Multiplication in Polar Form

Multiplication in Polar Form

## Formula

If:

$$z_1=r_1(\cos\theta_1+i\sin\theta_1)$$

and:

$$z_2=r_2(\cos\theta_2+i\sin\theta_2)$$

then:

$$z_1z_2=r_1r_2[\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)]$$

In exponential form:

$$z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}$$

## Conditions / Special Cases

Moduli multiply:

$$|z_1z_2|=|z_1||z_2|$$

Arguments add:

$$\arg(z_1z_2)=\arg(z_1)+\arg(z_2)$$

up to multiples of $2\pi$.

## Usage

- Used to multiply complex numbers quickly in polar form.