) \\\\\\hline [ANS] & 4. & 1 \\\\\\hline [ANS] & 5. & \\tan(x) \\\\\\hline\\end{array}$", "answer_v1": [ "A", "E", "B", "D", "C" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For each trigonometric expression in the lefthand column, choose the expression from the righthand column that completes a fundamental identity. Enter the appropriate letter (A,B,C,D, or E) in each blank.\n$\\begin{array}{ccccccccccccccccccc}\\hline & [ANS] 1.1 1 [ANS] 2.\\cot(x) \\cot(x) [ANS] 3.\\tan(x) \\tan(x) [ANS] 4.1-\\cos^2 (x) 1-\\cos^2 (x) [ANS] 5.\\sec^2 (x) \\sec^2 (x) & [ANS] & 1. & 1 & [ANS] & 2. & \\cot(x) & [ANS] & 3. & \\tan(x) & [ANS] & 4. & 1-\\cos^2 (x) & [ANS] & 5. & \\sec^2 (x) & & A. \\tan^2 (x)+1 B. \\frac{\\sin (x)}{\\cos (x)} C. \\sin^2 (x) D. \\frac{\\cos (x)}{\\sin(x)} E. \\sin^2 (x)+\\cos^2 (x) \\\\\\hline [ANS] & 1. & 1 \\\\\\hline [ANS] & 2. & \\cot(x) \\\\\\hline [ANS] & 3. & \\tan(x) \\\\\\hline [ANS] & 4. & 1-\\cos^2 (x) \\\\\\hline [ANS] & 5. & \\sec^2 (x) \\\\\\hline\\end{array}$", "answer_v2": [ "E", "D", "B", "C", "A" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For each trigonometric expression in the lefthand column, choose the expression from the righthand column that completes a fundamental identity. Enter the appropriate letter (A,B,C,D, or E) in each blank.\n$\\begin{array}{ccccccccccccccccccc}\\hline & [ANS] 1.\\sec^2 (x) \\sec^2 (x) [ANS] 2.\\cot(x) \\cot(x) [ANS] 3.\\tan(x) \\tan(x) [ANS] 4.1 1 [ANS] 5.1-\\cos^2 (x) 1-\\cos^2 (x) & [ANS] & 1. & \\sec^2 (x) & [ANS] & 2. & \\cot(x) & [ANS] & 3. & \\tan(x) & [ANS] & 4. & 1 & [ANS] & 5. & 1-\\cos^2 (x) & & A. \\sin^2 (x) B. \\tan^2 (x)+1 C. \\sin^2 (x)+\\cos^2 (x) D. \\frac{\\sin (x)}{\\cos (x)} E. \\frac{\\cos (x)}{\\sin(x)} \\\\\\hline [ANS] & 1. & \\sec^2 (x) \\\\\\hline [ANS] & 2. & \\cot(x) \\\\\\hline [ANS] & 3. & \\tan(x) \\\\\\hline [ANS] & 4. & 1 \\\\\\hline [ANS] & 5. & 1-\\cos^2 (x) \\\\\\hline\\end{array}$", "answer_v3": [ "B", "E", "D", "C", "A" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0141", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving basic identities", "level": "3", "keywords": [ "trigonometry", "trigonometric identities", "precalculus", "identities" ], "problem_v1": "Each expression simplifies to a constant, a single trigonometric function or a power of a single trigonometric function. Use fundamental identities to simplify each expression.\n${\\tan (x) \\cos (x)=}$ [ANS]\n${\\frac{\\sin (x) \\tan (x)}{\\cos (x)}=}$ [ANS]\n${(\\sec (x))^2-1=}$ [ANS]\n${(\\tan (x))^2+\\sin (x) \\csc (x)=}$ [ANS]\n${\\sec (x) \\cos (x)=}$ [ANS]", "answer_v1": [ "sin(x)", "[tan(x)]^2", "[tan(x)]^2", "[sec(x)]^2", "1" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Each expression simplifies to a constant, a single trigonometric function or a power of a single trigonometric function. Use fundamental identities to simplify each expression.\n${\\sec (x) \\cos (x)=}$ [ANS]\n${(\\tan (x))^2+\\sin (x) \\csc (x)=}$ [ANS]\n${(\\sec (x))^2-1=}$ [ANS]\n${\\frac{\\sin (x) \\tan (x)}{\\cos (x)}=}$ [ANS]\n${\\tan (x) \\cos (x)=}$ [ANS]", "answer_v2": [ "1", "[sec(x)]^2", "[tan(x)]^2", "[tan(x)]^2", "sin(x)" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Each expression simplifies to a constant, a single trigonometric function or a power of a single trigonometric function. Use fundamental identities to simplify each expression.\n${(\\sec (x))^2-1=}$ [ANS]\n${\\tan (x) \\cos (x)=}$ [ANS]\n${\\sec (x) \\cos (x)=}$ [ANS]\n${(\\tan (x))^2+\\sin (x) \\csc (x)=}$ [ANS]\n${\\frac{\\sin (x) \\tan (x)}{\\cos (x)}=}$ [ANS]", "answer_v3": [ "[tan(x)]^2", "sin(x)", "1", "[sec(x)]^2", "[tan(x)]^2" ], "answer_type_v3": [ "EX", "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0142", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving basic identities", "level": "3", "keywords": [ "trigonometry", "identity" ], "problem_v1": "Write each expression in terms of sines and/or cosines and then simplify. $\\frac{\\sin(x)}{\\csc(x)}-\\sec(x)\\cos(x)$=[ANS]\n$\\frac{\\sin(x)}{\\csc(x)}+\\cos^2(x)$=[ANS]\n$\\frac{\\cos(x)}{\\sec(x)}+\\cos^2(x)$=[ANS]", "answer_v1": [ "-(cos(x))^2", "1", "2(cos(x))^2" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Write each expression in terms of sines and/or cosines and then simplify. $\\frac{\\sec(x)}{\\tan(x)}$=[ANS]\n$\\frac{1}{\\sin^2(x)}-\\frac{1}{\\tan^2(x)}$=[ANS]\n$\\frac{\\cot(x)}{\\csc(x)}$=[ANS]", "answer_v2": [ "1/sin(x)", "1", "cos(x)" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Write each expression in terms of sines and/or cosines and then simplify. $\\frac{\\cot(x)}{\\csc(x)}$=[ANS]\n$\\frac{\\sin(x)}{\\csc(x)}-\\sec(x)\\cos(x)$=[ANS]\n$\\frac{\\sin(x)}{\\csc(x)}+\\cos^2(x)$=[ANS]", "answer_v3": [ "cos(x)", "-(cos(x))^2", "1" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0143", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving basic identities", "level": "2", "keywords": [ "trigonometry", "identity" ], "problem_v1": "Match each given expression with one of the expressions from below (A-L): [ANS] 1. $\\sin(-x)$ [ANS] 2. $\\cos(-x)$ [ANS] 3. $\\csc(-x)$ [ANS] 4. $\\sec(x)$ [ANS] 5. $\\sec^2(x)$ [ANS] 6. $\\tan^2(x)$\nAnswers:\nA. $1-\\sin^2(x)$ B. $1+\\tan^2(x)$ C. ${\\frac{\\sin(x)}{\\cos(x)}}$ D. $1+\\cot^2(x)$ E. ${\\frac{\\cos(x)}{\\sin(x)}}$ F. ${\\frac{1}{\\cos(x)}}$ G. $\\csc^2(x)-1$ H. $\\sec^2(x)-1$ I. ${\\frac{1}{\\sec(x)}}$ J. $-\\csc(x)$ K. $-\\sin(x)$ L. ${\\frac{1}{\\csc(x)}}$", "answer_v1": [ "K", "I", "J", "F", "B", "H" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ] ], "problem_v2": "Match each given expression with one of the expressions from below (A-L): [ANS] 1. $\\sin(x)$ [ANS] 2. $\\cos^2(x)$ [ANS] 3. $\\cot(-x)$ [ANS] 4. $\\csc^2(x)$ [ANS] 5. $\\sec^2(x)$ [ANS] 6. $\\tan^2(x)$\nAnswers:\nA. ${\\frac{1}{\\cos(x)}}$ B. $1+\\tan^2(x)$ C. $\\sec^2(x)-1$ D. $-\\tan(x)$ E. $\\csc^2(x)-1$ F. $-\\cot(x)$ G. $-\\csc(x)$ H. ${\\frac{\\cos(x)}{\\sin(x)}}$ I. $1-\\sin^2(x)$ J. $1+\\cot^2(x)$ K. ${\\frac{1}{\\csc(x)}}$ L. $1-\\cos^2(x)$", "answer_v2": [ "K", "I", "F", "J", "B", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ] ], "problem_v3": "Match each given expression with one of the expressions from below (A-L): [ANS] 1. $\\cos^2(x)$ [ANS] 2. $\\tan(x)$ [ANS] 3. $\\sec(x)$ [ANS] 4. $\\sin(-x)$ [ANS] 5. $\\tan(-x)$ [ANS] 6. $\\csc(-x)$\nAnswers:\nA. ${\\frac{1}{\\sec(x)}}$ B. $-\\csc(x)$ C. ${\\frac{1}{\\cos(x)}}$ D. ${\\frac{1}{\\sin(x)}}$ E. ${\\frac{\\sin(x)}{\\cos(x)}}$ F. $1-\\sin^2(x)$ G. $\\csc^2(x)-1$ H. $1+\\tan^2(x)$ I. $-\\tan(x)$ J. $1+\\cot^2(x)$ K. ${\\frac{1}{\\csc(x)}}$ L. $-\\sin(x)$", "answer_v3": [ "F", "E", "C", "L", "I", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ], [ "A", "B", "C", "D", "E", "F", "H", "I", "J", "K", "L" ] ] }, { "id": "Trigonometry_0144", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving basic identities", "level": "3", "keywords": [ "trigonometry", "proving trig identities", "multipart" ], "problem_v1": "Three part problem: To verify any given identity, it is often best to manipulate one side of the equation using logical steps until it is transformed into the other side of the formula. (This type of thinking is reflective of the way you might also solve any logical problem in real life. Indeed, showing that certain raw products input into a manufacturing process lead to a desired output doesn't allow for you to cancel inputs with outputs to get something like 1=1). In this multi-part problem, we will use algebra starting with one side and moving to the other to verify the identity\n$\\left(5\\sin(t)-5\\cos(t) \\right)^2-25=-50\\sin(t)\\cos(t).$ To get started, it is often best to start with the \"worst\" side...that is, the side that looks like it can really use some simplification. For this problem, the lets focus on the squared term on the left. $\\left (5\\sin(t)-5\\cos(t) \\right)^2-25=\\left (5\\sin(t)-5\\cos(t) \\right) \\cdot ($ [ANS] $)-25$", "answer_v1": [ "5*sin(t)-5*cos(t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Three part problem: To verify any given identity, it is often best to manipulate one side of the equation using logical steps until it is transformed into the other side of the formula. (This type of thinking is reflective of the way you might also solve any logical problem in real life. Indeed, showing that certain raw products input into a manufacturing process lead to a desired output doesn't allow for you to cancel inputs with outputs to get something like 1=1). In this multi-part problem, we will use algebra starting with one side and moving to the other to verify the identity\n$\\left(2\\sin(t)-2\\cos(t) \\right)^2-4=-8\\sin(t)\\cos(t).$ To get started, it is often best to start with the \"worst\" side...that is, the side that looks like it can really use some simplification. For this problem, the lets focus on the squared term on the left. $\\left (2\\sin(t)-2\\cos(t) \\right)^2-4=\\left (2\\sin(t)-2\\cos(t) \\right) \\cdot ($ [ANS] $)-4$", "answer_v2": [ "2*sin(t)-2*cos(t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Three part problem: To verify any given identity, it is often best to manipulate one side of the equation using logical steps until it is transformed into the other side of the formula. (This type of thinking is reflective of the way you might also solve any logical problem in real life. Indeed, showing that certain raw products input into a manufacturing process lead to a desired output doesn't allow for you to cancel inputs with outputs to get something like 1=1). In this multi-part problem, we will use algebra starting with one side and moving to the other to verify the identity\n$\\left(3\\sin(t)-3\\cos(t) \\right)^2-9=-18\\sin(t)\\cos(t).$ To get started, it is often best to start with the \"worst\" side...that is, the side that looks like it can really use some simplification. For this problem, the lets focus on the squared term on the left. $\\left (3\\sin(t)-3\\cos(t) \\right)^2-9=\\left (3\\sin(t)-3\\cos(t) \\right) \\cdot ($ [ANS] $)-9$", "answer_v3": [ "3*sin(t)-3*cos(t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0145", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving basic identities", "level": "2", "keywords": [ "trigonometry", "identity", "simplify" ], "problem_v1": "Complete the following fundamental identities:\n$\\begin{array}{cccc}\\hline 1. & Pythagorean Identity: 1-\\cos^2(A) &=& [ANS] \\\\\\hline 2. & Ratio Identity: \\cot(A) &=& [ANS] \\\\\\hline 3. & Pythagorean Identity: 1+\\tan^2(A) &=& [ANS] \\\\\\hline 4. & Reciprocal Identity: \\cot(A) &=& [ANS] \\\\\\hline 5. & Reciprocal Identity: \\csc(A) &=& [ANS] \\\\\\hline\\end{array}$", "answer_v1": [ "[sin(A)]^2", "[cos(A)]/[sin(A)]", "[sec(A)]^2", "1/[tan(A)]", "1/[sin(A)]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Complete the following fundamental identities:\n$\\begin{array}{cccc}\\hline 1. & Reciprocal Identity: \\cot(A) &=& [ANS] \\\\\\hline 2. & Ratio Identity: \\cot(A) &=& [ANS] \\\\\\hline 3. & Reciprocal Identity: \\csc(A) &=& [ANS] \\\\\\hline 4. & Pythagorean Identity: 1-\\cos^2(A) &=& [ANS] \\\\\\hline 5. & Pythagorean Identity: 1+\\tan^2(A) &=& [ANS] \\\\\\hline\\end{array}$", "answer_v2": [ "1/[tan(A)]", "[cos(A)]/[sin(A)]", "1/[sin(A)]", "[sin(A)]^2", "[sec(A)]^2" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Complete the following fundamental identities:\n$\\begin{array}{cccc}\\hline 1. & Pythagorean Identity: 1+\\tan^2(A) &=& [ANS] \\\\\\hline 2. & Ratio Identity: \\cot(A) &=& [ANS] \\\\\\hline 3. & Reciprocal Identity: \\csc(A) &=& [ANS] \\\\\\hline 4. & Reciprocal Identity: \\cot(A) &=& [ANS] \\\\\\hline 5. & Pythagorean Identity: 1-\\cos^2(A) &=& [ANS] \\\\\\hline\\end{array}$", "answer_v3": [ "[sec(A)]^2", "[cos(A)]/[sin(A)]", "1/[sin(A)]", "1/[tan(A)]", "[sin(A)]^2" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Trigonometry_0146", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving general identities", "level": "3", "keywords": [ "calculus", "trigonometric functions", "inverse functions", "trigonometric identities" ], "problem_v1": "Refer to the appropriate triangle or trigonometric identity to compute the given value. $\\cot(\\tan^{-1}(39))=$ [ANS]", "answer_v1": [ "0.025641" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Refer to the appropriate triangle or trigonometric identity to compute the given value. $\\cot(\\tan^{-1}(8))=$ [ANS]", "answer_v2": [ "0.125" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Refer to the appropriate triangle or trigonometric identity to compute the given value. $\\cot(\\tan^{-1}(19))=$ [ANS]", "answer_v3": [ "0.0526316" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0147", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving