# De Moivre's Formula/Proof 1

## Theorem

Let $z \in \C$ be a complex number expressed in complex form:

$z = r \paren {\cos x + i \sin x}$

Then:

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

## Proof

 $\ds \paren {r \paren {\cos x + i \sin x} }^\omega$ $=$ $\ds \paren {r e^{i x} }^\omega$ Euler's Formula $\ds$ $=$ $\ds r^\omega e^{i \omega x}$ Power of Power $\ds$ $=$ $\ds r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }$ Euler's Formula

$\blacksquare$

## Source of Name

This entry was named for Abraham de Moivre.