De Moivre's Formula/Positive Integer Index

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

Then:
 * $\forall n \in \Z_{>0}: \left({r \left({\cos x + i \sin x}\right)}\right)^n = r^n \left({\cos \left({n x}\right) + i \sin \left({n x}\right)}\right)$

Also defined as
This result is also often presented in the simpler form:


 * $\forall n \in \Z_{> 0}:\left({\cos x + i \sin x}\right)^n = \cos \left({n x}\right) + i \sin \left({n x}\right)$

Also known as
De Moivre's Theorem.