Complex Numbers in Polar Form

When writing complex numbers, there are several different forms you can write them in.

Rectangular Form:

z = a+bi

Polar Form:

z=r*(cos(\Theta)+i*sin(\Theta))

To convert:

  • r = \mid z\mid= \sqrt{a^{2}+b^{2}}
  • a = r*cos(\Theta)
  • b = r*sin(\Theta)
  • \Theta = \tan^{-1}({\dfrac{b}{a}}) for a > 0
  • \Theta = \tan^{-1}({\dfrac{b}{a}}) + \pi for a < 0
  • When a = 0 then z = r

Example: Convert   -5+6i

 r = \sqrt{a^{2}+b^{2}} = \sqrt{(-5)^{2}+(6)^{2}} = \sqrt{25+36} = \sqrt{61} = 7.81

Since -5 < 0, we will use the second formula for calculating theta:

\Theta  = \tan^{-1}({\dfrac{b}{a}}) + \pi = \tan^{-1}({\dfrac{6}{-5}}) + \pi2.27

So the polar form is z = 7.81 * (\cos{(2.27)}+i*\sin{(2.27)})