16. Relation to the complex exponential function

 (Euler's formula),

${\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x}$

${\displaystyle e^{i\pi }=-1}$ (Euler's identity),

${\displaystyle e^{2\pi i}=1}$

${\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}$

${\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}$

and hence the corollary:

${\displaystyle \tan x={\frac {\sin x}{\cos x}}={\frac {e^{ix}-e^{-ix}}{i({e^{ix}+e^{-ix}})}}}$

where i² = −1.

Mastering Trigonometry: The Best Self-Study Guide and Essential Formulas on Amazon