De Moivre's Formula/Integer Index/Corollary

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Corollary to De Moivre's Formula: Positive Integer Index

$\forall n \in \Z: \paren {\cos x + i \sin x}^n = \map \cos {n x} + i \map \sin {n x}$


$\cos x + i \sin x$ is a complex number expressed in polar form $\polar {r, \theta}$ whose complex modulus is $1$ and whose argument is $x$.

From De Moivre's Formula: Integer Index:

$\forall n \in \Z: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$

The result follows by setting $r = 1$.


Source of Name

This entry was named for Abraham de Moivre.