Trigonometric Functions

Definition

$\sin (x)=\frac{e^{ix}-e^{-ix}}{2i}$ $\cos (x)=\frac{e^{ix}+e^{-ix}}{2i}$ $\tan (x)=\frac{\sin (x)}{\cos (x)}=\frac{e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}=\frac{e^{2ix}-1}{e^{2ix}+1}$

Properties

Simple Algebraic Values

radius	angle	$\sin (\theta )$	$\cos (\theta )$	$\tan \theta$
$0$	$0^{\circ }$	$0$	$1$	$0$
$\frac{\pi }{6}$	$30^{\circ }$	$\frac{1}{2}$	$\frac{\sqrt{(}3)}{2}$	$\frac{\sqrt{3}}{3}$
$\frac{\pi }{4}$	$45^{\circ }$	$\frac{\sqrt{(}2)}{2}$	$\frac{\sqrt{(}2)}{2}$	$1$
$\frac{\pi }{3}$	$60^{\circ }$	$\frac{\sqrt{(}3)}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$\frac{\pi }{2}$	$90^{\circ }$	$1$	$0$

Identities

Parity

$\sin (-x)=-\sin (x)$ $\cos (-x)=\cos (x)$ $\tan (-x)=-\tan (x)$ $\csc (-x)=-\csc (x)$ $\sec (-x)=\sec (x)$ $\cot (-x)=-\cot (x)$

Periods

$\sin (x+2k\pi )=\sin (x)$ $\cos (x+2k\pi )=\cos (x)$ $\tan (x+k\pi )=\tan (x)$ $\csc (x+2k\pi )=\csc (x)$ $\sec (x+2k\pi )=\sec (x)$ $\cot (x+k\pi )=\cot (x)$ where $k\in \mathbb{Z}$

Pythagorean Identity

${\sin }^2(x)+{\cos }^2(x)=1$ ${\tan }^2(x)+1={\sec }^2(x)$ ${\cot }^2(x)+1={\csc }^2(x)$

Sum and Difference Formulas

Sum

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

Difference

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

Double-Angle Formulas

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

Half-Angle Formulas

${\sin }^2(\frac{\theta }{2})=\frac{1-\cos (\theta )}{2}$ ${\cos }^2(\frac{\theta }{2})=\frac{1+\cos (\theta )}{2}$ ${\tan }^2(\frac{\theta }{2})=\frac{1-\cos (\theta )}{1+\cos (\theta )}$

Expression using Euler’s Formula

$e^{ix}=\cos (x)+i\sin (x)$ $e^{-ix}=\cos (x)-i\sin (x)$

$\sin (x)=\frac{e^{ix}-e^{-ix}}{2i}$ $\cos (x)=\frac{e^{ix}+e^{-ix}}{2}$

Derivatives and Antiderivatives

$f(x)$	$\frac{d}{dx}f(x)$	$\int f(x)dx$
$\sin (x)$	$\cos (x)$	$-\cos (x)+C$
$\cos (x)$	$-\sin (x)$	$\sin (x)+C$
$\tan (x)$	${\sec }^2(x)$	$-\ln \Vert \cos (x)\Vert +C$
$\csc (x)$	$-\csc (x)\cot (x)$	$\ln \Vert \csc (x)-\cot (x)\Vert +C$
$\sec (x)$	$\sec (x)\tan (x)$	$-\ln \Vert \sec (x)-\tan (x)\Vert +C$
$\cot (x)$	$-{\csc }^2(x)$	$\ln \Vert \sin (x)\Vert +C$
$\arcsin (x)={\sin }^{-1}(x)$	$\frac{1}{\sqrt{1-x^2}}$
$\arccos (x)={\cos }^{-1}(x)$	$-\frac{1}{\sqrt{1-x^2}}$
$\arctan (x)={\tan }^{-1}(x)$	$\frac{1}{x^2+1}$