general identities", "level": "4", "keywords": [ "trig" ], "problem_v1": "If $x+4=5 \\sin(\\theta)$ for $0 < \\theta < \\pi/2$ express $\\cos(2\\theta)$ in terms of $x$. $\\cos(2\\theta)$=[ANS]", "answer_v1": [ "1-2*[(x+4)/5]^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $x+6=2 \\sin(\\theta)$ for $0 < \\theta < \\pi/2$ express $\\cos(2\\theta)$ in terms of $x$. $\\cos(2\\theta)$=[ANS]", "answer_v2": [ "1-2*[(x+6)/2]^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $x+4=3 \\sin(\\theta)$ for $0 < \\theta < \\pi/2$ express $\\cos(2\\theta)$ in terms of $x$. $\\cos(2\\theta)$=[ANS]", "answer_v3": [ "1-2*[(x+4)/3]^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0148", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving general identities", "level": "4", "keywords": [ "trigonometry", "inverse trigonometric function", "integration", "inverse trigonometric functions", "trig substitution", "Substitution", "Trig Substitution", "Trigonometric Substitution" ], "problem_v1": "Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank.\nA. $\\tan(\\arcsin(x/8))$ B. $\\cos(\\arcsin(x/8))$ C. $(1/2)\\sin(2\\arcsin(x/8))$ D. $\\sin(\\arctan(x/8))$ E. $\\cos(\\arctan(x/8))$ [ANS] 1. $ \\frac{x}{64}\\sqrt{64-x^2}$ [ANS] 2. $ \\frac{x}{\\sqrt{64+x^2}}$ [ANS] 3. $ \\frac{8}{\\sqrt{64+x^2}}$ [ANS] 4. $ \\frac{x}{\\sqrt{64-x^2}}$ [ANS] 5. $ \\frac{\\sqrt{64-x^2}}{8}$", "answer_v1": [ "C", "D", "E", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank.\nA. $\\tan(\\arcsin(x/2))$ B. $\\cos(\\arcsin(x/2))$ C. $(1/2)\\sin(2\\arcsin(x/2))$ D. $\\sin(\\arctan(x/2))$ E. $\\cos(\\arctan(x/2))$ [ANS] 1. $ \\frac{2}{\\sqrt{4+x^2}}$ [ANS] 2. $ \\frac{x}{\\sqrt{4-x^2}}$ [ANS] 3. $ \\frac{x}{4}\\sqrt{4-x^2}$ [ANS] 4. $ \\frac{x}{\\sqrt{4+x^2}}$ [ANS] 5. $ \\frac{\\sqrt{4-x^2}}{2}$", "answer_v2": [ "E", "A", "C", "D", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank.\nA. $\\tan(\\arcsin(x/4))$ B. $\\cos(\\arcsin(x/4))$ C. $(1/2)\\sin(2\\arcsin(x/4))$ D. $\\sin(\\arctan(x/4))$ E. $\\cos(\\arctan(x/4))$ [ANS] 1. $ \\frac{x}{\\sqrt{16+x^2}}$ [ANS] 2. $ \\frac{\\sqrt{16-x^2}}{4}$ [ANS] 3. $ \\frac{x}{16}\\sqrt{16-x^2}$ [ANS] 4. $ \\frac{x}{\\sqrt{16-x^2}}$ [ANS] 5. $ \\frac{4}{\\sqrt{16+x^2}}$", "answer_v3": [ "D", "B", "C", "A", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Trigonometry_0149", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving general identities", "level": "2", "keywords": [ "trigonometry", "factorization" ], "problem_v1": "Factor each trigonometric expression. $-2 \\sec^2(x)-\\sec(x)+1$=[ANS] $\\cdot$ [ANS]\n$16 \\tan^2(x)+24 \\tan(x)+9$=[ANS] $\\cdot$ [ANS]", "answer_v1": [ "-2*sec(x)+1", "sec(x)+1", "4*tan(x)+3", "4*tan(x)+3" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Factor each trigonometric expression. $\\sin^2(x)-5 \\sin(x)-24$=[ANS] $\\cdot$ [ANS]\n$9 \\tan^2(x)+6 \\tan(x)+1$=[ANS] $\\cdot$ [ANS]", "answer_v2": [ "1*sin(x)+-8", "sin(x)+3", "3*tan(x)+1", "3*tan(x)+1" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Factor each trigonometric expression. $3 \\cos^2(x)+10 \\cos(x)+7$=[ANS] $\\cdot$ [ANS]\n$49 \\sec^2(x)-84 \\sec(x)+36$=[ANS] $\\cdot$ [ANS]", "answer_v3": [ "3*cos(x)+7", "cos(x)+1", "7*sec(x)+-6", "7*sec(x)+-6" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0150", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Using and proving general identities", "level": "1", "keywords": [ "trigonometry", "identity" ], "problem_v1": "Complete the following formulas:\n$\\sin(2C)=$ [ANS]\n$\\cos(A+B)=$ [ANS]\n$\\tan(2C)=$ [ANS]\n$\\sin(A-B)=$ [ANS]\n$\\sin(A+B)=$ [ANS]\n$\\cos(A-B)=$ [ANS]\n$\\tan(x/2)=$ [ANS]\n$\\cos(2C)=$ [ANS]\nBe certain to use the specified variables for each formula. Be certain to use the specified variables for each formula.", "answer_v1": [ "2*sin(C)*cos(C)", "cos(A)*cos(B)-sin(A)*sin(B)", "2*tan(C)/(1-[tan(C)]^2)", "sin(A)*cos(B)-cos(A)*sin(B)", "sin(A)*cos(B)+cos(A)*sin(B)", "cos(A)*cos(B)+sin(A)*sin(B)", "[1-cos(x)]/[sin(x)]", "[cos(C)]^2-[sin(C)]^2" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Complete the following formulas:\n$\\tan(x/2)=$ [ANS]\n$\\sin(A-B)=$ [ANS]\n$\\sin(2C)=$ [ANS]\n$\\cos(A-B)=$ [ANS]\n$\\cos(2C)=$ [ANS]\n$\\tan(2C)=$ [ANS]\n$\\cos(A+B)=$ [ANS]\n$\\sin(A+B)=$ [ANS]\nBe certain to use the specified variables for each formula. Be certain to use the specified variables for each formula.", "answer_v2": [ "[1-cos(x)]/[sin(x)]", "sin(A)*cos(B)-cos(A)*sin(B)", "2*sin(C)*cos(C)", "cos(A)*cos(B)+sin(A)*sin(B)", "[cos(C)]^2-[sin(C)]^2", "2*tan(C)/(1-[tan(C)]^2)", "cos(A)*cos(B)-sin(A)*sin(B)", "sin(A)*cos(B)+cos(A)*sin(B)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Complete the following formulas:\n$\\cos(A+B)=$ [ANS]\n$\\sin(A-B)=$ [ANS]\n$\\tan(x/2)=$ [ANS]\n$\\tan(2C)=$ [ANS]\n$\\cos(A-B)=$ [ANS]\n$\\sin(A+B)=$ [ANS]\n$\\sin(2C)=$ [ANS]\n$\\cos(2C)=$ [ANS]\nBe certain to use the specified variables for each formula. Be certain to use the specified variables for each formula.", "answer_v3": [ "cos(A)*cos(B)-sin(A)*sin(B)", "sin(A)*cos(B)-cos(A)*sin(B)", "[1-cos(x)]/[sin(x)]", "2*tan(C)/(1-[tan(C)]^2)", "cos(A)*cos(B)+sin(A)*sin(B)", "sin(A)*cos(B)+cos(A)*sin(B)", "2*sin(C)*cos(C)", "[cos(C)]^2-[sin(C)]^2" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Trigonometry_0151", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations exactly", "level": "2", "keywords": [ "arctan", "arcsin", "arccos", "reference angle" ], "problem_v1": "Find all solutions to the equation below for $0 \\leq \\alpha \\leq 2 \\pi$. If there is more than one answer, enter your solutions in a comma-separated list, and be sure to give exact answers without any rounding.\n2 \\sin{(\\alpha)}=1 $\\alpha=$ [ANS]", "answer_v1": [ "(0.523599, 2.61799)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all solutions to the equation below for $0 \\leq \\alpha \\leq 2 \\pi$. If there is more than one answer, enter your solutions in a comma-separated list, and be sure to give exact answers without any rounding.-2 \\cos{(\\alpha)}=1 $\\alpha=$ [ANS]", "answer_v2": [ "(2.0944, 4.18879)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all solutions to the equation below for $0 \\leq \\alpha \\leq 2 \\pi$. If there is more than one answer, enter your solutions in a comma-separated list, and be sure to give exact answers without any rounding.