# What is the trigonometric form of -1+3i?

Feb 20, 2018

The trignometric form is $z = \sqrt{10} \left(\cos \left({108.4}^{\circ}\right) + i \sin \left({108.4}^{\circ}\right)\right)$, $\left[\mod {360}^{\circ}\right]$

#### Explanation:

The complex number is

$z = - 1 + 3 i$

The polar form of $z = a + i b$

is

$z = r \left(\cos \theta + i \sin \theta\right)$

Where,

$r = | z |$

$\cos \theta = \frac{a}{|} z |$

$\sin \theta = \frac{b}{|} z |$

Here,

$| z | = r = \sqrt{{\left(- 1\right)}^{2} + {\left(3\right)}^{2}} = \sqrt{10}$

$\cos \theta = - \frac{1}{\sqrt{10}}$

$\sin \theta = \frac{3}{\sqrt{10}}$

$\theta = {108.4}^{\circ}$

Therefore,

$z = \sqrt{10} \left(\cos \left({108.4}^{\circ}\right) + i \sin \left({108.4}^{\circ}\right)\right)$, $\left[\mod {360}^{\circ}\right]$