\begin {align*} \cos \left (i\theta \right ) & =\cosh \left (\theta \right ) \\ \sin \left (i\theta \right ) & =i\sinh \left (\theta \right ) \end {align*}
\begin {align*} \cos ^{2}\left (\theta \right ) +\sin ^{2}\left (\theta \right ) & =1\\ \tan ^{2}\left (\theta \right ) & =\frac {1}{\cos ^{2}\left (\theta \right ) }-1\\ & =\sec ^{2}\left (\theta \right ) -1\\ \frac {\cos ^{2}\left (\theta \right ) }{\sin ^{2}\left (\theta \right ) }+1 & =\frac {1}{\sin ^{2}\left (\theta \right ) }\\ \frac {1}{\tan ^{2}\left (\theta \right ) } & =\frac {1}{\sin ^{2}\left ( \theta \right ) }-1\\ \cot ^{2}\left (\theta \right ) & =\csc ^{2}\left (\theta \right ) -1\\ \cosh ^{2}\left (\theta \right ) -\sinh ^{2}\left (\theta \right ) & =1 \end {align*}
\begin {align*} \sin \left (2\theta \right ) & =2\sin \left (\theta \right ) \cos \left ( \theta \right ) \\ \cos \left (2\theta \right ) & =\cos ^{2}\left (\theta \right ) -\sin ^{2}\left (\theta \right ) \\ & =2\cos ^{2}\left (\theta \right ) -1\\ & =1-2\sin ^{2}\left (\theta \right ) \\ \tan \left (2\theta \right ) & =\frac {2\tan \left (\theta \right ) }{1-\tan ^{2}\left (\theta \right ) }\\ \sinh \left (2\theta \right ) & =2\sinh \left (\theta \right ) \cosh \left ( \theta \right ) \\ \cosh \left (2\theta \right ) & =2\cosh ^{2}\left (\theta \right ) -1\\ \tanh \left (2\theta \right ) & =\frac {2\tanh \left (\theta \right ) }{1+\tanh ^{2}\left (\theta \right ) } \end {align*}
\begin {align*} \sin \left (\theta \right ) & =\cos \left (\frac {\pi }{2}-\theta \right ) \\ \cos \left (\theta \right ) & =\sin \left (\frac {\pi }{2}-\theta \right ) \end {align*}
\begin {align*} \sin \left (A+B\right ) & =\sin A\cos B+\cos A\sin B\\ \sin \left (A-B\right ) & =\sin A\cos B-\cos A\sin B\\ \cos \left (A+B\right ) & =\cos A\cos B-\sin A\sin B\\ \cos \left (A-B\right ) & =\cos A\cos B+\sin A\sin B\\ \tan \left (A+B\right ) & =\frac {\tan A+\tan B}{1-\tan A\tan B}\\ \tan \left (A-B\right ) & =\frac {\tan A+\tan B}{1+\tan A\tan B} \end {align*}
\begin {align*} \sin ^{2}\left (\theta \right ) & =\frac {1}{2}\left (1-\cos \left ( 2\theta \right ) \right ) \\ \cos ^{2}\left (\theta \right ) & =\frac {1}{2}\left (1+\cos \left ( 2\theta \right ) \right ) \\ \tan ^{2}\left (\theta \right ) & =\frac {1-\cos \left (2\theta \right ) }{1+\cos \left (2\theta \right ) } \end {align*}
\begin {align*} \sin A+\sin B & =2\sin \left (\frac {A+B}{2}\right ) \cos \left (\frac {A-B}{2}\right ) \\ \sin A-\sin B & =2\sin \left (\frac {A-B}{2}\right ) \cos \left (\frac {A+B}{2}\right ) \\ \cos A+\cos B & =2\cos \left (\frac {A+B}{2}\right ) \cos \left (\frac {A-B}{2}\right ) \\ \cos A-\cos B & =-2\sin \left (\frac {A+B}{2}\right ) \sin \left (\frac {A-B}{2}\right ) \end {align*}
\begin {align*} \sin A\sin B & =\frac {1}{2}\left (\cos \left (A-B\right ) -\cos \left ( A+B\right ) \right ) \\ \cos A\cos B & =\frac {1}{2}\left (\cos \left (A-B\right ) +\cos \left ( A+B\right ) \right ) \\ \sin A\cos B & =\frac {1}{2}\left (\sin \left (A+B\right ) +\sin \left ( A-B\right ) \right ) \\ \cos A\sin B & =\frac {1}{2}\left (\sin \left (A+B\right ) -\sin \left ( A-B\right ) \right ) \end {align*}
\begin {align*} a\cos \left (\omega t\right ) +b\sin \left (\omega t\right ) & =A\sin \left ( \omega t+\phi \right ) \\ & =A\cos \left (\omega t-\phi \right ) \\ A & =\sqrt {a^{2}+b^{2}}\\ \phi & =\arctan \left (\frac {B}{A}\right ) \\ \cos x+\sin x & =\sqrt {2}\sin \left (x+\frac {\pi }{4}\right ) \\ \cos x+\sin x & =\sqrt {2}\cos \left (x-\frac {\pi }{4}\right ) \end {align*}
Laws of sines (\(a,b,c\)) are lengths of triangle sides and \(A,B,C\) are facing angles.\[ \frac {a}{\sin A}=\frac {b}{\sin B}=\frac {c}{\sin C}\] laws of cosine\[ a^{2}=b^{2}+c^{2}-2bc\cos A \]
\begin {align*} \Gamma \relax (n) & =\left (n-1\right ) !\\ \Gamma \left (n+1\right ) & =n\left (n-1\right ) !\\ & =n\Gamma \relax (n) \end {align*}
For \(n\gg 1\)\[ \Gamma \left (n+1\right ) =n!=\sqrt {2\pi }n^{n+\frac {1}{2}}e^{-n}\]