Unit Circle Examples: 45 Problems with Answers

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: On the unit circle, a point at angle $\theta$ has coordinates $(\cos\theta,\sin\theta)$ , defining trig for every angle.

Common stuck point: The procedure for unit circle is the easy part; the trap is swapping $\sin$ and $\cos$ in the coordinates. Asking "Are you reading sine and cosine of an angle as coordinates on a circle of radius 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are you reading sine and cosine of an angle as coordinates on a circle of radius 1?

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

Answer

Point is on unit circle;

\theta = \dfrac{\pi}{6}

(30°)

First step

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.

Full solution

2
Identify angle: $\cos\theta=\frac{\sqrt{3}}{2}$ and $\sin\theta=\frac{1}{2}$ (both positive → first quadrant).
3
$\cos\theta=\frac{\sqrt{3}}{2}$ and $\sin\theta=\frac{1}{2}$ corresponds to $\theta=\frac{\pi}{6}$ (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.

Example 3

medium

Find all

\theta

[0, 2\pi)

with

\cos\theta = -\dfrac{\sqrt{3}}{2}

Example 4

medium

Find the reference angle for

\theta = \dfrac{11\pi}{6}

, then compute

\sin\dfrac{11\pi}{6}

Example 5

medium

Use the unit circle to find all solutions in

[0, 2\pi)

\sin\theta = \dfrac{\sqrt{2}}{2}

Example 6

hard

Find all

\theta

[0, 2\pi)

satisfying

2\cos^2\theta - 1 = 0

Example 7

hard

Show that the unit-circle point at angle

-\theta

has coordinates

(\cos\theta, -\sin\theta)

, and conclude

\cos(-\theta) = \cos\theta

and

\sin(-\theta) = -\sin\theta

Example 8

hard

Use the unit-circle identity to simplify

\sin\theta\,\cos(-\theta) + \cos\theta\,\sin(-\theta)

Example 9

challenge

Find all

\theta \in [0, 2\pi)

satisfying

2\sin^2\theta - \sin\theta - 1 = 0

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

Example 3

easy

On the unit circle, what are the coordinates of the point at angle

\theta = 0

Example 4

easy

What are the coordinates of the point on the unit circle at

\theta = 90°

Example 5

easy

What is the radius of the unit circle?

Example 6

easy

Find

\cos 180°

using the unit circle.

Example 7

easy

In which quadrant is the angle

\theta = 210°

Example 8

easy

What is the sign of

\sin\theta

when

\theta

is in Quadrant II?

Example 9

easy

Find

\sin 270°

using the unit circle.

Example 10

easy

What is

\cos 90°

Example 11

medium

Find the exact coordinates on the unit circle at

\theta = 45°

Example 12

medium

Find

\cos 120°

exactly.

Example 13

medium

Find

\sin 330°

exactly.

Example 14

medium

A point on the unit circle is

\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

. Find

\theta

[0°,360°)

Example 15

medium

Find

\tan 135°

exactly.

Example 16

medium

Find

\cos 240°

exactly.

Example 17

medium

\cos\theta = \frac{3}{5}

and

\theta

is in Quadrant IV, find

\sin\theta

Example 18

medium

Find the coordinates on the unit circle at

\theta = 300°

Example 19

medium

Find

\tan 210°

exactly.

Example 20

challenge

Show that the points at

\theta

and

\theta + 180°

are reflections through the origin, and use this to relate

\cos(\theta+180°)

\cos\theta

Example 21

challenge

A regular hexagon is inscribed in the unit circle with one vertex at

(1,0)

. Find the coordinates of the vertex at

\theta = 120°

Example 22

challenge

For which angles

\theta

[0°,360°)

does the unit-circle point satisfy

x = y

Example 23

easy

On the unit circle, what are the coordinates at

\theta = 360^\circ

Example 24

easy

Find the unit-circle coordinates at

\theta = \dfrac{\pi}{4}

Example 25

easy

What is the sign of

\cos\theta

when

\theta

is in Quadrant III?

Example 26

easy

Find

\tan 0

using the unit circle.

Example 27

easy

Verify that

\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)

lies on the unit circle.

Example 28

medium

Find

\sin\dfrac{7\pi}{6}

using the unit circle.

Example 29

medium

Evaluate

\tan\dfrac{2\pi}{3}

using the unit circle.

Example 30

medium

Compute

\cos\dfrac{5\pi}{3}+\sin\dfrac{5\pi}{3}

Example 31

medium

\sin\theta = \dfrac{3}{5}

and

\theta

is in Quadrant II, find

\cos\theta

Example 32

medium

Evaluate

\sin\dfrac{\pi}{2}+\cos\pi+\tan\pi

Example 33

medium

Find

\sec\dfrac{\pi}{3}

from unit-circle coordinates.

Example 34

hard

\tan\theta = -1

and

\sin\theta > 0

, find

\theta

[0, 2\pi)

Example 35

hard

Compute

\cos\dfrac{13\pi}{6}

by first reducing modulo

2\pi

Example 36

hard

\cos\theta = -\dfrac{\sqrt{2}}{2}

and

\tan\theta > 0

, find

\theta

[0, 2\pi)

← Back to Unit Circle Formula Explained Practice Mode

Related Concepts

Trigonometric Functions Circles Radian Measure Pythagorean Trigonometric Identities Trigonometric Function Graphs

Background Knowledge

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

trigonometric functionscircles