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Reed College Department of Mathematics & Statistics

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Worksheet Degrees and Radians

1.

Convert the given angle in degrees to radians. Enter answer as a fraction.
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\(120^{\circ} =\) ::
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Answer.
\(2.0944\)
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2.

Convert the given angle in degrees to radians. Enter answer as a fraction.
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\(-480^{\circ}=\) ::
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Answer.
\(-8.37758\)
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3.

Convert \(134^{\circ}\) into radian measure:
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Convert \(45^{\circ}\) into radian measure:
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Convert \(283^{\circ}\) into radian measure:
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Convert \(212^{\circ}\) into radian measure:
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  • Enter your answer as a fraction of pi
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  • You may also use decimal approximation, but your answer must be accurate to at least 3 decimal places.
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Hint.
If you’re familiar with unit conversion, this is pretty much the same thing.
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You want to multiply by a fraction that’s equivalent to 1 (because multiplying by 1 won’t change the value).
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But the fraction will have different units in the numerator and denominator.
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In the denominator you put the units that you want to remove, and in the numerator you put the units that you want to end up with.
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In this case, you want to remove degrees, and have radians instead.
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So you need two equivalent measurements - one in radians, and one in degrees.
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Answer 1.
\(\frac{67\pi }{90}\)
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Answer 2.
\(\frac{1\pi }{4}\)
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Answer 3.
\(\frac{283\pi }{180}\)
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Answer 4.
\(\frac{53\pi }{45}\)
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Solution.
For converting between radians and degrees, we rely on the fact that \(360^{\circ} = 2\pi\) radians.
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This means we can use the conversion factors: \(\frac{2\pi}{360^{\circ}}\) or \(\frac{360^{\circ}}{2\pi}\)
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To convert from degrees, to radians, we need to cancel out the existing degrees.
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That means putting \(360^{\circ}\) in the denominator.
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So we’ll be using \(\frac{2\pi}{360^{\circ}}\text{,}\) which is often reduced: \(\frac{\pi}{180^{\circ}}\)
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Note that both fractions are equivalent to 1, since we’re taking a measurement divided by the same measurement.
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If you’re having trouble seeing that, think of \(\frac{1 \text{ft}}{12 \text{in}}\) or \(\frac{60 \text{min}}{1 \text{hr}}\text{.}\) They’re both equivalent to 1 as well.
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  1. \(\displaystyle 134^{\circ} \rightarrow 134^{\circ} \times \frac{\pi\text{ radians}}{180^{\circ}} \rightarrow \frac{134\pi\text{ radians}}{180} \rightarrow {\frac{67\pi }{90}}\text{ radians}\)
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  2. \(\displaystyle 45^{\circ} \rightarrow 45^{\circ} \times \frac{\pi\text{ radians}}{180^{\circ}} \rightarrow \frac{45\pi\text{ radians}}{180} \rightarrow {\frac{1\pi }{4}}\text{ radians}\)
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  3. \(\displaystyle 283^{\circ} \rightarrow 283^{\circ} \times \frac{\pi\text{ radians}}{180^{\circ}} \rightarrow \frac{283\pi\text{ radians}}{180} \rightarrow {\frac{283\pi }{180}}\text{ radians}\)
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  4. \(\displaystyle 212^{\circ} \rightarrow 212^{\circ} \times \frac{\pi\text{ radians}}{180^{\circ}} \rightarrow \frac{212\pi\text{ radians}}{180} \rightarrow {\frac{53\pi }{45}}\text{ radians}\)
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4.

