Reference Angles: How to Find Them

A reference angle is the acute angle between an angle’s terminal side and the x-axis. Any trig value of an angle equals the trig value of its reference angle, with a plus or minus sign set by the quadrant. For example, 150° has a reference angle of 30°, so $\sin 150^\circ = \sin 30^\circ = \frac{1}{2}$ .

Before this lesson: You should know Trig Functions of Any Angle, Trig Values of Special Angles and Coterminal Angles.

Why Reference Angles Work

Draw 150° in standard position. Its terminal side is 30° above the negative x-axis. Now draw 30°. Its terminal side is 30° above the positive x-axis. The two sides are mirror images across the y-axis.

A point on the 150° side, like $(-\sqrt{3}, 1)$ , matches a point on the 30° side, $(\sqrt{3}, 1)$ , except for the sign of $x$ . So the trig values of 150° and 30° are the same size. Only the signs can differ, and the quadrant tells you the sign.

Four small coordinate planes, one for each quadrant. Quadrant I shows an angle of 40 degrees with reference angle 40 degrees. Quadrant II shows 150 degrees with a green reference angle of 30 degrees next to the negative x-axis, found as 180 minus theta. Quadrant III shows 225 degrees with reference angle 45 degrees, found as theta minus 180. Quadrant IV shows 300 degrees with reference angle 60 degrees, found as 360 minus theta. — The reference angle is always measured from the terminal side to the nearest part of the x-axis.

Reference Angle Formulas

Quadrant of θ	Reference angle (degrees)	Reference angle (radians)
I	$\theta$	$\theta$
II	$180^\circ - \theta$	$\pi - \theta$
III	$\theta - 180^\circ$	$\theta - \pi$
IV	$360^\circ - \theta$	$2\pi - \theta$

If the angle is not between 0° and 360°, find a coterminal angle that is.
Find the quadrant of the terminal side.
Use the formula for that quadrant to find the reference angle.
Find the trig value of the reference angle.
Attach the sign for the quadrant (All Students Take Calculus).

Worked Examples

Example 1: A Quadrant II angle

Find the reference angle of 150° and the exact value of $\sin 150^\circ$ .

Find the quadrant: $90^\circ \lt 150^\circ \lt 180^\circ$ , so Quadrant II

180^\circ - 150^\circ = 30^\circ

Use the Quadrant II formula.

\sin 30^\circ = \frac{1}{2}

Find the value for the reference angle.

\sin 150^\circ = \frac{1}{2}

Sine is positive in Quadrant II.

Answer: reference angle 30°, and $\sin 150^\circ = \frac{1}{2}$

Example 2: A Quadrant III angle

Find the exact value of $\cos 225^\circ$ .

Find the quadrant: $180^\circ \lt 225^\circ \lt 270^\circ$ , so Quadrant III

225^\circ - 180^\circ = 45^\circ

Use the Quadrant III formula.

\cos 45^\circ = \frac{\sqrt{2}}{2}

Find the value for the reference angle.

\cos 225^\circ = -\frac{\sqrt{2}}{2}

Cosine is negative in Quadrant III.

Answer: $\cos 225^\circ = -\frac{\sqrt{2}}{2}$

Example 3: An angle in radians

Find the exact value of $\tan \frac{5\pi}{3}$ .

Find the quadrant: $\frac{5\pi}{3}$ is between $\frac{3\pi}{2}$ and $2\pi$ , so Quadrant IV

2\pi - \frac{5\pi}{3}

Use the Quadrant IV formula.

\frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}

Rewrite $2\pi$ with denominator 3.

\tan \frac{\pi}{3} = \sqrt{3}

Find the value for the reference angle.

\tan \frac{5\pi}{3} = -\sqrt{3}

Tangent is negative in Quadrant IV.

Answer: $\tan \frac{5\pi}{3} = -\sqrt{3}$

Example 4: A negative angle

Find the exact value of $\sin(-120^\circ)$ .

-120^\circ + 360^\circ = 240^\circ

Find a coterminal angle (see coterminal angles)

Find the quadrant: $180^\circ \lt 240^\circ \lt 270^\circ$ , so Quadrant III

240^\circ - 180^\circ = 60^\circ

Use the Quadrant III formula.

\sin 60^\circ = \frac{\sqrt{3}}{2}

Find the value for the reference angle.

\sin(-120^\circ) = -\frac{\sqrt{3}}{2}

Sine is negative in Quadrant III.

Answer: $\sin(-120^\circ) = -\frac{\sqrt{3}}{2}$

Example 5: Word problem with a robot arm

A robot arm 2 meters long pivots at the origin. It turns 210° counterclockwise from the positive x-axis. Find the exact coordinates of the tip of the arm, $(2 \cos 210^\circ, 2 \sin 210^\circ)$ .

Find the quadrant: $180^\circ \lt 210^\circ \lt 270^\circ$ , so Quadrant III

210^\circ - 180^\circ = 30^\circ

Find the reference angle.

\cos 210^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}

Cosine is negative in Quadrant III.

\sin 210^\circ = -\sin 30^\circ = -\frac{1}{2}

Sine is negative in Quadrant III.

2 \cos 210^\circ = 2 \cdot (-\frac{\sqrt{3}}{2}) = -\sqrt{3}

Find x.

2 \sin 210^\circ = 2 \cdot (-\frac{1}{2}) = -1

Find y.

Answer: The tip is at $(-\sqrt{3}, -1) \approx (-1.73, -1)$ , below and to the left of the pivot.

