The Unit Circle

The unit circle has radius 1 centered at the origin. A point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$ .

The unit circle: radius = 1, center at origin

Key angles (memorize!):

$\theta$	$\cos\theta$	$\sin\theta$
$0°$	$1$	$0$
$30°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$45°$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$60°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$90°$	$0$	$1$
$180°$	$-1$	$0$
$270°$	$0$	$-1$
$360°$	$1$	$0$

45° angle: sin(45°) = √2/2, cos(45°) = √2/2

In other quadrants, use reference angles and the sign rules (ASTC: All Students Take Calculus $\to$ which trig functions are positive in each quadrant).

Example 1

Find $\sin(150°)$ and $\cos(150°)$ .

Reference angle: $180° - 150° = 30°$

Quadrant II: sin is positive, cos is negative

$\sin(150°) = \sin(30°) = \frac{1}{2}$

$\cos(150°) = -\cos(30°) = -\frac{\sqrt{3}}{2}$

\sin(150°) = \frac{1}{2}

\cos(150°) = -\frac{\sqrt{3}}{2}

Theory

Key Trig Identities

Pythagorean: $\sin^2\theta + \cos^2\theta = 1$

Reciprocal:
- $\sec\theta = \frac{1}{\cos\theta}$
- $\csc\theta = \frac{1}{\sin\theta}$
- $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

Quotient: $\tan\theta = \frac{\sin\theta}{\cos\theta}$

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