Convert \({3.85}\) radians into degrees:
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Convert \({6.15}\) radians into degrees:
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Convert \({0.31}\) radians into degrees:
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Convert \({2.63}\) radians into degrees:
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  • You may use decimal approximation, but your answer must be accurate to at least 3 decimal places.
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Hint.
If you’re familiar with unit conversion, this is pretty much the same thing.
🔗
You want to multiply by a fraction that’s equivalent to 1 (because multiplying by 1 won’t change the value).
🔗
But the fraction will have different units in the numerator and denominator.
🔗
In the denominator you put the units that you want to remove, and in the numerator you put the units that you want to end up with.
🔗
In this case, you want to remove radians, and have degrees instead.
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So you need two equivalent measurements - one in radians, and one in degrees.
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Answer 1.
\(220.589\)
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Answer 2.
\(352.369\)
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Answer 3.
\(17.7617\)
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Answer 4.
\(150.688\)
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Solution.
For converting between radians and degrees, we rely on the fact that \(360^{\circ} = 2\pi\) radians.
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This means we can use the conversion factors: \(\frac{2\pi}{360^{\circ}}\) or \(\frac{360^{\circ}}{2\pi}\)
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To convert from radians, to degrees, we need to cancel out the existing radians.
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That means putting \(2 \pi\) radians in the denominator.
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So we’ll be using \(\frac{360^{\circ}}{2\pi\text{ radians}}\text{,}\) which is often reduced: \(\frac{180^{\circ}}{\pi\text{ radians}}\)
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Note that both fractions are equivalent to 1, since we’re taking a measurement divided by the same measurement.
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If you’re having trouble seeing that, think of \(\frac{1 \text{ft}}{12 \text{in}}\) or \(\frac{60 \text{min}}{1 \text{hr}}\text{.}\) They’re both equivalent to 1 as well.
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  1. \(\displaystyle {3.85} \text{ radians} \rightarrow {3.85} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{693^{\circ}}{\pi} \approx {220.589}^{\circ}\)
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  2. \(\displaystyle {6.15} \text{ radians} \rightarrow {6.15} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{1107^{\circ}}{\pi} \approx {352.369}^{\circ}\)
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  3. \(\displaystyle {0.31} \text{ radians} \rightarrow {0.31} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{55.8^{\circ}}{\pi} \approx {17.7617}^{\circ}\)
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  4. \(\displaystyle {2.63} \text{ radians} \rightarrow {2.63} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{473.4^{\circ}}{\pi} \approx {150.688}^{\circ}\)
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5.

Convert \(\displaystyle{ {\frac{5\pi }{3}} }\) radians into degrees:
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Convert \(\displaystyle{ {\frac{3\pi }{2}} }\) radians into degrees:
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Convert \(\displaystyle{ {\frac{11\pi }{6}} }\) radians into degrees:
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Convert \(\displaystyle{ {\frac{3\pi }{4}} }\) radians into degrees:
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Hint.
If you’re familiar with unit conversion, this is pretty much the same thing.
🔗
You want to multiply by a fraction that’s equivalent to 1 (because multiplying by 1 won’t change the value).
🔗
But the fraction will have different units in the numerator and denominator.
🔗
In the denominator you put the units that you want to remove, and in the numerator you put the units that you want to end up with.
🔗
In this case, you want to remove radians, and have degrees instead.
🔗
So you need two equivalent measurements - one in radians, and one in degrees.
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Answer 1.
\(300\)
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Answer 2.
\(270\)
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Answer 3.
\(330\)
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Answer 4.
\(135\)
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Solution.
For converting between radians and degrees, we rely on the fact that \(360^{\circ} = 2\pi\) radians.
🔗
This means we can use the conversion factors: \(\frac{2\pi}{360^{\circ}}\) or \(\frac{360^{\circ}}{2\pi}\)
🔗
To convert from radians, to degrees, we need to cancel out the existing radians.
🔗
That means putting \(2 \pi\) radians in the denominator.
🔗
So we’ll be using \(\frac{360^{\circ}}{2\pi\text{ radians}}\text{,}\) which is often reduced: \(\frac{180^{\circ}}{\pi\text{ radians}}\)
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Note that both fractions are equivalent to 1, since we’re taking a measurement divided by the same measurement.
🔗
If you’re having trouble seeing that, think of \(\frac{1 \text{ft}}{12 \text{in}}\) or \(\frac{60 \text{min}}{1 \text{hr}}\text{.}\) They’re both equivalent to 1 as well.
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  1. \(\displaystyle {\frac{5\pi }{3}} \text{ radians} \rightarrow {\frac{5\pi }{3}} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{900^{\circ}\pi}{3\pi} \rightarrow 300^{\circ}\)
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  2. \(\displaystyle {\frac{3\pi }{2}} \text{ radians} \rightarrow {\frac{3\pi }{2}} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{540^{\circ}\pi}{2\pi} \rightarrow 270^{\circ}\)
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  3. \(\displaystyle {\frac{11\pi }{6}} \text{ radians} \rightarrow {\frac{11\pi }{6}} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{1980^{\circ}\pi}{6\pi} \rightarrow 330^{\circ}\)
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  4. \(\displaystyle {\frac{3\pi }{4}} \text{ radians} \rightarrow {\frac{3\pi }{4}} \text{ radians} \times \frac{180^{\circ}}{\pi\text{ radians}} \rightarrow \frac{540^{\circ}\pi}{4\pi} \rightarrow 135^{\circ}\)
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6.

