The Unit Circle: Complete Mastery

By the end of this lesson, you will know all 16 standard angle positions on the unit circle from memory and be able to derive any entry quickly using reference angles and quadrant signs.

Why Memorize the Unit Circle?

Every trig calculation in pre-calculus and calculus begins here. SAT problems, calculus integrals, and physics problems all draw on the same 16 values. Knowing them instantly — without a calculator — is the difference between spending 30 seconds and 3 minutes on every trig problem.

The Complete Unit Circle (Key Angles)

Angle (rad)	Angle (°)	$\cos\theta$	$\sin\theta$
$0$	$0°$	$1$	$0$
$\pi/6$	$30°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\pi/4$	$45°$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$\pi/3$	$60°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\pi/2$	$90°$	$0$	$1$
$2\pi/3$	$120°$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$3\pi/4$	$135°$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$5\pi/6$	$150°$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\pi$	$180°$	$-1$	$0$
$7\pi/6$	$210°$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$
$5\pi/4$	$225°$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$
$4\pi/3$	$240°$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$
$3\pi/2$	$270°$	$0$	$-1$
$5\pi/3$	$300°$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$
$7\pi/4$