Practice
The Radian Measure of an Angle Practice
Fifty original questions on radians, exact conversions, signed rotations, coterminal angles, and periodicity.
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Question 1
Explanation
Multiply by \(\pi/180^\circ\): \(15^\circ\cdot\pi/180^\circ=\frac{\pi}{12}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 2
Explanation
Multiply by \(\pi/180^\circ\): \(20^\circ\cdot\pi/180^\circ=\frac{\pi}{9}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 3
Explanation
Multiply by \(\pi/180^\circ\): \(72^\circ\cdot\pi/180^\circ=\frac{2\pi}{5}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 4
Explanation
Multiply by \(\pi/180^\circ\): \(105^\circ\cdot\pi/180^\circ=\frac{7\pi}{12}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 5
Explanation
Multiply by \(\pi/180^\circ\): \(120^\circ\cdot\pi/180^\circ=\frac{2\pi}{3}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 6
Explanation
Multiply by \(\pi/180^\circ\): \(135^\circ\cdot\pi/180^\circ=\frac{3\pi}{4}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 7
Explanation
Multiply by \(\pi/180^\circ\): \(150^\circ\cdot\pi/180^\circ=\frac{5\pi}{6}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 8
Explanation
Multiply by \(\pi/180^\circ\): \(210^\circ\cdot\pi/180^\circ=\frac{7\pi}{6}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 9
Explanation
Multiply by \(\pi/180^\circ\): \(300^\circ\cdot\pi/180^\circ=\frac{5\pi}{3}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 10
Explanation
Multiply by \(\pi/180^\circ\): \(330^\circ\cdot\pi/180^\circ=\frac{11\pi}{6}\).
- Reasoning
Choose the conversion factor whose degree unit cancels, then reduce the fraction.
Question 11
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(18^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 12
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(22.5^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 13
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(50^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 14
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(105^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 15
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(72^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 16
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(150^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 17
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(210^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 18
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(220^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 19
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(315^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 20
Explanation
Multiply by \(180^\circ/\pi\); the factor \(\pi\) cancels, leaving \(285^\circ\).
- Reasoning
Use the factor with radians in the denominator so the old unit cancels.
Question 21
Explanation
Subtract one full turn: 470°−360°=110°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 22
Explanation
Add 720°: −725°+720°=−5°; add one more 360° to obtain 355°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 23
Explanation
Subtract 360°: 125°−360°=−235°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 24
Explanation
Add 360° to −40° to get 320°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 25
Explanation
Subtract 2π=12π/6: 17π/6−12π/6=5π/6.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 26
Explanation
Subtract 2π=10π/5: 3π/5−10π/5=−7π/5.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 27
Explanation
Degree-measure coterminal angles differ by integer multiples of 360°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 28
Explanation
Radian-measure coterminal angles differ by integer multiples of 2π.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 29
Explanation
Subtract 360° from 120° to obtain −240°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 30
Explanation
Add 360°: −225°+360°=135°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 31
Explanation
The angles are 405° and 45°, so they differ by one full revolution, 360°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 32
Explanation
Adding 360° gives 580°, so the terminal side is unchanged.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 33
Explanation
The angle is −510°. Add 720° to obtain 210°.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 34
Explanation
Two full turns equal 4π=16π/4, leaving 3π/4.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 35
Explanation
Add 4π=24π/6 to −23π/6, leaving π/6.
- Reasoning
Add or subtract complete turns until the angle is in the requested range.
Question 36
Explanation
Sine repeats after 2π radians.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 37
Explanation
Cosine repeats after 360°.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 38
Explanation
Tangent repeats after π radians.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 39
Explanation
The angles differ by 360°, one sine period.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 40
Explanation
20π/9−2π/9=18π/9=2π, one cosine period.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 41
Explanation
The angles differ by 180°, one tangent period.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 42
Explanation
Subtract 4π=24π/6, leaving 5π/6.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 43
Explanation
Add 6π=24π/4 to obtain 7π/4.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 44
Explanation
Subtract 4π=28π/7; tangent allows multiples of π, and 31π/7−28π/7=3π/7.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 45
Explanation
Sine completes one full repeat every 2π.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 46
Explanation
Tangent repeats every half-turn, π radians.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 47
Explanation
Five π-radian shifts are five tangent periods, so the value is unchanged.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 48
Explanation
The shift −8π is four complete cosine periods.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 49
Explanation
Coterminal angles share sine, cosine, and tangent values, but need not have equal measures or signs.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
Question 50
Explanation
Tangent has a 180° period; sine and cosine require 360°.
- Reasoning
Identify the function's fundamental period and remove an integer number of periods.
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Questions to review
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- Question 1Degree-to-radian conversionEasy
- Question 2Degree-to-radian conversionEasy
- Question 3Degree-to-radian conversionEasy
- Question 4Degree-to-radian conversionEasy
- Question 5Degree-to-radian conversionEasy
- Question 6Degree-to-radian conversionEasy
- Question 7Degree-to-radian conversionEasy
- Question 8Degree-to-radian conversionEasy
- Question 9Degree-to-radian conversionEasy
- Question 10Degree-to-radian conversionEasy
- Question 11Radian-to-degree conversionEasy
- Question 12Radian-to-degree conversionEasy
- Question 13Radian-to-degree conversionEasy
- Question 14Radian-to-degree conversionEasy
- Question 15Radian-to-degree conversionEasy
- Question 16Radian-to-degree conversionMedium
- Question 17Radian-to-degree conversionMedium
- Question 18Radian-to-degree conversionMedium
- Question 19Radian-to-degree conversionMedium
- Question 20Radian-to-degree conversionMedium
- Question 21Coterminal anglesMedium
- Question 22Coterminal anglesMedium
- Question 23Coterminal anglesMedium
- Question 24Coterminal anglesMedium
- Question 25Coterminal anglesMedium
- Question 26Coterminal anglesMedium
- Question 27Coterminal anglesMedium
- Question 28Coterminal anglesMedium
- Question 29Standard-position diagram interpretationMedium
- Question 30Standard-position diagram interpretationMedium
- Question 31Standard-position diagram interpretationMedium
- Question 32Coterminal anglesMedium
- Question 33Coterminal anglesMedium
- Question 34Coterminal anglesMedium
- Question 35Coterminal anglesMedium
- Question 36Trigonometric periodicityMedium
- Question 37Trigonometric periodicityMedium
- Question 38Trigonometric periodicityMedium
- Question 39Trigonometric periodicityMedium
- Question 40Trigonometric periodicityMedium
- Question 41Trigonometric periodicityHard
- Question 42Trigonometric periodicityHard
- Question 43Trigonometric periodicityHard
- Question 44Trigonometric periodicityHard
- Question 45Trigonometric periodicityHard
- Question 46Trigonometric periodicityHard
- Question 47Trigonometric periodicityHard
- Question 48Trigonometric periodicityHard
- Question 49Trigonometric periodicityHard
- Question 50Trigonometric periodicityHard