Convert the following degree measures to radians.
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(Write your answers as angles \(0 \le \theta \lt 2\pi\text{.}\) Write as a fraction using \(\fbox{pi}\) for \(\pi\text{.}\))
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\(-540\ \text{degrees}\ = \ \) \(\text{radians}\)
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\(-315\ \text{degrees}\ = \ \) \(\text{radians}\)
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\(-420\ \text{degrees}\ = \ \) \(\text{radians}\)
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\(\phantom{-}210\ \text{degrees}\ = \ \) \(\text{radians}\)
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\(-630\ \text{degrees}\ = \ \) \(\text{radians}\)
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Answer 1.
\(1\frac{\pi }{1}\)
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Answer 2.
\(1\frac{\pi }{4}\)
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Answer 3.
\(5\frac{\pi }{3}\)
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Answer 4.
\(7\frac{\pi }{6}\)
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Answer 5.
\(1\frac{\pi }{2}\)
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7.

Convert the following radian measures to degrees.
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(Write your answers as angles \(0^\circ \le \theta \lt 360^\circ\text{.}\))
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\(\phantom{x}\frac{2\pi}{3}\ \text{radians}\ = \ \) \(\text{degrees}\)
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\(\phantom{x}\frac{5\pi}{4}\ \text{radians}\ = \ \) \(\text{degrees}\)
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\(-3\pi\ \text{radians}\ = \ \) \(\text{degrees}\)
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\(\phantom{x}\frac{11\pi}{6}\ \text{radians}\ = \ \) \(\text{degrees}\)
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\(\phantom{x}\frac{5\pi}{2}\ \text{radians}\ = \ \) \(\text{degrees}\)
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Answer 1.
\(120\)
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Answer 2.
\(225\)
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Answer 3.
\(180\)
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Answer 4.
\(330\)
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Answer 5.
\(90\)
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8.

Consider the pictures below. Click on the pictures to see them more clearly. Each angle \(\theta\) is an integer when measured in radians. Give the radian measure of the angle.
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1.
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radian measure =
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2.
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radian measure =
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3.
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radian measure =
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Answer 1.
\(1\)
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Answer 2.
\(3\)
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Answer 3.
\(6\)
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9.

Determine the exact radian measure for the angle \(285^{\circ}\text{.}\) Do not give a decimal approximation, and recall in order to enter \(\pi\) you must type pi.
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\(285^{\circ} =\) radians
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Answer.
\(\frac{285\pi }{180}\)
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Solution.
SOLUTION\(285^{\circ}\)\(\frac{2 \pi}{360^{\circ}}\text{:}\)
\begin{equation*} 285^{\circ} \cdot \left( \frac{2 \pi}{360^{\circ}} \right) = \frac{ 285 \pi}{180} = \frac{19\pi}{12} \mbox{ radians}. \end{equation*}
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10.

Determine the exact degree measure for the angle \(\frac{5 \pi}{4}\) radians.
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\(\frac{5 \pi}{4}\) radians = degrees
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Answer.
\(225\)
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Solution.
SOLUTION\(\frac{360^{\circ}}{2 \pi}\text{,}\)
\begin{equation*} \frac{5 \pi}{4} \cdot \frac{360^{\circ}}{2 \pi} = 225^{\circ}. \end{equation*}
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11.

For each angle (in degrees) below, determine the quadrant in which the terminal side of the angle is found.
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[NOTE: Enter ’1’ for quadrant I, ’2’ for quadrant II, ’3’ for quadrant III, and ’4’ for quadrant IV.]
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(a) \(-198^\circ\) is found in quadrant
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(b) \(-317^\circ\) is found in quadrant
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(c) \(-415^\circ\) is found in quadrant
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(d) \(187^\circ\) is found in quadrant
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Answer 1.
\(2\)
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Answer 2.
\(1\)
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Answer 3.
\(4\)
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Answer 4.
\(3\)
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12.

