DeMoivre's (Euler's) Theorem etc

1.

DeMoivre's theorem:

$$e^{\imath \alpha} = \cos \alpha + \imath \sin \alpha$$

obtained by Taylor expansion of both sides.

2.

so that

$$\sin \alpha = \frac{e^{\imath \alpha} - e^{-\imath \alpha}}{2\imath}$$

and

$$\cos \alpha = \frac{e^{\imath \alpha} + e^{-\imath \alpha}}{2}$$

Hyperbolic Functions

1.

$$\sinh \alpha = \frac{e^{ \alpha} - e^{- \alpha}}{2}$$

2.

$$\cosh \alpha = \frac{e^{ \alpha} + e^{- \alpha}}{2}$$