# De Moivre's Formula/Integer Index/Corollary

## Corollary to De Moivre's Formula: Positive Integer Index

$\forall n \in \Z: \paren {\cos x + i \sin x}^n = \map \cos {n x} + i \map \sin {n x}$

## Proof

$\cos x + i \sin x$ is a complex number expressed in polar form $\polar {r, \theta}$ whose complex modulus is $1$ and whose argument is $x$.

$\forall n \in \Z: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$

The result follows by setting $r = 1$.

$\blacksquare$

## Source of Name

This entry was named for Abraham de Moivre.