De Moivre's formula

De Moivre's formula (also known as de Moivre's theorem or de Moivre's identity) is a theorem in complex analysis which states

$(\cos(\theta)+i\sin(\theta))^n=\text{cis}^n(\theta)=\cos(n\theta)+i\sin(n\theta)$

This makes computing powers of any complex number very simple.

$z^n=(re^{\theta i})^n=r^n\text{cis}(n\theta)$

Derivation

By Euler's formula,

$(\cos(\theta)+i\sin(\theta))^n=(e^{\theta i})^n=e^{n\theta i}=\cos(n\theta)+i\sin(n\theta)$