Signals and Systems/Table of Trigonometric Identities

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Useful Mathematical Identities[edit]

 \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}