Unit Circle Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Unit Circle.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every angle corresponds to a unique point on the unit circle, and the coordinates of that point ARE the cosine and sine values.

Common stuck point: Students often try to memorize coordinates without understanding the pattern. Focus on the reference angle approach: find the angle's position in its quadrant, then assign the correct signs based on which quadrant you're in.

Sense of Study hint: Sketch the unit circle, mark the angle, and drop a vertical line to the x-axis. The legs of that right triangle give you cos and sin.

Worked Examples

Example 1

easy
Verify that the point \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) lies on the unit circle and identify the angle \theta.

Solution

  1. 1
    Check x^2+y^2=1: \left(\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2 = \frac{3}{4}+\frac{1}{4}=1. ✓ On the unit circle.
  2. 2
    Identify angle: \cos\theta=\frac{\sqrt{3}}{2} and \sin\theta=\frac{1}{2} (both positive → first quadrant).
  3. 3
    \cos\theta=\frac{\sqrt{3}}{2} and \sin\theta=\frac{1}{2} corresponds to \theta=\frac{\pi}{6} (30°).

Answer

Point is on unit circle; \theta = \dfrac{\pi}{6} (30°)
Every point on the unit circle satisfies x^2+y^2=1, with x=\cos\theta and y=\sin\theta. Recognizing standard values of cosine and sine allows immediate identification of the corresponding angle.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Using the unit circle, find the exact values of \sin, \cos, and \tan for \theta=\dfrac{\pi}{2} and \theta=\pi.

Example 2

hard
Use the unit circle to prove the Pythagorean identity \sin^2\theta+\cos^2\theta=1 and derive 1+\tan^2\theta=\sec^2\theta.
← Back to Unit Circle Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

trigonometric functionscircles