Signals and Systems/Table of Trigonometric Identities

Useful Mathematical Identities

sin 2θ + cos 2θ = 1	1 + tan 2θ = sec 2θ
$\sin ( \frac{ \pi}{2}- \theta)= \cos \theta$	$\cos ( \frac{ \pi}{2}- \theta)= \sin \theta$
$\sec ( \frac{ \pi}{2}- \theta)= \csc \theta$	$\csc ( \frac{ \pi}{2}- \theta)= \sec \theta$
sin( − θ) = − sin θ	cos( − θ) = sin θ
sin 2θ = 2sin θcos θ	cos 2θ = cos 2 − sin 2 = 2cos 2θ − 1 = 1 − 2sin 2θ
$\sin^2 \theta= \frac{1- \cos 2 \theta}{2}$	$\cos^2 \theta= \frac{1+ \cos 2 \theta}{2}$
$\sin \alpha + \sin \beta = 2 \sin (\frac{ \alpha + \beta}{2}) \cos (\frac{ \alpha - \beta}{2})$	$\sin \alpha - \sin \beta = 2 \cos (\frac{ \alpha + \beta}{2}) \sin (\frac{ \alpha - \beta}{2})$
$\cos \alpha + \cos \beta = 2 \cos (\frac{ \alpha + \beta}{2}) \cos (\frac{ \alpha - \beta}{2})$	$\cos \alpha - \cos \beta = -2 \sin (\frac{ \alpha + \beta}{2}) \sin (\frac{ \alpha - \beta}{2})$
$\sin \alpha \sin \beta = \frac{1}{2}[\cos( \alpha - \beta) - \cos( \alpha + \beta)]$	$\cos \alpha \cos \beta = \frac{1}{2}[\cos( \alpha - \beta) + \cos( \alpha + \beta)]$
$\sin \alpha \cos \beta = \frac{1}{2}[\sin( \alpha + \beta) + \sin( \alpha - \beta)]$	1 + cot 2 = csc 2
ejθ = cos θ + jsin θ	$\cos \theta = \frac{e^{j \theta} + e^{-j \theta} } {2}$
e − jθ = cos θ − jsin θ	$\sin \theta = \frac{e^{j \theta} - e^{-j \theta}} {2j}$
$\tan ( \frac{ \pi}{2} - \theta)= \cot \theta$	$\cot ( \frac{ \pi}{2}- \theta)= \tan \theta$
tan( − θ) = − tan θ
$\tan^2 \theta= \frac{1- \cos 2 \theta}{1+ \cos 2 \theta}$	$\tan 2 \theta= \frac{2 \tan \theta}{1-tan^2 \theta}$