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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) $
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