For each angle listed in the table below, select the letter of the corresponding point on the unit circle, the value of the \(x\)-coordinate of the point, and the value of the \(y\)-coordinate of the point. Round the coordinates of the point to 3 decimal places. Don’t enter sin or cos. (You must approximate your answers.)
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(Click on graph to enlarge)
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Answer 1.
\(-0.866\)
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Answer 2.
\(0.5\)
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Answer 3.
\(-0.866\)
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Answer 4.
\(-0.5\)
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Answer 5.
\(0.766\)
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Answer 6.
\(-0.643\)
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Answer 7.
\(-0.342\)
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Answer 8.
\(-0.94\)
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Answer 9.
\(0.707\)
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Answer 10.
\(0.707\)
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Answer 11.
\(0.174\)
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Answer 12.
\(0.985\)
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Solution.
SOLUTION
Angle (in degrees) Point (enter letter A-F) \(x\)-coordinate \(y\)-coordinate
\(150^{\circ}\) C -0.866 0.5
\(-150^{\circ}\) D -0.866 -0.5
\(-400^{\circ}\) F 0.766 -0.643
\(250^{\circ}\) E -0.342 -0.94
\(45^{\circ}\) A 0.707 0.707
\(800^{\circ}\) B 0.174 0.985
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13.

Find two angles (one positive and one negative) that are coterminal with \(\displaystyle{ {\frac{5\pi }{4}} }\text{.}\)
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is a positive angle that is coterminal with \(\displaystyle{ {\frac{5\pi }{4}} }\text{.}\)
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is a negative angle that is coterminal with \(\displaystyle{ {\frac{5\pi }{4}} }\text{.}\)
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  • You’ll have an easier time with this problem if you use fractions and pi instead of decimal approximations.
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  • If you use decimal approximations, your answer must be correct to at least 3 decimal places.
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Hint.
Coterminal angles are angles that differ by a full rotation.
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In degrees, a full rotation is \(360^{\circ}\text{.}\)
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In radians, a full rotation is \(2\pi\text{.}\)
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Try adding or subtracting full rotations to find angles that are coterminal with \({\frac{5\pi }{4}}\text{.}\)
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Answer 1.
\(\frac{5\pi }{4}\)
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Answer 2.
\(\frac{5\pi }{4}\)
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Solution.
There are many positive coterminal angles with \({\frac{5\pi }{4}}\text{.}\)
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To find positive coterminal angles, we add full rotations.
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You can think of this like spinning around in a full circle.
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You end up facing exactly the same direction you were before you spun around.
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Add one full rotation:
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\({\frac{5\pi }{4}} + 2\pi \rightarrow {\frac{5\pi }{4}} + \frac{8\pi}{4} \rightarrow \frac{13\pi}{4}\)
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Add two full rotations:
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\({\frac{5\pi }{4}} + 4\pi \rightarrow {\frac{5\pi }{4}} + \frac{16\pi}{4} \rightarrow \frac{21\pi}{4}\)
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You could even add more full rotations, there are infinitely many positive coterminal angles with \({\frac{5\pi }{4}}\text{.}\)
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In other words, it doesn’t matter how many times you spin around,
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as long as you end up facing the same direction you were before you started.
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To find negative coterminal angles, we instead subtract full rotations.
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Here, we’re just spinning circles in the opposite direction.
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Subtract one full rotation:
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\({\frac{5\pi }{4}} - 2\pi \rightarrow {\frac{5\pi }{4}} - \frac{8\pi}{4} \rightarrow \frac{-3\pi}{4}\)
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Subtract two full rotations:
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\({\frac{5\pi }{4}} - 4\pi \rightarrow {\frac{5\pi }{4}} - \frac{16\pi}{4} \rightarrow \frac{-11\pi}{4}\)
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Again, we could subtract any number of full rotations.
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There are an infinite number of negative coterminal angles as well.
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In general: Let \(k\) be the number of times to spin around.
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\(k\) can be positive or negative (depending on which direction you want to spin).
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\({\frac{5\pi }{4}} + k(2\pi) \rightarrow {\frac{5\pi }{4}} + \frac{8k\pi}{4} \rightarrow \frac{(5+8k)\pi}{4}\)
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Do you see how this relates to our answers above?
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What happens when \(k\) is 1? Or -1?
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What about when \(k\) is 2? Or -2?
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