\n2 \\cos{(\\alpha)}=1 $\\alpha=$ [ANS]", "answer_v3": [ "(1.0472, 5.23599)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0152", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations exactly", "level": "3", "keywords": [ "arctan", "arcsin", "arccos", "reference angle" ], "problem_v1": "Find all solutions to $ \\tan(\\theta)=-1$ in the interval $0 \\leq \\theta \\leq 2\\pi$. If there is more than one answer, enter your answers as a comma separated list. Your answers should be exact values (given as fractions, not decimal approximations). $\\theta$=[ANS]", "answer_v1": [ "(2.35619, 5.49779)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all solutions to $ \\tan(\\theta)=\\frac{1}{\\sqrt{3}}$ in the interval $0 \\leq \\theta \\leq 2\\pi$. If there is more than one answer, enter your answers as a comma separated list. Your answers should be exact values (given as fractions, not decimal approximations). $\\theta$=[ANS]", "answer_v2": [ "(0.523599, 3.66519)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all solutions to $ \\tan(\\theta)=1$ in the interval $0 \\leq \\theta \\leq 2\\pi$. If there is more than one answer, enter your answers as a comma separated list. Your answers should be exact values (given as fractions, not decimal approximations). $\\theta$=[ANS]", "answer_v3": [ "(0.785398, 3.92699)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0153", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations exactly", "level": "2", "keywords": [ "precalculus", "trigonometric equation", "trigonometry", "equation" ], "problem_v1": "Solve the following equations in the interval [0,2 $\\pi$]. Note: Give the answer as a multiple of $\\pi$. Do not use decimal numbers. The answer should be a fraction or an integer. Note that $\\pi$ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is $\\pi/2$ you should enter 1/2. If there is more than one answer enter them separated by commas. $\\sin(t)=\\frac{\\sqrt{3}}{2}$ t=[ANS] $\\pi$ $\\sin(t)=-\\frac{1}{2}$ t=[ANS] $\\pi$", "answer_v1": [ "(0.333333333333333, 0.666666666666667)", "(1.16666666666667, 1.83333333333333)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Solve the following equations in the interval [0,2 $\\pi$]. Note: Give the answer as a multiple of $\\pi$. Do not use decimal numbers. The answer should be a fraction or an integer. Note that $\\pi$ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is $\\pi/2$ you should enter 1/2. If there is more than one answer enter them separated by commas. $\\sin(t)=\\frac{\\sqrt{2}}{2}$ t=[ANS] $\\pi$ $\\sin(t)=-\\frac{\\sqrt{3}}{2}$ t=[ANS] $\\pi$", "answer_v2": [ "(0.25, 0.75)", "(1.33333333333333, 1.66666666666667)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Solve the following equations in the interval [0,2 $\\pi$]. Note: Give the answer as a multiple of $\\pi$. Do not use decimal numbers. The answer should be a fraction or an integer. Note that $\\pi$ is already included in the answer so you just have to enter the appropriate multiple. E.g. if the answer is $\\pi/2$ you should enter 1/2. If there is more than one answer enter them separated by commas. $\\sin(t)=-\\frac{\\sqrt{2}}{2}$ t=[ANS] $\\pi$ $\\sin(t)=-\\frac{1}{2}$ t=[ANS] $\\pi$", "answer_v3": [ "(1.25, 1.75)", "(1.16666666666667, 1.83333333333333)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0154", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations exactly", "level": "4", "keywords": [ "identity", "trigonometry", "equation" ], "problem_v1": "Find all angles $x$ (in radians) that satisfy the equation \\cos\\!\\left(x\\right)\\tan\\!\\left(x\\right)+\\sqrt{3}\\cos\\!\\left(x\\right)=0. Your answers should be formulas that look like x=\\text{angle}+k*\\text{period} where $k=\\cdots,-2,-1,0,1,2,\\cdots$ ranges over the integers, $0\\leq \\text{angle} < \\text{period}$, and period is the smallest positive \"period\" that makes the formula work. Note: decimal answers are not allowed, so, for example, if $\\pi$ appears in your answer call it pi not 3.14159. The solutions of the equation are: $x$=[ANS]\n(Warning: division by zero is not allowed!)", "answer_v1": [ "2*pi/3+k*pi" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find all angles $x$ (in radians) that satisfy the equation 3\\sin\\!\\left(x\\right)\\tan\\!\\left(x\\right)+\\sqrt{3}\\sin\\!\\left(x\\right)=0. Your answers should be formulas that look like x=\\text{angle}+k*\\text{period} where $k=\\cdots,-2,-1,0,1,2,\\cdots$ ranges over the integers, $0\\leq \\text{angle} < \\text{period}$, and period is the smallest positive \"period\" that makes the formula work. Note: decimal answers are not allowed, so, for example, if $\\pi$ appears in your answer call it pi not 3.14159. The solutions of the equation are: $x$=[ANS]\nor $x$=[ANS].", "answer_v2": [ "k*pi", "5*pi/6+k*pi" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find all angles $x$ (in radians) that satisfy the equation \\sin\\!\\left(x\\right)\\tan\\!\\left(x\\right)+\\sqrt{3}\\sin\\!\\left(x\\right)=0. Your answers should be formulas that look like x=\\text{angle}+k*\\text{period} where $k=\\cdots,-2,-1,0,1,2,\\cdots$ ranges over the integers, $0\\leq \\text{angle} < \\text{period}$, and period is the smallest positive \"period\" that makes the formula work. Note: decimal answers are not allowed, so, for example, if $\\pi$ appears in your answer call it pi not 3.14159. The solutions of the equation are: $x$=[ANS]\nor $x$=[ANS].", "answer_v3": [ "k*pi", "2*pi/3+k*pi" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0155", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "4", "keywords": [ "calculus", "trigonometry", "angles", "radians" ], "problem_v1": "Solve for $0\\leq \\theta<2\\pi$: \\cos 4 \\theta+\\cos 6 \\theta=0. Separate your answers with commas. $\\theta=$ [ANS]", "answer_v1": [ "(0.314159, 0.942478, 1.5708, 2.19911, 2.82743, 3.45575, 4.08407, 4.71239, 5.34071, 5.96903)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve for $0\\leq \\theta<2\\pi$: \\sin 2 \\theta+\\sin 5 \\theta=0. Separate your answers with commas. $\\theta=$ [ANS]", "answer_v2": [ "(0, 0.897598, 1.0472, 1.7952, 2.69279, 3.14159, 3.59039, 4.48799, 5.23599, 5.38559)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve for $0\\leq \\theta<2\\pi$: \\sin 2 \\theta+\\sin 4 \\theta=0. Separate