The Unit Circle
The Unit Circle
The unit circle is a circle of radius 1 centered at the origin, and it extends trigonometry beyond acute angles to any angle — positive or negative, large or small. On the unit circle, the cosine of an angle is the x-coordinate and the sine is the y-coordinate of the corresponding point.
Memorizing the key angles — 0°, 30°, 45°, 60°, 90° and their radian equivalents — along with their sine and cosine values is one of the most valuable investments you can make in your mathematical education.
This lesson covers the unit circle, reference angles, signs of trigonometric functions in each quadrant, and the exact values of sine and cosine at standard angles.
The unit circle is a circle of radius 1 centered at the origin. It extends trigonometric functions to all angles, not just acute ones.
For any angle $\theta$ in standard position, let $(x, y)$ be the point where the terminal side intersects the unit circle. Then:
$$\cos\theta = x \qquad \sin\theta = y \qquad \tan\theta = \frac{y}{x}$$
Key Unit Circle Values
| $\theta$ (degrees) | $\theta$ (radians) | $\cos\theta$ | $\sin\theta$ |
|---|---|---|---|
| $0°$ | $0$ | $1$ | $0$ |
| $30°$ | $\frac{\pi}{6}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ |
| $45°$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ |
| $60°$ | $\frac{\pi}{3}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ |
| $90°$ | $\frac{\pi}{2}$ | $0$ | $1$ |
| $120°$ | $\frac{2\pi}{3}$ | $-\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ |
| $135°$ | $\frac{3\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ |
| $150°$ | $\frac{5\pi}{6}$ | $-\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ |
| $180°$ | $\pi$ | $-1$ | $0$ |
| $210°$ | $\frac{7\pi}{6}$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ |
| $225°$ | $\frac{5\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ |
| $240°$ | $\frac{4\pi}{3}$ | $-\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ |
| $270°$ | $\frac{3\pi}{2}$ | $0$ | $-1$ |
| $300°$ | $\frac{5\pi}{3}$ | $\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ |
| $315°$ | $\frac{7\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ |
| $330°$ | $\frac{11\pi}{6}$ | $\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ |
Signs by Quadrant
| Quadrant | $\sin$ | $\cos$ | $\tan$ | Memory |
|---|---|---|---|---|
| I (0° – 90°) | $+$ | $+$ | $+$ | All positive |
| II (90° – 180°) | $+$ | $-$ | $-$ | Sine positive |
| III (180° – 270°) | $-$ | $-$ | $+$ | Tangent positive |
| IV (270° – 360°) | $-$ | $+$ | $-$ | Cosine positive |
Reference Angles
The reference angle $\theta_R$ is the acute angle formed between the terminal side and the $x$-axis.
Find the exact value of $\sin 150°$.
Reference angle: $180° - 150° = 30°$. QII → sine is positive.
$$\sin 150° = +\sin 30° = \frac{1}{2}$$
Find $\cos\dfrac{5\pi}{4}$.
$\frac{5\pi}{4}$ is in QIII. Reference angle: $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$. Cosine is negative in QIII.
$$\cos\frac{5\pi}{4} = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$
Find $\tan 300°$.
QIV. Reference angle: $360° - 300° = 60°$. Tangent is negative in QIV.
$$\tan 300° = -\tan 60° = -\sqrt{3}$$
Practice Problems
Find the exact value.
Show Answer Key
1. $-\frac{1}{2}$
2. $\frac{\sqrt{2}}{2}$
3. $-\sqrt{3}$
4. $\frac{\sqrt{3}}{2}$
5. $-\frac{\sqrt{3}}{2}$
6. $-\sqrt{3}$
7. $-1$
8. $-1$
9. $\sec 240° = \frac{1}{\cos 240°} = \frac{1}{-1/2} = -2$
10. $\frac{1}{\sin(\pi/6)} = \frac{1}{1/2} = 2$
11. $\cot 135° = \frac{1}{\tan 135°} = \frac{1}{-1} = -1$
12. $250° - 180° = 70°$
13. $-\frac{1}{2}$ (sine is odd)
14. $\frac{\sqrt{2}}{2}$ (cosine is even)
15. $1$