De Moivre's Theorem

De Moivre's Theorem

## Formula

For integer $$n$$:

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

Therefore, if:

$$z=r(\cos\theta+i\sin\theta)$$

then:

$$z^n=r^n[\cos(n\theta)+i\sin(n\theta)]$$

In exponential form:

$$z^n=r^ne^{in\theta}$$

## Conditions / Special Cases

For negative integers:

$$[\cos\theta+i\sin\theta]^{-n}=\cos(n\theta)-i\sin(n\theta)$$

## Usage

- Used to find powers of complex numbers.

- Used to derive multiple-angle identities.