Complex Trigonometric Identities and Functions

• $sin(a+ib)=sinacosh\text{ }b+icosasinhb$ where $sinhb=\frac{e^b-e^{-b}}{2}$ and $coshb=\frac{e^b+e^{-b}}{2}$
• $cos\text{ }z=\frac{e^{iz}+e^{-iz}}{2}$
• $sin\text{ }z=\frac{e^{iz}-e^{-iz}}{2}$
• $tanz=\frac{1}{i}\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}$

Hyperbolic

• $sin\text{ }ib=sinhb$
• $cos\text{ }ib=coshb$

Arc

• $arctanz=\sum \_{n=0}^{\infty }(-1{)}^n\frac{z^{2n+1}}{2n+1}$
• $arcsinz=-iln(iz+\sqrt{1-z^2})$

Special Cases

• When $w=tanz$: $arctanw=\frac{1}{2i}ln(\frac{i-w}{i+w})$