# Signals and Systems/Table of Trigonometric Identities

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