Trigonometric Functions

Right Triangle Definitions

For a right triangle with angle $\theta$, opposite side (opp), adjacent side (adj), and hypotenuse (hyp):

Function	Definition	Abbreviation
Sine	$\sin\theta = \frac{\text{opp}}{\text{hyp}}$	SOH
Cosine	$\cos\theta = \frac{\text{adj}}{\text{hyp}}$	CAH
Tangent	$\tan\theta = \frac{\text{opp}}{\text{adj}}$	TOA
Cosecant	$\csc\theta = \frac{\text{hyp}}{\text{opp}}$
Secant	$\sec\theta = \frac{\text{hyp}}{\text{adj}}$
Cotangent	$\cot\theta = \frac{\text{adj}}{\text{opp}}$

Memory Aid: SOH-CAH-TOA

Unit Circle Definitions

For a point $(x, y)$ on the unit circle at angle $\theta$ from the positive $x$-axis:

Function	Definition
$\sin\theta$	$y$
$\cos\theta$	$x$
$\tan\theta$	$\frac{y}{x}$
$\csc\theta$	$\frac{1}{y}$
$\sec\theta$	$\frac{1}{x}$
$\cot\theta$	$\frac{x}{y}$

Reciprocal Relationships

$\csc\theta = \frac{1}{\sin\theta}$

$\sec\theta = \frac{1}{\cos\theta}$

$\cot\theta = \frac{1}{\tan\theta}$

Quotient Relationships

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

$\cot\theta = \frac{\cos\theta}{\sin\theta}$

Domains and Ranges

Function	Domain	Range
$\sin\theta$	All real numbers	$[-1, 1]$
$\cos\theta$	All real numbers	$[-1, 1]$
$\tan\theta$	$\theta \neq \frac{\pi}{2} + n\pi$	All real numbers
$\csc\theta$	$\theta \neq n\pi$	$(-\infty, -1] \cup [1, \infty)$
$\sec\theta$	$\theta \neq \frac{\pi}{2} + n\pi$	$(-\infty, -1] \cup [1, \infty)$
$\cot\theta$	$\theta \neq n\pi$	All real numbers

Signs by Quadrant

Quadrant	$\sin$	$\cos$	$\tan$
I ($0°$ to $90°$)	$+$	$+$	$+$
II ($90°$ to $180°$)	$+$	$-$	$-$
III ($180°$ to $270°$)	$-$	$-$	$+$
IV ($270°$ to $360°$)	$-$	$+$	$-$

Memory Aid: "All Students Take Calculus"

• All positive in Quadrant I
• Sine positive in Quadrant II
• Tangent positive in Quadrant III
• Cosine positive in Quadrant IV

Periods

Function	Period
$\sin\theta$, $\cos\theta$	$2\pi$
$\tan\theta$, $\cot\theta$	$\pi$
$\csc\theta$, $\sec\theta$	$2\pi$

Even and Odd Functions

Even (symmetric about y-axis):

$\cos(-\theta) = \cos\theta$

$\sec(-\theta) = \sec\theta$

Odd (symmetric about origin):

$\sin(-\theta) = -\sin\theta$

$\tan(-\theta) = -\tan\theta$

$\csc(-\theta) = -\csc\theta$

$\cot(-\theta) = -\cot\theta$

Cofunction Relationships

$\sin\theta = \cos(90° - \theta)$

$\cos\theta = \sin(90° - \theta)$

$\tan\theta = \cot(90° - \theta)$

$\cot\theta = \tan(90° - \theta)$

$\sec\theta = \csc(90° - \theta)$

$\csc\theta = \sec(90° - \theta)$