Section 10.5 (c) - Polar Form of a Complex Number

Section Objectives

1. Write complex numbers in polar (trigonometric) form.
2. Multiply and divide complex numbers in polar form.

Polar or Trigonometric Form

The polar (or trigonometric) form of the complex number $z = x+yi$ is

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

where $r=\sqrt{x^2+y^2}$ and $\tan \theta= y/x$. $r$ is the absolute value (or modulus) of $z$, and $\theta$ is called the argument of $z$. Normally, $\theta$ is restricted so that $0 \le \theta < 2\pi$.

Examples

• Write $z=6-6i$ in polar form.

• Write $6+2i$ in polar form.

• Write $z=4(\cos 120^{\circ} + i \, \sin 120^{\circ})$ in standard form.

Multiplication and Division in Polar Form

It is fairly easy to prove the following useful results.

Let $z_1=r_1(\cos \theta_1 + i \, \sin \theta_1)$ and $z_2=r_2(\cos \theta_2 + i \, \sin \theta_2)$.

• Let $z_1=2(\cos 120^{\circ} + i \, \sin 120^{\circ})$ and $z_2=8(\cos 330^{\circ} + i \, \sin 330^{\circ})$. Compute $z_1 z_2$ and $z_2/z_1$.