your answers with commas. $\\theta=$ [ANS]", "answer_v3": [ "(0, 1.0472, 1.5708, 2.0944, 3.14159, 4.18879, 4.71239, 5.23599)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0156", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "3", "keywords": [ "arctan", "arcsin", "arccos", "reference angle" ], "problem_v1": "Find a solution for $\\theta$ in radians. If no solution exists, enter NONE. $ \\tan{(\\theta-4)}=0.27$ $\\theta=$ [ANS]\n(enter your answer accurate to at least four decimal places)", "answer_v1": [ "atan(0.27)+4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find a solution for $\\theta$ in radians. If no solution exists, enter NONE. $ \\tan{(\\theta-1)}=0.38$ $\\theta=$ [ANS]\n(enter your answer accurate to at least four decimal places)", "answer_v2": [ "atan(0.38)+1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find a solution for $\\theta$ in radians. If no solution exists, enter NONE. $ \\tan{(\\theta-2)}=0.28$ $\\theta=$ [ANS]\n(enter your answer accurate to at least four decimal places)", "answer_v3": [ "atan(0.28)+2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0157", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "2", "keywords": [ "arctan", "arcsin", "arccos", "reference angle" ], "problem_v1": "If possible, find a solution to $ \\sin(\\theta)=\\frac{7}{10}$. If no solution exists, enter NONE. NONE. $\\theta=$ [ANS] radians", "answer_v1": [ "asin(7/10)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If possible, find a solution to $ \\sin(\\theta)=\\frac{1}{10}$. If no solution exists, enter NONE. NONE. $\\theta=$ [ANS] radians", "answer_v2": [ "asin(1/10)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If possible, find a solution to $ \\sin(\\theta)=\\frac{3}{8}$. If no solution exists, enter NONE. NONE. $\\theta=$ [ANS] radians", "answer_v3": [ "asin(3/8)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0158", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "3", "keywords": [ "arctan", "arcsin", "arccos", "reference angle" ], "problem_v1": "Find all solutions to the equation $ 8 \\cos(x+3)=1$ in the interval $0 \\leq x \\leq 2 \\pi$. If there is more than one answer, enter your answers as a comma separated list. $x=$ [ANS]", "answer_v1": [ "(4.72865, 1.83772)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all solutions to the equation $ 3 \\cos(x+4)=1$ in the interval $0 \\leq x \\leq 2 \\pi$. If there is more than one answer, enter your answers as a comma separated list. $x=$ [ANS]", "answer_v2": [ "(3.51414, 1.05223)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all solutions to the equation $ 5 \\cos(x+3)=1$ in the interval $0 \\leq x \\leq 2 \\pi$. If there is more than one answer, enter your answers as a comma separated list. $x=$ [ANS]", "answer_v3": [ "(4.65262, 1.91375)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0159", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "3", "keywords": [ "trigonometry", "trigonometric equation", "precalculus" ], "problem_v1": "Approximate, to three decimal places, the solutions to the equation. If there is more than one solution write them separated by commas. $\\cos x=4x^2$ $x=$ [ANS]", "answer_v1": [ "(-0.4718876482, 0.4718876482)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Approximate, to three decimal places, the solutions to the equation. If there is more than one solution write them separated by commas. $\\cos x=x^2$ $x=$ [ANS]", "answer_v2": [ "(-0.8241323, 0.8241323)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Approximate, to three decimal places, the solutions to the equation. If there is more than one solution write them separated by commas. $\\cos x=\\frac{x^2}{2}$ $x=$ [ANS]", "answer_v3": [ "(-1.021689954, 1.021689954)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0160", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "2", "keywords": [ "function' 'graph' 'equation", "functions", "Newton's method" ], "problem_v1": "Find the positive value of $x$ which satisfies $x=3.8 \\cos(x)$. Give the answer to 2 decimal places. [ANS]", "answer_v1": [ "1.23874289906441" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the positive value of $x$ which satisfies $x=0.5 \\cos(x)$. Give the answer to 2 decimal places. [ANS]", "answer_v2": [ "0.450183611294874" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the positive value of $x$ which satisfies $x=1.6 \\cos(x)$. Give the answer to 2 decimal places. [ANS]", "answer_v3": [ "0.941596635990477" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0161", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "4", "keywords": [ "trigonometry", "inverse trigonometric function", "inverse trigonometric functions" ], "problem_v1": "Solve the equation in the interval $[0,\\, 2 \\pi]$. If there is more than one solution write them separated by commas.\nHint: To solve this problem you will have to use the quadratic formula, inverse trigonometric functions and the symmetry of the unit circle. (\\tan x) ^2-1 \\tan(x)-4.16=0\n$x=$ [ANS]", "answer_v1": [ "(1.20362249297668, 2.12939564213846, 4.34521514656647, 5.27098829572825)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Solve the equation in the interval $[0,\\, 2 \\pi]$. If there is more than one solution write them separated by commas.\nHint: To solve this problem you will have to use the quadratic formula, inverse trigonometric functions and the symmetry of the unit circle. (\\tan x) ^2-0.8 \\tan(x)-0.65=0\n$x=$ [ANS]", "answer_v2": [ "(0.91510070055336, 2.67794504458899, 4.05669335414315, 5.81953769817878)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Solve the equation in the interval $[0,\\, 2 \\pi]$. If there is more than one solution write them separated by commas.\nHint: To solve this problem you will have to use the quadratic formula, inverse trigonometric functions and the symmetry of the unit circle. (\\tan x) ^2-0.1 \\tan(x)-2.72=0\n$x=$ [ANS]", "answer_v3": [ "(1.03907225953609, 2.12939564213846, 4.18066491312588, 5.27098829572825)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0162", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "2", "keywords": [ "Trig", "Inverse" ], "problem_v1": "Determine all solutions for $\\tan(\\theta)=4.76$ $\\theta=$ [ANS]+[ANS] $n$ where $n$ is any integer and where the value in each first blank lies between $-\\pi/2$ and $\\pi/2$. Use radian measure and enter your answers using at least four significant digits. Use radian measure and enter your answers using at least four significant digits.", "answer_v1": [ "atan(4.76)", "pi" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine all solutions for $\\tan(\\theta)=-8.12$ $\\theta=$ [ANS]+[ANS] $n$ where $n$ is any integer and where the value in each first blank lies between $-\\pi/2$ and $\\pi/2$. Use radian measure and enter your answers using at least four significant digits. Use radian measure and enter your answers using at least four significant digits.", "answer_v2": [ "atan(-8.12)", "pi" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine all solutions for $\\tan(\\theta)=-3.52$ $\\theta=$ [ANS]+[ANS] $n$ where $n$ is any integer and where the value in each first blank lies between $-\\pi/2$ and $\\pi/2$. Use radian measure and enter your answers using at least four significant digits. Use radian measure and enter your answers using at least four significant digits.", "answer_v3": [ "atan(-3.52)", "pi" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0163", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Solving trigonometric equations numerically", "level": "3", "keywords": [ "sage", "sagelet", "trig", "solving", "equations" ], "problem_v1": "var('x')\n@interact def _(x0=slider(0,3.14/(2*1.07),0.01,0.5)):\nG=plot(sin(1.07*x),(x,0,pi/1.07)) G+=point2d((x0,sin(1.07*x0)),size=20,color='red') G+=point2d((x0+1.75/1.07,sin(1.07*x0+1.75)),size=20,color='red') G+=line([(0,sin(1.07*x0)),(pi/1.07,sin(1.07*x0))],alpha=0.1,color='red') G+=line([(0,sin(1.07*x0+1.75)),(pi/1.07,sin(1.07*x0+1.75))],alpha=0.1,color='red') show(G+text('sin(bx)',(x0,sin(1.07*x0)))+text('sin(bx+a)',(x0+1.75/1.07,sin(1.07*x0+1.75))))\n\\$(function () {sagecell.makeSagecell({inputLocation: '#sagecell', template: sagecell.templates.minimal, autoeval: true, evalButtonText: 'Start/Restart'});}); Using the interactive application estimate a solution $x$ to the equation: \\sin(1.07x)=\\sin(1.07x+1.75) $x=$ [ANS]", "answer_v1": [ "0.650277" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "var('x')\n@interact def _(x0=slider(0,3.14/(2*1.56),0.01,0.5)):\nG=plot(sin(1.56*x),(x,0,pi/1.56)) G+=point2d((x0,sin(1.56*x0)),size=20,color='red') G+=point2d((x0+0.5/1.56,sin(1.56*x0+0.5)),size=20,color='red') G+=line([(0,sin(1.56*x0)),(pi/1.56,sin(1.56*x0))],alpha=0.1,color='red') G+=line([(0,sin(1.56*x0+0.5)),(pi/1.56,sin(1.56*x0+0.5))],alpha=0.1,color='red') show(G+text('sin(bx)',(x0,sin(1.56*x0)))+text('sin(bx+a)',(x0+0.5/1.56,sin(1.56*x0+0.5))))\n\\$(function () {sagecell.makeSagecell({inputLocation: '#sagecell', template: sagecell.templates.minimal, autoeval: true, evalButtonText: 'Start/Restart'});}); Using the interactive application estimate a solution $x$ to the equation: \\sin(1.56x)=\\sin(1.56x+0.5) $x=$ [ANS]", "answer_v2": [ "0.846664" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "var('x')\n@interact def _(x0=slider(0,3.14/(2*1.14),0.01,0.5)):\nG=plot(sin(1.14*x),(x,0,pi/1.14)) G+=point2d((x0,sin(1.14*x0)),size=20,color='red') G+=point2d((x0+1/1.14,sin(1.14*x0+1)),size=20,color='red') G+=line([(0,sin(1.14*x0)),(pi/1.14,sin(1.14*x0))],alpha=0.1,color='red') G+=line([(0,sin(1.14*x0+1)),(pi/1.14,sin(1.14*x0+1))],alpha=0.1,color='red') show(G+text('sin(bx)',(x0,sin(1.14*x0)))+text('sin(bx+a)',(x0+1/1.14,sin(1.14*x0+1))))\n\\$(function () {sagecell.makeSagecell({inputLocation: '#sagecell', template: sagecell.templates.minimal, autoeval: true, evalButtonText: 'Start/Restart'});}); Using the interactive application estimate a solution $x$ to the equation: \\sin(1.14x)=\\sin(1.14x+1) $x=$ [ANS]", "answer_v3": [ "0.939295" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0164", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Product-to-sum & sum-to-product formulas", "level": "2", "keywords": [ "trig" ], "problem_v1": "Write $-8 \\sin(5 t)+7 \\cos(5 t)$ in the form $A \\sin(B t+\\phi)$ using sum or difference formulas. $-8 \\sin(5 t)+7 \\cos(5 t)$=[ANS]", "answer_v1": [ "sqrt(8^2+7^2)*sin(5*t+atan(-7/8)+pi)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write $-2 \\sin(9 t)+3 \\cos(9 t)$ in the form $A \\sin(B t+\\phi)$ using sum or difference formulas. $-2 \\sin(9 t)+3 \\cos(9 t)$=[ANS]", "answer_v2": [ "sqrt(2^2+3^2)*sin(9*t+atan(-3/2)+pi)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write $-4 \\sin(7 t)+5 \\cos(7 t)$ in the form $A \\sin(B t+\\phi)$ using sum or difference formulas. $-4 \\sin(7 t)+5 \\cos(7 t)$=[ANS]", "answer_v3": [ "sqrt(4^2+5^2)*sin(7*t+atan(-5/4)+pi)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0165", "subject": "Trigonometry", "topic": "Analytic trigonometry", "subtopic": "Product-to-sum & sum-to-product formulas", "level": "3", "keywords": [ "algebra", "analytic trigonometry", "sum-to-product" ], "problem_v1": "Use the sum-to-product formula to simplify the expression: If $\\sin 40^\\circ+\\sin 20^\\circ=\\sin A^\\circ, 0 0$ and $\\pi < \\theta < \\frac{3 \\pi}{2}.$ $r=$ [ANS]\nb.) Find cartesian coordinates for point $P.$ $x=$ [ANS]\n$y=$ [ANS]\nc.) How may times does the curve pass through the origin when $0 < \\theta < 2 \\pi?$ Answer: [ANS]", "answer_v1": [ "11 + 4*cos( (4/(2*11) +1)*pi)", "(11 + 4*cos( (4/(2*11) +1)*pi) )*cos( (4/(2*11) +1)*pi )", "(11 + 4*cos( (4/(2*11) +1)*pi) )*sin( (4/(2*11) +1)*pi )", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A curve in polar coordinates is given by: $r=7+5 \\cos \\theta.$ Point $P$ is at $\\theta=\\frac{19 \\pi}{14}.$\na.) Find polar coordinate $r$ for $P$, with $r > 0$ and $\\pi < \\theta < \\frac{3 \\pi}{2}.$ $r=$ [ANS]\nb.) Find cartesian coordinates for point $P.$ $x=$ [ANS]\n$y=$ [ANS]\nc.) How may times does the curve pass through the origin when $0 < \\theta < 2 \\pi?$ Answer: [ANS]", "answer_v2": [ "7 + 5*cos( (5/(2*7) +1)*pi)", "(7 + 5*cos( (5/(2*7) +1)*pi) )*cos( (5/(2*7) +1)*pi )", "(7 + 5*cos( (5/(2*7) +1)*pi) )*sin( (5/(2*7) +1)*pi )", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A curve in polar coordinates is given by: $r=8+4 \\cos \\theta.$ Point $P$ is at $\\theta=\\frac{20 \\pi}{16}.$\na.) Find polar coordinate $r$ for $P$, with $r > 0$ and $\\pi < \\theta < \\frac{3 \\pi}{2}.$ $r=$ [ANS]\nb.) Find cartesian coordinates for point $P.$ $x=$ [ANS]\n$y=$ [ANS]\nc.) How may times does the curve pass through the origin when $0 < \\theta < 2 \\pi?$ Answer: [ANS]", "answer_v3": [ "8 + 4*cos( (4/(2*8) +1)*pi)", "(8 + 4*cos( (4/(2*8) +1)*pi) )*cos( (4/(2*8) +1)*pi )", "(8 + 4*cos( (4/(2*8) +1)*pi) )*sin( (4/(2*8) +1)*pi )", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0170", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Polar and rectangular coordinates", "level": "4", "keywords": [ "Integration", "Arc length", "parametric", "calculus", "integral' 'polar' 'match", "polar' 'curve' 'length", "polar coordinates" ], "problem_v1": "A circle $C$ has center at the origin and radius $8$. Another circle $K$ has a diameter with one end at the origin and the other end at the point $(0,15)$. The circles $C$ and $K$ intersect in two points. Let $P$ be the point of intersection of $C$ and $K$ which lies in the first quadrant. Let $(r, \\theta)$ be the polar coordinates of $P$, chosen so that $r$ is positive and $0 \\leq \\theta \\leq 2$. Find $r$ and $\\theta$.\n$r=$ [ANS]\n$\\theta=$ [ANS]", "answer_v1": [ "8", "0.562536" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A circle $C$ has center at the origin and radius $3$. Another circle $K$ has a diameter with one end at the origin and the other end at the point $(0,19)$. The circles $C$ and $K$ intersect in two points. Let $P$ be the point of intersection of $C$ and $K$ which lies in the first quadrant. Let $(r, \\theta)$ be the polar coordinates of $P$, chosen so that $r$ is positive and $0 \\leq \\theta \\leq 2$. Find $r$ and $\\theta$.\n$r=$ [ANS]\n$\\theta=$ [ANS]", "answer_v2": [ "3", "0.158558" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A circle $C$ has center at the origin and radius $5$. Another circle $K$ has a diameter with one end at the origin and the other end at the point $(0,16)$. The circles $C$ and $K$ intersect in two points. Let $P$ be the point of intersection of $C$ and $K$ which lies in the first quadrant. Let $(r, \\theta)$ be the polar coordinates of $P$, chosen so that $r$ is positive and $0 \\leq \\theta \\leq 2$. Find $r$ and $\\theta$.\n$r=$ [ANS]\n$\\theta=$ [ANS]", "answer_v3": [ "5", "0.317824" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0171", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Polar and rectangular coordinates", "level": "4", "keywords": [], "problem_v1": "A bullet is fired into the air with an initial velocity of 1100 feet per second at an angle of $67^o$ from the horizontal. The horizontal and vertical components of the velocity vector are:\n$<$ [ANS], [ANS] $>$ (Express your answers using at least three decimal places.)", "answer_v1": [ "429.804", "1012.555" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A bullet is fired into the air with an initial velocity of 700 feet per second at an angle of $79^o$ from the horizontal. The horizontal and vertical components of the velocity vector are:\n$<$ [ANS], [ANS] $>$ (Express your answers using at least three decimal places.)", "answer_v2": [ "133.566", "687.139" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A bullet is fired into the air with an initial velocity of 850 feet per second at an angle of $67^o$ from the horizontal. The horizontal and vertical components of the velocity vector are:\n$<$ [ANS], [ANS] $>$ (Express your answers using at least three decimal places.)", "answer_v3": [ "332.121", "782.429" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Trigonometry_0172", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Convert $\\left(xy\\right)^{6}=8$ to an equation in polar coordinates. [ANS] $=r^{12}$ Note: use \"t\" for $\\theta$", "answer_v1": [ "8*[2*csc(2*t)]^6" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Convert $\\left(xy\\right)^{9}=2$ to an equation in polar coordinates. [ANS] $=r^{18}$ Note: use \"t\" for $\\theta$", "answer_v2": [ "2*[2*csc(2*t)]^9" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Convert $\\left(xy\\right)^{6}=4$ to an equation in polar coordinates. [ANS] $=r^{12}$ Note: use \"t\" for $\\theta$", "answer_v3": [ "4*[2*csc(2*t)]^6" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0173", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find a polar equation of the hyperbola $\\left(\\frac{x}{8}\\right)^2-\\left(\\frac{y}{9}\\right)^2=1$. $r^2=$ [ANS]\nNote: use t for $\\theta$.", "answer_v1": [ "5184/(81*[cos(t)]^2-64*[sin(t)]^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a polar equation of the hyperbola $\\left(\\frac{x}{3}\\right)^2-\\left(\\frac{y}{5}\\right)^2=1$. $r^2=$ [ANS]\nNote: use t for $\\theta$.", "answer_v2": [ "225/(25*[cos(t)]^2-9*[sin(t)]^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a polar equation of the hyperbola $\\left(\\frac{x}{5}\\right)^2-\\left(\\frac{y}{8}\\right)^2=1$. $r^2=$ [ANS]\nNote: use t for $\\theta$.", "answer_v3": [ "1600/(64*[cos(t)]^2-25*[sin(t)]^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0174", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the equation in polar coordinates of the line through the origin with slope $\\frac{1}{8}$. $\\theta=$ [ANS]", "answer_v1": [ "0.124355" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the equation in polar coordinates of the line through the origin with slope $\\frac{1}{2}$. $\\theta=$ [ANS]", "answer_v2": [ "0.463648" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the equation in polar coordinates of the line through the origin with slope $\\frac{1}{4}$. $\\theta=$ [ANS]", "answer_v3": [ "0.244979" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0175", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "The following polar equation describes a circle in rectangular coordinates: r=14\\cos\\theta Locate its center on the $xy$-plane, and find the circle's radius. $(x_0,y_0)=$ ([ANS], [ANS]) $R=$ [ANS]", "answer_v1": [ "7", "0", "7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The following polar equation describes a circle in rectangular coordinates: r=2\\cos\\theta Locate its center on the $xy$-plane, and find the circle's radius. $(x_0,y_0)=$ ([ANS], [ANS]) $R=$ [ANS]", "answer_v2": [ "1", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The following polar equation describes a circle in rectangular coordinates: r=6\\cos\\theta Locate its center on the $xy$-plane, and find the circle's radius. $(x_0,y_0)=$ ([ANS], [ANS]) $R=$ [ANS]", "answer_v3": [ "3", "0", "3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Trigonometry_0176", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "2", "keywords": [ "polar' 'curve' 'conics" ], "problem_v1": "Match each polar equation below to the best description. Each answer should be C, E, F, H, I, L, O, P, T, or W.\nDESCRIPTIONS\nC. Cardioid, E. Ellipse, F. Rose with four petals, H. Hyperbola, I. Inwardly spiraling spiral, L. Line, P. Parabola, O. Outwardly spiraling spiral, T. Rose with three petals,\nPOLAR EQUATIONS [ANS] 1. $r=\\sin 3 \\theta$ [ANS] 2. $r=\\sin 2 \\theta$ [ANS] 3. $r^2=\\csc 2 \\theta$ [ANS] 4. $r=8-8 \\sin \\theta$ [ANS] 5. $r=6 \\theta, r>0$ [ANS] 6. $r=\\frac{2}{\\theta}, r>0$ [ANS] 7. $1=\\tan \\theta$", "answer_v1": [ "T", "F", "H", "C", "O", "I", "L" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ] ], "problem_v2": "Match each polar equation below to the best description. Each answer should be C, E, F, H, I, L, O, P, T, or W.\nDESCRIPTIONS\nC. Cardioid, E. Ellipse, F. Rose with four petals, H. Hyperbola, I. Inwardly spiraling spiral, L. Line, P. Parabola, O. Outwardly spiraling spiral, T. Rose with three petals,\nPOLAR EQUATIONS [ANS] 1. $r^2=\\csc 2 \\theta$ [ANS] 2. $1=\\tan \\theta$ [ANS] 3. $r=\\sin 2 \\theta$ [ANS] 4. $r=\\frac{2}{\\theta}, r>0$ [ANS] 5. $r=9 \\theta, r>0$ [ANS] 6. $r=\\sin 3 \\theta$ [ANS] 7. $r=2-2 \\sin \\theta$", "answer_v2": [ "H", "L", "F", "I", "O", "T", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ] ], "problem_v3": "Match each polar equation below to the best description. Each answer should be C, E, F, H, I, L, O, P, T, or W.\nDESCRIPTIONS\nC. Cardioid, E. Ellipse, F. Rose with four petals, H. Hyperbola, I. Inwardly spiraling spiral, L. Line, P. Parabola, O. Outwardly spiraling spiral, T. Rose with three petals,\nPOLAR EQUATIONS [ANS] 1. $r=6 \\theta, r>0$ [ANS] 2. $r=\\sin 2 \\theta$ [ANS] 3. $r^2=\\csc 2 \\theta$ [ANS] 4. $1=\\tan \\theta$ [ANS] 5. $r=\\sin 3 \\theta$ [ANS] 6. $r=4-4 \\sin \\theta$ [ANS] 7. $r=\\frac{2}{\\theta}, r>0$", "answer_v3": [ "O", "F", "H", "L", "T", "C", "I" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ], [ "C", "E", "F", "H", "I", "L", "O", "P", "T", "W" ] ] }, { "id": "Trigonometry_0177", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "3", "keywords": [ "polar' 'curve" ], "problem_v1": "A curve with polar equation r=\\frac{32}{8 \\sin \\theta+49 \\cos \\theta} represents a line. This line has a Cartesian equation of the form $y=mx+b$,where $m$ and $b$ are constants. Give the formula for $y$ in terms of $x$. For example, if the line had equation $y=2x+3$ then the answer would be $2*x+3$. [ANS]\nHint: multiply both sides by the denominator on the right hand side and use $r \\cos \\theta=x$ and $r \\sin \\theta=y$.", "answer_v1": [ "(32/8) - ( 49/8)*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A curve with polar equation r=\\frac{9}{3 \\sin \\theta+25 \\cos \\theta} represents a line. This line has a Cartesian equation of the form $y=mx+b$,where $m$ and $b$ are constants. Give the formula for $y$ in terms of $x$. For example, if the line had equation $y=2x+3$ then the answer would be $2*x+3$. [ANS]\nHint: multiply both sides by the denominator on the right hand side and use $r \\cos \\theta=x$ and $r \\sin \\theta=y$.", "answer_v2": [ "(9/3) - ( 25/3)*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A curve with polar equation r=\\frac{15}{5 \\sin \\theta+31 \\cos \\theta} represents a line. This line has a Cartesian equation of the form $y=mx+b$,where $m$ and $b$ are constants. Give the formula for $y$ in terms of $x$. For example, if the line had equation $y=2x+3$ then the answer would be $2*x+3$. [ANS]\nHint: multiply both sides by the denominator on the right hand side and use $r \\cos \\theta=x$ and $r \\sin \\theta=y$.", "answer_v3": [ "(15/5) - ( 31/5)*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Trigonometry_0178", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "3", "keywords": [ "Polar", "Rectangular", "Trigonometric", "Sin", "Cos" ], "problem_v1": "Write each equation in polar coordinates. Express as a function of $\\small{t}$. Assume that $\\small{r > 0}$.\n(a) $\\small{y=5}$\n$\\quad \\small{r=}$ [ANS]\n(b) $\\small{x^2+y^2=7}$\n$\\quad \\small{r=}$ [ANS]\n(c) $\\small{x^2+y^2+3x=0}$\n$\\quad \\small{r=}$ [ANS]\n(d) $\\small{x^2(x^2+y^2)=8 y^2}$\n$\\quad \\small{r=}$ [ANS]", "answer_v1": [ "5/[sin(t)]", "2.64575", "-3*cos(t)", "2.82843*tan(t)" ], "answer_type_v1": [ "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Write each equation in polar coordinates. Express as a function of $\\small{t}$. Assume that $\\small{r > 0}$.\n(a) $\\small{y=-9}$\n$\\quad \\small{r=}$ [ANS]\n(b) $\\small{x^2+y^2=10}$\n$\\quad \\small{r=}$ [ANS]\n(c) $\\small{x^2+y^2-7x=0}$\n$\\quad \\small{r=}$ [ANS]\n(d) $\\small{x^2(x^2+y^2)=5 y^2}$\n$\\quad \\small{r=}$ [ANS]", "answer_v2": [ "-9/[sin(t)]", "3.16228", "7*cos(t)", "2.23607*tan(t)" ], "answer_type_v2": [ "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Write each equation in polar coordinates. Express as a function of $\\small{t}$. Assume that $\\small{r > 0}$.\n(a) $\\small{y=-4}$\n$\\quad \\small{r=}$ [ANS]\n(b) $\\small{x^2+y^2=7}$\n$\\quad \\small{r=}$ [ANS]\n(c) $\\small{x^2+y^2-5x=0}$\n$\\quad \\small{r=}$ [ANS]\n(d) $\\small{x^2(x^2+y^2)=6 y^2}$\n$\\quad \\small{r=}$ [ANS]", "answer_v3": [ "-4/[sin(t)]", "2.64575", "5*cos(t)", "2.44949*tan(t)" ], "answer_type_v3": [ "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Trigonometry_0179", "subject": "Trigonometry", "topic": "Polar coordinates & vectors", "subtopic": "Curves", "level": "2", "keywords": [ "polar coordinates" ], "problem_v1": "The equation $r=8 \\sin \\theta$ represents a circle. Find the cartesian coordinates of the center: x=[ANS]\ny=[ANS]\nand the radius: r=[ANS]", "answer_v1": [ "0", "4", "4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The equation $r=2 \\sin \\theta$ represents a circle. Find the cartesian coordinates of the center: x=[ANS]\ny=[ANS]\nand the radius: r=[ANS]", "answer_v2": [ "0", "1", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The equation $r=4 \\sin \\theta$ represents a circle. Find the cartesian coordinates of the center: x=[ANS]\ny=[ANS]\nand the radius: r=[ANS]", "answer_v3": [ "0", "2", "2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] } ]