Trigonometry

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Definitions of Functions

$\sin \theta =\frac{y}{r}=\frac{\text{opposite}}{\text{hypotenuse}}$	$\csc \theta =\frac{r}{y}=\frac{\text{hypotenuse}}{\text{opposite}}$
$\cos \theta =\frac{x}{r}=\frac{\text{adjacent}}{\text{hypotenuse}}$	$\sec \theta =\frac{r}{x}=\frac{\text{hypotenuse}}{\text{adjacent}}$
$\tan \theta =\frac{y}{x}=\frac{\text{opposite}}{\text{adjacent}}$	$\cot \theta =\frac{x}{y}=\frac{\text{adjacent}}{\text{opposite}}$

$\sin \theta$ is an odd function: $\sin (-\theta )=-\sin \theta$
$\cos \theta$ is an even function: $\cos (-\theta )=\cos \theta$

Both the $\sin \theta$ and $\cos \theta$ functions are between -1 and 1.

Basic/Core Identities

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

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

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

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

${\sin }^2\theta +{\cos }^2\theta =1$

${\tan }^2\theta +1={\sec }^2\theta$

$1+{\cot }^2\theta ={\csc }^2\theta$

Important Limits

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$

$\underset{x\to 0}{\lim }\frac{\cos x-1}{x}=0$

Addition & Subtraction Formulas

Addition	Subtraction
$\sin (x+y)=\sin x\cos y+\cos x\sin y$	$\sin (x-y)=\sin x\cos y-\cos x\sin y$
$\cos (x+y)=\cos x\cos y-\sin x\sin y$	$\cos (x-y)=\cos x\cos y+\sin x\sin y$
$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}$	$\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x\tan y}$

Double-Angle Formulas

Derived by evaluating $\sin (x+x)$ and $\cos (x+x)$. The second and third identities for $\cos 2x$ are derived by substituting ${\sin }^{2}x=1-{\cos }^{2}x$ and ${\cos }^{2}x=1-{\sin }^{2}x$ based on the basic Pythagorean identities.

$\sin 2x=2\sin x\cos x$	$\cos 2x=co{s}^{2}x-si{n}^{2}x$
	$\cos 2x=2{\cos }^{}x-1$
	$\cos 2x=1-2{\sin }^{}x$

Half-Angle Formulas

Derived by solving the second and third cosine double angle ($\cos 2x$) formulas for ${\sin }^{2}x$ and ${\cos }^{2}x$.

${\sin }^{2}x=\frac{1-\cos 2x}{2}$

${\cos }^{2}x=\frac{1+\cos 2x}{2}$

Product Formulas

Formula for $\sin x\cos y$ derived by adding the sine addition and subtraction equations and solving for $\sin x\cos x$. Formulas for $\cos x\cos y$ and $\sin x\sin y$ derived by adding the cosine addition and subtraction equations and solving for $\cos x\cos y$ and $\sin x\sin y$, respectively.

$\sin x\sin y=\frac{1}{2}[\cos (x-y)-\cos (x+y)]$	$\cos x\cos y=\frac{1}{2}[\cos (x+y)+\cos (x-y)]$
$\sin x\cos y=\frac{1}{2}[\sin (x+y)+\sin (x-y)]$

Derivatives of Inverse Functions

$\frac{d}{dx}\left({\sin }^{-1}x\right)=\frac{1}{\sqrt{1-x^2}}$	$\frac{d}{dx}\left({\cos }^{-1}x\right)=-\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\left({\sec }^{-1}x\right)=\frac{1}{x\sqrt{x^2-1}}$	$\frac{d}{dx}\left({\csc }^{-1}x\right)=-\frac{1}{x\sqrt{x^2-1}}$
$\frac{d}{dx}\left({\tan }^{-1}x\right)=\frac{1}{1+x^2}$	$\frac{d}{dx}\left({\cot }^{-1}x\right)=-\frac{1}{1+x^2}$

Integrals

The table below lists the integrals of the six basic trig functions and their squares.

$\int \sin xdx=-\cos x+C$	$\int {\sin }^{2}xdx=\int \frac{1}{2}(1-\cos \left(2x\right))dx=\frac{1}{2}(x-\frac{1}{2}\sin 2x)+C$
$\int \cos xdx=\sin x+C$	$\int {\cos }^{2}xdx=\int \frac{1}{2}(1+\cos \left(2x\right))dx=\frac{1}{2}(x+\frac{1}{2}\sin 2x)+C$
$\int \tan xdx=\ln |\sec x|+C$	$\int {\tan }^{2}xdx=\int {\sec }^{2}x-1dx=\tan x-x+C$
$\int \cot xdx=\ln |\sin x|+C$	$\int {\cot }^{2}xdx=\int {\csc }^{2}x-1dx=-\cot x-x+C$
$\int \sec xdx=\ln |\sec x+\tan x|+C$	$\int {\sec }^{2}xdx=\tan x+C$
$\int \csc xdx=\ln |\csc x-\cot x|+C$	$\int {\csc }^{2}xdx=-\cot x+C$

Derivatives

$\frac{d}{dx}\sin x=\cos x$	$\frac{d}{dx}\csc x=-\csc x\cot x$
$\frac{d}{dx}\cos x=-\sin x$	$\frac{d}{dx}\sec x=\sec x\tan x$
$\frac{d}{dx}\tan x={\sec }^{}x$	$\frac{d}{dx}\cot x=-{\csc }^{}x$