De Moivre's Formula/Rational Index

Theorem
Let $z \in \C$ be a complex number expressed in complex form:
 * $z = r \paren {\cos x + i \sin x}$

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
Write $p = \dfrac a b$, where $a, b \in \Z$, $b \ne 0$.

Then:

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.