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 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.

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 θ\theta has coordinates (cos⁡θ,sin⁡θ)(\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⁡\sin and cos⁡\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 (32,12)\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) lies on the unit circle and identify the angle θ\theta.

Answer

Point is on unit circle; θ=π6\theta = \dfrac{\pi}{6} (30°)

First step

1
Check x2+y2=1x^2+y^2=1: (32)2+(12)2=34+14=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

  1. 2
    Identify angle: cos⁡θ=32\cos\theta=\frac{\sqrt{3}}{2} and sin⁡θ=12\sin\theta=\frac{1}{2} (both positive → first quadrant).
  2. 3
    cos⁡θ=32\cos\theta=\frac{\sqrt{3}}{2} and sin⁡θ=12\sin\theta=\frac{1}{2} corresponds to θ=π6\theta=\frac{\pi}{6} (30°).
Every point on the unit circle satisfies x2+y2=1x^2+y^2=1, with x=cos⁡θx=\cos\theta and y=sin⁡θy=\sin\theta. Recognizing standard values of cosine and sine allows immediate identification of the corresponding angle.

Example 2

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 3

medium
Find all θ\theta in [0,2π)[0, 2\pi) with cos⁡θ=−32\cos\theta = -\dfrac{\sqrt{3}}{2}.

Example 4

medium
Find the reference angle for θ=11π6\theta = \dfrac{11\pi}{6}, then compute sin⁡11π6\sin\dfrac{11\pi}{6}.

Example 5

medium
Use the unit circle to find all solutions in [0,2π)[0, 2\pi) of sin⁡θ=22\sin\theta = \dfrac{\sqrt{2}}{2}.

Example 6

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

Example 7

hard
Show that the unit-circle point at angle −θ-\theta has coordinates (cos⁡θ,−sin⁡θ)(\cos\theta, -\sin\theta), and conclude cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta and sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta.

Example 8

hard
Use the unit-circle identity to simplify sin⁡θ cos⁡(−θ)+cos⁡θ sin⁡(−θ)\sin\theta\,\cos(-\theta) + \cos\theta\,\sin(-\theta).

Example 9

challenge
Find all θ∈[0,2π)\theta \in [0, 2\pi) satisfying 2sin⁡2θ−sin⁡θ−1=02\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⁡\sin, cos⁡\cos, and tan⁡\tan for θ=π2\theta=\dfrac{\pi}{2} and θ=π\theta=\pi.

Example 2

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 3

easy
On the unit circle, what are the coordinates of the point at angle θ=0\theta = 0?

Example 4

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

Example 5

easy
What is the radius of the unit circle?

Example 6

easy
Find cos⁡180°\cos 180° using the unit circle.

Example 7

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

Example 8

easy
What is the sign of sin⁡θ\sin\theta when θ\theta is in Quadrant II?

Example 9

easy
Find sin⁡270°\sin 270° using the unit circle.

Example 10

easy
What is cos⁡90°\cos 90°?

Example 11

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

Example 12

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

Example 13

medium
Find sin⁡330°\sin 330° exactly.

Example 14

medium
A point on the unit circle is (−32,12)\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right). Find θ\theta in [0°,360°)[0°,360°).

Example 15

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

Example 16

medium
Find cos⁡240°\cos 240° exactly.

Example 17

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

Example 18

medium
Find the coordinates on the unit circle at θ=300°\theta = 300°.

Example 19

medium
Find tan⁡210°\tan 210° exactly.

Example 20

challenge
Show that the points at θ\theta and θ+180°\theta + 180° are reflections through the origin, and use this to relate cos⁡(θ+180°)\cos(\theta+180°) to cos⁡θ\cos\theta.

Example 21

challenge
A regular hexagon is inscribed in the unit circle with one vertex at (1,0)(1,0). Find the coordinates of the vertex at θ=120°\theta = 120°.

Example 22

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

Example 23

easy
On the unit circle, what are the coordinates at θ=360∘\theta = 360^\circ?

Example 24

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

Example 25

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

Example 26

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

Example 27

easy
Verify that (−12,32)\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) lies on the unit circle.

Example 28

medium
Find sin⁡7π6\sin\dfrac{7\pi}{6} using the unit circle.

Example 29

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

Example 30

medium
Compute cos⁡5π3+sin⁡5π3\cos\dfrac{5\pi}{3}+\sin\dfrac{5\pi}{3}.

Example 31

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

Example 32

medium
Evaluate sin⁡π2+cos⁡π+tan⁡π\sin\dfrac{\pi}{2}+\cos\pi+\tan\pi.

Example 33

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

Example 34

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 35

hard
Compute cos⁡13π6\cos\dfrac{13\pi}{6} by first reducing modulo 2π2\pi.

Example 36

hard
If cos⁡θ=−22\cos\theta = -\dfrac{\sqrt{2}}{2} and tan⁡θ>0\tan\theta > 0, find θ\theta in [0,2π)[0, 2\pi).
← Back to Unit Circle Formula Explained Practice Mode

Background Knowledge

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

trigonometric functionscircles