1.13: de Moivre's formula

Moivre’s Formula

For positive integers $n$ we have de Moivre’s formula:

$$(\cos (\theta) + i \sin (\theta))^n = \cos (n \theta) + i \sin (n \theta) \nonumber$$

Proof

This is a simple consequence of Euler’s formula:

$(\cos (\theta) + i \sin (\theta))^n = (e^{i \theta})^n = e^{i n \theta} = \cos (n \theta) + i \sin (n \theta)$

The reason this simple fact has a name is that historically de Moivre stated it before Euler’s formula was known. Without Euler’s formula there is not such a simple proof.