Common Mistakes

Mistake	Example of the error	Fix
Measuring to the y-axis	Reference angle of 150° is 60° (to the y-axis)	Always measure to the x-axis: $180^\circ - 150^\circ = 30^\circ$ .
Using the wrong quadrant formula	Reference angle of 225° is $180^\circ - 225^\circ = -45^\circ$	In Quadrant III, use $\theta - 180^\circ = 45^\circ$ . A reference angle is never negative.
Forgetting the sign	$\cos 225^\circ = \frac{\sqrt{2}}{2}$	Cosine is negative in Quadrant III: $-\frac{\sqrt{2}}{2}$ .
Mixing degrees and radians	Reference angle of $\frac{5\pi}{3}$ is $360^\circ - \frac{5\pi}{3}$	Use $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$ in radians.
Skipping the coterminal step	Using $180^\circ - (-120^\circ)$ for −120°	First add 360° to get 240°, then use the Quadrant III formula.

Related Methods

Picture the nearest x-axis

You don’t need the formulas if you sketch the angle. Ask how far the terminal side is from the closer side of the x-axis. For 300°, it is 60° short of 360°, so the reference angle is 60°.

Non-special angles

Reference angles work for any angle, not just special ones. For example, $\cos 160^\circ = -\cos 20^\circ \approx -0.9397$ . A calculator gives this directly, but the reference angle explains the sign.

The unit circle

The next lesson puts all the special angles and their reference angles on one circle of radius 1, so you can read sine and cosine values at a glance.

Practice Problems

Find the reference angle of 160°.
Find the reference angle of 250°.
Find the reference angle of 315°.
Find the reference angle of $\frac{7\pi}{6}$ .
Find the reference angle of $\frac{3\pi}{4}$ .
Find the exact value of $\sin 300^\circ$ .
Find the exact value of $\cos 135^\circ$ .
Find the exact value of $\tan 240^\circ$ .
Find the exact value of $\sin 495^\circ$ .
The minute hand of a clock is 10 centimeters long. At one moment, it points at 330° in standard position, with its center at the origin. Find the exact coordinates of its tip, $(10 \cos 330^\circ, 10 \sin 330^\circ)$ .

Show answers

180^\circ - 160^\circ = 20^\circ

Quadrant II: subtract the angle from 180°.

Answer: 20°

250^\circ - 180^\circ = 70^\circ

Quadrant III: subtract 180° from the angle.

Answer: 70°

360^\circ - 315^\circ = 45^\circ

Quadrant IV: subtract the angle from 360°.

Answer: 45°

Quadrant III, since $\pi \lt \frac{7\pi}{6} \lt \frac{3\pi}{2}$

\frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}

Subtract.

Answer: $\frac{\pi}{6}$

Quadrant II, since $\frac{\pi}{2} \lt \frac{3\pi}{4} \lt \pi$

\pi - \frac{3\pi}{4} = \frac{4\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{4}

Subtract.

Answer: $\frac{\pi}{4}$

Quadrant IV: reference angle $360^\circ - 300^\circ = 60^\circ$

\sin 60^\circ = \frac{\sqrt{3}}{2}

Find the value for the reference angle.

Sine is negative in Quadrant IV
Answer: $\sin 300^\circ = -\frac{\sqrt{3}}{2}$

Quadrant II: reference angle $180^\circ - 135^\circ = 45^\circ$

\cos 45^\circ = \frac{\sqrt{2}}{2}

Find the value for the reference angle.

Cosine is negative in Quadrant II
Answer: $\cos 135^\circ = -\frac{\sqrt{2}}{2}$

Quadrant III: reference angle $240^\circ - 180^\circ = 60^\circ$

\tan 60^\circ = \sqrt{3}

Find the value for the reference angle.

Tangent is positive in Quadrant III
Answer: $\tan 240^\circ = \sqrt{3}$

495^\circ - 360^\circ = 135^\circ

Find a coterminal angle.

Quadrant II: reference angle $180^\circ - 135^\circ = 45^\circ$
Value: $\sin 45^\circ = \frac{\sqrt{2}}{2}$ , and sine is positive in Quadrant II
Answer: $\sin 495^\circ = \frac{\sqrt{2}}{2}$

10.

Quadrant IV: reference angle $360^\circ - 330^\circ = 30^\circ$

\cos 330^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}

Cosine is positive.

\sin 330^\circ = -\sin 30^\circ = -\frac{1}{2}

Sine is negative.

10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}

Find x.

10 \cdot (-\frac{1}{2}) = -5

Find y.

Answer: $(5\sqrt{3}, -5) \approx (8.66, -5)$

FAQ

What is a reference angle?

It is the acute angle between the terminal side of an angle and the x-axis. It is always between 0° and 90°, or 0 and $\frac{\pi}{2}$ .

How do you find a reference angle in radians?

Use $\theta$ in Quadrant I, $\pi - \theta$ in Quadrant II, $\theta - \pi$ in Quadrant III and $2\pi - \theta$ in Quadrant IV.

Can a reference angle be negative?

No. A reference angle is a size, so it is always positive. For a negative angle, find a positive coterminal angle first.

Why do we use reference angles?

They let you find the trig value of any angle from the values of acute angles. You only need to memorize the special angles from 0° to 90° and the sign rules.

Summary

The reference angle is the acute angle from the terminal side to the x-axis.
Quadrant II: $180^\circ - \theta$ ; Quadrant III: $\theta - 180^\circ$ ; Quadrant IV: $360^\circ - \theta$ .
In radians, replace 180° with $\pi$ and 360° with $2\pi$ .
A trig value of $\theta$ equals the value of its reference angle, with the sign for the quadrant.
For angles outside 0° to 360°, find a coterminal angle first.

Previous lesson: Trig Functions of Any Angle

Next lesson: The Unit Circle

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