# How do you find a trigonometric form of a complex number?

Feb 22, 2015

Let $z = x + i y$ a complex number in algebraic form.

$z = r \left(\cos \phi + i \sin \phi\right)$ is its trigonometric form, where:

$r = \sqrt{{x}^{2} + {y}^{2}}$ is the modulus of the number and

• if $x > 0$

$\phi = \arctan \left(\frac{y}{x}\right)$ ,

• if $x < 0$

$\phi = \arctan \left(\frac{y}{x}\right) + \pi$,

• if $x = 0$ and $y > 0$

$\phi = \frac{\pi}{2}$,

• if $x = 0$ and $y < 0$

$\phi = \frac{3}{2} \pi$

• if $x = y = 0$

It's all zero!