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Math notes/Pre-Calculus/Trigonometry

The Unit Circle — Cheat sheet

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Read them off the circle

Angle	Radians	Point	$\cos$	$\sin$	$\tan$	$\cot$
$0^\circ$	$0$	$(1, 0)$	$1$	$0$	$0$	undefined
$90^\circ$	$\dfrac{\pi}{2}$	$(0, 1)$	$0$	$1$	undefined	$0$
$180^\circ$	$\pi$	$(-1, 0)$	$-1$	$0$	$0$	undefined
$270^\circ$	$\dfrac{3\pi}{2}$	$(0, -1)$	$0$	$-1$	undefined	$0$
$360^\circ$	$2\pi$	$(1, 0)$	$1$	$0$	$0$	undefined

Tangent and cotangent break in opposite places

$\tan x = \dfrac{\sin x}{\cos x}$ divides by the cosine. $\cot x = \dfrac{\cos x}{\sin x}$ divides by the sine.

At	$\tan$	$\cot$
$0^\circ$ , $180^\circ$ , $360^\circ$	$0$	undefined
$90^\circ$ , $270^\circ$	undefined	$0$

A zero on the top gives $0$ . A zero on the bottom gives nothing at all.

Why it matters later

Fact	Consequence
$\cos x = 0$ at $90^\circ$ , $270^\circ$	tangent is undefined there - its asymptotes
$\sin x = 0$ at $0^\circ$ , $180^\circ$ , $360^\circ$	cotangent is undefined there instead
$\cos x$ is the first coordinate	no triangle needed on an axis
$360^\circ$ returns to $0^\circ$	periodicity: $\sin(x + 2\pi) = \sin x$