Signals and Systems/Table of Trigonometric Identities

Current revision (unreviewed)

Signals and Systems

[edit] Useful Mathematical Identities

$sin 2 + cos 2 = 1$	$1 + tan 2 = sec 2$
$\sin ( \frac{ \pi}{2}- \theta)= \cos \theta$	$\cos ( \frac{ \pi}{2}- \theta)= \sin \theta$
$\sec ( \frac{ \pi}{2}- \theta)= \csc \theta$	$\csc ( \frac{ \pi}{2}- \theta)= \sec \theta$
$sin( - θ) = - sinθ$	$cos( - θ) = sinθ$
$sin2θ = 2sinθcosθ$	$cos2θ = cos 2 - sin 2 = 2cos 2 θ - 1 = 1 - 2sin 2 θ$
$\sin^2 \theta= \frac{1- \cos 2 \theta}{2}$	$\cos^2 \theta= \frac{1+ \cos 2 \theta}{2}$
$\sin \alpha + \sin \beta = 2 \sin (\frac{ \alpha + \beta}{2}) \cos (\frac{ \alpha - \beta}{2})$	$\sin \alpha - \sin \beta = 2 \cos (\frac{ \alpha + \beta}{2}) \sin (\frac{ \alpha - \beta}{2})$
$\cos \alpha + \cos \beta = 2 \cos (\frac{ \alpha + \beta}{2}) \cos (\frac{ \alpha - \beta}{2})$	$\cos \alpha - \cos \beta = -2 \sin (\frac{ \alpha + \beta}{2}) \sin (\frac{ \alpha - \beta}{2})$
$\sin \alpha \sin \beta = \frac{1}{2}[\cos( \alpha - \beta) - \cos( \alpha + \beta)]$	$\cos \alpha \cos \beta = \frac{1}{2}[\cos( \alpha - \beta) + \cos( \alpha + \beta)]$
$\sin \alpha \cos \beta = \frac{1}{2}[\sin( \alpha + \beta) + \sin( \alpha - \beta)]$	$1 + cot 2 = csc 2$
$e j θ = cosθ + j sinθ$	$\cos \theta = \frac{e^{j \theta} + e^{-j \theta} } {2}$
$\tan ( \frac{ \pi}{2} - \theta)= \cot \theta$	$\cot ( \frac{ \pi}{2}- \theta)= \tan \theta$
$tan( - θ) = cotθ$	$\tan 2 \theta= \frac{2 \tan \theta}{1-tan^2 \theta}$
$\tan^2 \theta= \frac{1- \cos 2 \theta}{1+ \cos 2 \theta}$	$\sin \theta = \frac{e^{j \theta} - e^{-j \theta} } {j2}$