Question

How do you write `-3+4i` in trigonometric form?

Answer 1

You need the module and the argument of the complex number.

Explanation:

In order to have the trigonometric form of this complex number, we first need its module. Let's say `z = -3 +4i`.

`absz = sqrt((-3)^2 + 4^2) = sqrt(25) = 5`

In `RR^2`, this complex number is represented by `(-3,4)`. So the argument of this complex number seen as a vector in `RR^2` is `arctan(4/-3) + pi = -arctan(4/3) + pi`. We add `pi` because `-3 < 0`.

So the trigonometric form of this complex number is `5e^(i(pi - arctan(4/3))`