Exact Values of Trig Functions

First Quadrant (0° to 90°)

Common Angles Table

Degrees	Radians	$\sin$	$\cos$	$\tan$
$0°$	$0$	$0$	$1$	$0$
$30°$	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45°$	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90°$	$\frac{\pi}{2}$	$1$	$0$	undefined

Memory Pattern for Sine

For $0°, 30°, 45°, 60°, 90°$ :

$\sin = \frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2} = 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$

Memory Pattern for Cosine

Reverse of sine pattern:

$\cos = \frac{\sqrt{4}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{0}}{2} = 1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0$

Complete Unit Circle Values

Second Quadrant (90° to 180°)

Degrees	Radians	$\sin$	$\cos$	$\tan$
$120°$	$\frac{2\pi}{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$
$135°$	$\frac{3\pi}{4}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$
$150°$	$\frac{5\pi}{6}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
$180°$	$\pi$	$0$	$-1$	$0$

Third Quadrant (180° to 270°)

Degrees	Radians	$\sin$	$\cos$	$\tan$
$210°$	$\frac{7\pi}{6}$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$225°$	$\frac{5\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$1$
$240°$	$\frac{4\pi}{3}$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$\sqrt{3}$
$270°$	$\frac{3\pi}{2}$	$-1$	$0$	undefined

Fourth Quadrant (270° to 360°)

Degrees	Radians	$\sin$	$\cos$	$\tan$
$300°$	$\frac{5\pi}{3}$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$-\sqrt{3}$
$315°$	$\frac{7\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$-1$
$330°$	$\frac{11\pi}{6}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
$360°$	$2\pi$	$0$	$1$	$0$

Reciprocal Functions

$\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
$0°$	undef	$1$	undef
$30°$	$2$	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
$45°$	$\sqrt{2}$	$\sqrt{2}$	$1$
$60°$	$\frac{2\sqrt{3}}{3}$	$2$	$\frac{\sqrt{3}}{3}$
$90°$	$1$	undef	$0$

Using Reference Angles

To find exact values for any angle:

Find reference angle (acute angle to x-axis)
Evaluate at reference angle using table
Determine sign based on quadrant

Example: Find $\sin 240°$

Reference angle: $240° - 180° = 60°$
$\sin 60° = \frac{\sqrt{3}}{2}$
Quadrant III: sin is negative
$\sin 240° = -\frac{\sqrt{3}}{2}$

Exact Values Summary (Radians)

$\theta$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$\pi$
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{3}}{2}$	$-1$

Decimal Approximations

Value	Decimal
$\frac{\sqrt{2}}{2}$	$\approx 0.707$
$\frac{\sqrt{3}}{2}$	$\approx 0.866$
$\frac{\sqrt{3}}{3}$	$\approx 0.577$
$\sqrt{3}$	$\approx 1.732$
$\frac{1}{2}$	$= 0.5$