Engineering Handbook/Mathematics/Trigonometry

Trigonometry Identities Table

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 * $$ \sin^2+ \cos^2=1 $$ || $$ 1+ \tan^2 = \sec^2 $$
 * $$ \sin ( \frac{ \pi}{2}- \theta)= \cos \theta $$ || $$ \cos ( \frac{ \pi}{2}- \theta)= \sin \theta $$
 * $$ \sec ( \frac{ \pi}{2}- \theta)= \csc \theta $$ || $$ \csc ( \frac{ \pi}{2}- \theta)= \sec \theta $$
 * $$ \sin (- \theta)=- \sin \theta $$ || $$ \cos (- \theta)= \cos \theta $$
 * $$ \sin 2 \theta = 2 \sin \theta \cos \theta $$ || $$ \cos 2 \theta = \cos^2- \sin^2=2 \cos^2 \theta -1=1-2 \sin^2 \theta $$
 * $$ \sin^2 \theta= \frac{1- \cos 2 \theta}{2} $$ || $$ \cos^2 \theta= \frac{1+ \cos 2 \theta}{2} $$
 * $$ \sin \alpha + \sin \beta = 2 \sin (\frac{ \alpha + \beta}{2}) \cos (\frac{ \alpha - \beta}{2}) $$ || $$ \sin \alpha - \sin \beta = 2 \cos (\frac{ \alpha + \beta}{2}) \sin (\frac{ \alpha - \beta}{2}) $$
 * $$ \cos \alpha + \cos \beta = 2 \cos (\frac{ \alpha + \beta}{2}) \cos (\frac{ \alpha - \beta}{2}) $$ || $$ \cos \alpha - \cos \beta = -2 \sin (\frac{ \alpha + \beta}{2}) \sin (\frac{ \alpha - \beta}{2}) $$
 * $$ \sin \alpha \sin \beta = \frac{1}{2}[\cos( \alpha - \beta) - \cos( \alpha + \beta)] $$ || $$ \cos \alpha \cos \beta = \frac{1}{2}[\cos( \alpha - \beta) + \cos( \alpha + \beta)] $$
 * $$ \sin \alpha \cos \beta = \frac{1}{2}[\sin( \alpha + \beta) + \sin( \alpha - \beta)] $$ || $$ 1+ \cot^2= \csc^2 $$
 * $$ e^{j \theta}= \cos \theta + j \sin \theta $$ || $$ \cos \theta = \frac{e^{j \theta} + e^{-j \theta} } {2} $$
 * $$ \tan ( \frac{ \pi}{2} - \theta)= \cot \theta $$ || $$ \cot ( \frac{ \pi}{2}- \theta)= \tan \theta $$
 * $$ \tan (- \theta)= -\tan \theta $$ || $$ \tan 2 \theta= \frac{2 \tan \theta}{1-tan^2 \theta} $$
 * $$ \tan^2 \theta= \frac{1- \cos 2 \theta}{1+ \cos 2 \theta} $$ || $$ \sin \theta = \frac{e^{j \theta} - e^{-j \theta} } {j2} $$
 * }
 * $$ \sin \alpha \sin \beta = \frac{1}{2}[\cos( \alpha - \beta) - \cos( \alpha + \beta)] $$ || $$ \cos \alpha \cos \beta = \frac{1}{2}[\cos( \alpha - \beta) + \cos( \alpha + \beta)] $$
 * $$ \sin \alpha \cos \beta = \frac{1}{2}[\sin( \alpha + \beta) + \sin( \alpha - \beta)] $$ || $$ 1+ \cot^2= \csc^2 $$
 * $$ e^{j \theta}= \cos \theta + j \sin \theta $$ || $$ \cos \theta = \frac{e^{j \theta} + e^{-j \theta} } {2} $$
 * $$ \tan ( \frac{ \pi}{2} - \theta)= \cot \theta $$ || $$ \cot ( \frac{ \pi}{2}- \theta)= \tan \theta $$
 * $$ \tan (- \theta)= -\tan \theta $$ || $$ \tan 2 \theta= \frac{2 \tan \theta}{1-tan^2 \theta} $$
 * $$ \tan^2 \theta= \frac{1- \cos 2 \theta}{1+ \cos 2 \theta} $$ || $$ \sin \theta = \frac{e^{j \theta} - e^{-j \theta} } {j2} $$
 * }
 * $$ \tan ( \frac{ \pi}{2} - \theta)= \cot \theta $$ || $$ \cot ( \frac{ \pi}{2}- \theta)= \tan \theta $$
 * $$ \tan (- \theta)= -\tan \theta $$ || $$ \tan 2 \theta= \frac{2 \tan \theta}{1-tan^2 \theta} $$
 * $$ \tan^2 \theta= \frac{1- \cos 2 \theta}{1+ \cos 2 \theta} $$ || $$ \sin \theta = \frac{e^{j \theta} - e^{-j \theta} } {j2} $$
 * }
 * $$ \tan^2 \theta= \frac{1- \cos 2 \theta}{1+ \cos 2 \theta} $$ || $$ \sin \theta = \frac{e^{j \theta} - e^{-j \theta} } {j2} $$
 * }