DeMoivre's Theorem

From the basic formula

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

one derives several important trigonometric identitites. For example, since

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

it follows that

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

and

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