Practice Unit Circle in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
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 x-coordinate is \cos\theta and your y-coordinate is \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° 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.
Example 1
easyVerify that the point \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) lies on the unit circle and identify the angle \theta.
Example 2
mediumFind \sin, \cos, and \tan for \theta=\dfrac{3\pi}{4} using the unit circle. Identify which quadrant and the signs of each.
Example 3
easyUsing the unit circle, find the exact values of \sin, \cos, and \tan for \theta=\dfrac{\pi}{2} and \theta=\pi.
Example 4
hardUse the unit circle to prove the Pythagorean identity \sin^2\theta+\cos^2\theta=1 and derive 1+\tan^2\theta=\sec^2\theta.