# De Moivre's Formula/Rational Index

## Theorem

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

- $z = r \left({\cos x + i \sin x}\right)$

Then:

- $\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$

## Proof

## Also defined as

This result is also often presented in the simpler form:

- $\forall p \in \Q: \paren {\cos x + i \sin x}^p = \map \cos {p x} + i \, \map \sin {p x}$

## Also known as

**De Moivre's Theorem**.

## Source of Name

This entry was named for Abraham de Moivre.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $94 \ \text{(b)}$