Division in Polar Form

Division in Polar Form

## Formula

If:

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

and:

$$z_2=r_2(\cos\theta_2+i\sin\theta_2),\qquad z_2\ne0$$

then:

$$\frac{z_1}{z_2}=\frac{r_1}{r_2}[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)]$$

In exponential form:

$$\frac{z_1}{z_2}=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}$$

## Conditions / Special Cases

Moduli divide:

$$\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$$

Arguments subtract:

$$\arg\left(\frac{z_1}{z_2}\right)=\arg(z_1)-\arg(z_2)$$

up to multiples of $2\pi$.

## Usage

- Used to simplify division of complex numbers in polar form.