Polar Form of a Complex Number

Polar Form of a Complex Number

## Definition / Concept

A non-zero complex number can be represented using its modulus and argument.

## Formula

If:

$$z=x+iy$$

and:

$$r=|z|$$

then:

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

where:

$$x=r\cos\theta$$

$$y=r\sin\theta$$

Therefore:

$$r=\sqrt{x^2+y^2}$$

## Terminologies

- Modulus → $r=|z|$, the distance from the origin.

- Argument → Angle $$\theta$$ made by the line joining the point to the origin with the positive real axis.

## Usage

- Used to simplify multiplication, division and powers of complex numbers.