De Moivre's Formula/Positive Integer Index/Corollary/Proof 2
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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
| \(\ds \paren {\cos \theta + i \sin \theta}^n\) | \(=\) | \(\ds \paren {e^{i \theta} }^n\) | 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.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations