Question

How do you convert `-2 + 2i` to polar form?

Answer 1

`-2+2i = sqrt(8)[cos((3pi)/4) + isin((3pi)/4)]`

Explanation:

First we want to draw an Argand diagram.

The modulus (or length) of the complex number is simply given by using Pythagoras' theorem to find the hypotenuse, ie

`r = sqrt(x^2+y^2) = sqrt((-2)^2+2^2) = sqrt(8)`

The argument is defined as the angle from the x-axis in the counter-clockwise direction. In this case, the angle we find is in the second quadrant, so we find the 'positive' version and then need to take it away from `pi`.

`theta = pi - arctan(y/x) = pi - arctan((2)/(2)) = pi - pi/4 = (3pi)/4`