De Moivre's Formula/Positive Integer Index/Corollary

Corollary to De Moivre's Formula: Positive Integer Index

$\forall n \in \Z_{>0}: \paren {\cos x + i \sin x}^n = \map \cos {n x} + i \map \sin {n x}$

Proof 1

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

From De Moivre's Formula: Positive Integer Index:

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

The result follows by setting $r = 1$.

$\blacksquare$

Proof 2

\(\ds \paren {\cos \theta + i \sin \theta}^n\)

\(=\)

\(\ds \paren {e^{i \theta} }^n\)

Euler's Formula

\(\ds \)

\(=\)

\(\ds e^{i n \theta}\)

Exponential of Product

\(\ds \)

\(=\)

\(\ds \cos n \theta + i \sin n \theta\)

Euler's Formula

Source of Name

This entry was named for Abraham de Moivre.