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

Jan 9, 2016

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 $z = - 3 + 4 i$.

$\left\mid z \right\mid = \sqrt{{\left(- 3\right)}^{2} + {4}^{2}} = \sqrt{25} = 5$

In ${\mathbb{R}}^{2}$, this complex number is represented by $\left(- 3 , 4\right)$. So the argument of this complex number seen as a vector in ${\mathbb{R}}^{2}$ is $\arctan \left(\frac{4}{-} 3\right) + \pi = - \arctan \left(\frac{4}{3}\right) + \pi$. We add $\pi$ because $- 3 < 0$.

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