Trigonometry Practice

Radians vs Degrees Practice

Practice converting between degrees and radians, recognizing common angle equivalents, and finding coterminal angles.

Start Practicing Review the Guide

Try each problem on your own first. Then click Show solution to check the conversion factor, simplification, and final angle measure.

Keep These Rules Nearby

Radians vs Degrees Essentials

Core Relationship

\[ 180^\circ=\pi \]

Degrees → Radians

\[ \times \frac{\pi}{180} \]

Radians → Degrees

\[ \times \frac{180}{\pi} \]

Full Rotation

\[ 360^\circ=2\pi \]
Download Practice PDF Review Radians vs Degrees

Level 1

Convert Between Units

Use the correct conversion factor and simplify each result.

Problem 1

Degrees → Radians

Convert \(60^\circ\) to radians.

\[ 60^\circ \]

Step-by-Step Solution

To convert degrees to radians, multiply by \(\frac{\pi}{180}\).

\[ 60^\circ \times \frac{\pi}{180^\circ} \]

Simplify the fraction.

\[ \frac{60\pi}{180} = \frac{\pi}{3} \]
Answer: \[ \boxed{ \frac{\pi}{3} } \]

Problem 2

Radians → Degrees

Convert \(\frac{5\pi}{6}\) to degrees.

\[ \frac{5\pi}{6} \]

Step-by-Step Solution

To convert radians to degrees, multiply by \(\frac{180}{\pi}\).

\[ \frac{5\pi}{6} \times \frac{180^\circ}{\pi} \]

Cancel \(\pi\) and simplify.

\[ \frac{5(180^\circ)}{6} = 150^\circ \]
Answer: \[ \boxed{ 150^\circ } \]

Problem 3

Degrees → Radians

Convert \(225^\circ\) to radians.

\[ 225^\circ \]

Step-by-Step Solution

Multiply by \(\frac{\pi}{180}\).

\[ 225^\circ \times \frac{\pi}{180^\circ} \]

Simplify.

\[ \frac{225\pi}{180} = \frac{5\pi}{4} \]
Answer: \[ \boxed{ \frac{5\pi}{4} } \]

Level 2

Recognize Common Angle Equivalents

Use familiar degree-radian pairs to identify equivalent angle measures quickly.

Problem 4

Common Angles

What radian measure is equivalent to \(135^\circ\)?

\[ 135^\circ = \ ? \]

Step-by-Step Solution

Multiply by \(\frac{\pi}{180}\).

\[ 135^\circ \times \frac{\pi}{180^\circ} \]
\[ \frac{135\pi}{180} = \frac{3\pi}{4} \]
Answer: \[ \boxed{ \frac{3\pi}{4} } \]

Problem 5

Common Angles

What degree measure is equivalent to \(\frac{7\pi}{4}\)?

\[ \frac{7\pi}{4} = \ ? \]

Step-by-Step Solution

Multiply by \(\frac{180}{\pi}\).

\[ \frac{7\pi}{4} \times \frac{180^\circ}{\pi} \]

Cancel \(\pi\) and simplify.

\[ \frac{7(180^\circ)}{4} = 315^\circ \]
Answer: \[ \boxed{ 315^\circ } \]

Problem 6

Match the Equivalent

Which radian measure is equivalent to \(300^\circ\)?

\[ \frac{3\pi}{2} \]
\[ \frac{5\pi}{3} \]
\[ \frac{7\pi}{4} \]
\[ \frac{11\pi}{6} \]

Step-by-Step Solution

Convert \(300^\circ\) to radians.

\[ 300^\circ \times \frac{\pi}{180^\circ} \]
\[ \frac{300\pi}{180} = \frac{5\pi}{3} \]
Answer: \[ \boxed{ \frac{5\pi}{3} } \]

Level 3

Coterminal Angles & Mixed Review

Use full rotations to find coterminal angles, then combine the skills from the earlier levels.

Problem 7

Coterminal Degrees

Find one positive coterminal angle for \(-75^\circ\).

\[ -75^\circ \]

Step-by-Step Solution

Coterminal angles in degrees differ by complete rotations of \(360^\circ\).

Since the given angle is negative, add \(360^\circ\).

\[ -75^\circ + 360^\circ = 285^\circ \]
Answer: \[ \boxed{ 285^\circ } \]

Problem 8

Coterminal Radians

Find one positive coterminal angle for \(-\frac{5\pi}{6}\).

\[ -\frac{5\pi}{6} \]

Step-by-Step Solution

Coterminal angles in radians differ by complete rotations of \(2\pi\).

Add \(2\pi\) to the negative angle.

\[ -\frac{5\pi}{6} + 2\pi \]

Rewrite \(2\pi\) with a denominator of 6.

\[ -\frac{5\pi}{6} + \frac{12\pi}{6} = \frac{7\pi}{6} \]
Answer: \[ \boxed{ \frac{7\pi}{6} } \]

Problem 9

Mixed Review

Convert \(420^\circ\) to radians, then find a coterminal angle between \(0\) and \(2\pi\).

\[ 420^\circ \]

Step-by-Step Solution

First convert \(420^\circ\) to radians.

\[ 420^\circ \times \frac{\pi}{180^\circ} = \frac{420\pi}{180} \]

Simplify.

\[ \frac{420\pi}{180} = \frac{7\pi}{3} \]

Now subtract one complete rotation, \(2\pi\).

\[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} \]
Answer: \[ \boxed{ \frac{7\pi}{3} } \]

A coterminal angle between \(0\) and \(2\pi\) is \(\boxed{\frac{\pi}{3}}\).

Before You Check the Guide

Use the Same Process Every Time

  1. Identify the unit you are starting with.
  2. Choose the correct conversion factor.
  3. Multiply and cancel the matching unit.
  4. Simplify the fraction completely.
  5. For coterminal angles, add or subtract a full rotation.

Continue Learning

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