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 -coordinate is and your -coordinate is . Instead of being limited to right triangles, the unit circle lets you define sine and cosine for ANY angle—even angles bigger than 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
mediumFind , , and for using the unit circle. Identify which quadrant and the signs of each.
Example 2
mediumIf and is in Quadrant IV, find .
Example 3
hardWhat is the arc length on the unit circle from angle to ?
Example 4
mediumFind exactly.
Example 5
hardFind all in satisfying .
Example 6
easyFind the unit-circle coordinates at .
Example 7
mediumIf and is in Quadrant II, find .
Example 8
easyWhat is the sign of when is in Quadrant III?
Example 9
easyWhat are the coordinates of the point on the unit circle at ?
Example 10
mediumWhat is the period of on the unit circle?
Example 11
hardUse the unit circle to prove the Pythagorean identity and derive .
Example 12
mediumEvaluate using the unit circle.
Example 13
mediumFind exactly.
Example 14
easyFind using the unit circle.
Example 15
mediumFind from unit-circle coordinates.
Example 16
easyWhat is using the unit circle?
Example 17
hardIf and , find in .
Example 18
mediumFind the exact coordinates on the unit circle at .
Example 19
challengeFor which angles in does the unit-circle point satisfy ?
Example 20
easyIn which quadrant is the angle ?