De Moivre's Formula/Positive Integer Index/Corollary
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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.