Practice Unit Circle in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The circle of radius 1 centered at the origin in the coordinate plane, used to define trigonometric functions for all angles.

Imagine walking around a circle of radius 1. Your xx-coordinate is cos⁡θ\cos\theta and your yy-coordinate is sin⁡θ\sin\theta. Instead of being limited to right triangles, the unit circle lets you define sine and cosine for ANY angle—even angles bigger than 360°360° or negative angles. Every point on the circle is at distance 1 from the center, so the hypotenuse is always 1, and the trig ratios simplify to just coordinates.

Showing a random 20 of 50 problems.

Example 1

medium
Find sin⁡\sin, cos⁡\cos, and tan⁡\tan for θ=3π4\theta=\dfrac{3\pi}{4} using the unit circle. Identify which quadrant and the signs of each.

Example 2

medium
If cos⁡θ=35\cos\theta = \frac{3}{5} and θ\theta is in Quadrant IV, find sin⁡θ\sin\theta.

Example 3

hard
What is the arc length on the unit circle from angle θ=π6\theta = \dfrac{\pi}{6} to θ=5π6\theta = \dfrac{5\pi}{6}?

Example 4

medium
Find tan⁡135°\tan 135° exactly.

Example 5

hard
Find all θ\theta in [0,2π)[0, 2\pi) satisfying 2cos⁡2θ−1=02\cos^2\theta - 1 = 0.

Example 6

easy
Find the unit-circle coordinates at θ=π4\theta = \dfrac{\pi}{4}.

Example 7

medium
If sin⁡θ=35\sin\theta = \dfrac{3}{5} and θ\theta is in Quadrant II, find cos⁡θ\cos\theta.

Example 8

easy
What is the sign of cos⁡θ\cos\theta when θ\theta is in Quadrant III?

Example 9

easy
What are the coordinates of the point on the unit circle at θ=90°\theta = 90°?

Example 10

medium
What is the period of sin⁡θ\sin\theta on the unit circle?

Example 11

hard
Use the unit circle to prove the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1 and derive 1+tan⁡2θ=sec⁡2θ1+\tan^2\theta=\sec^2\theta.

Example 12

medium
Evaluate tan⁡2π3\tan\dfrac{2\pi}{3} using the unit circle.

Example 13

medium
Find cos⁡120°\cos 120° exactly.

Example 14

easy
Find tan⁡0\tan 0 using the unit circle.

Example 15

medium
Find sec⁡π3\sec\dfrac{\pi}{3} from unit-circle coordinates.

Example 16

easy
What is sin⁡0\sin 0 using the unit circle?

Example 17

hard
If tan⁡θ=−1\tan\theta = -1 and sin⁡θ>0\sin\theta > 0, find θ\theta in [0,2π)[0, 2\pi).

Example 18

medium
Find the exact coordinates on the unit circle at θ=45°\theta = 45°.

Example 19

challenge
For which angles θ\theta in [0°,360°)[0°,360°) does the unit-circle point satisfy x=yx = y?

Example 20

easy
In which quadrant is the angle θ=210°\theta = 210°?
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