# How do you find the trigonometric form of -7+7i?

Jul 16, 2018

The trigonometric form is $z = 7 \sqrt{2} \left(\cos \left(\frac{3}{4} \pi\right) + i \sin \left(\frac{3}{4} \pi\right)\right)$, $\left[\mod 2 \pi\right]$

#### Explanation:

To convert a complex number

$z = x + i y$

to the polar form

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

Apply the following :

$\left\{\begin{matrix}r = | z | = \sqrt{{x}^{2} + {y}^{2}} \\ \cos \theta = \frac{x}{| z |} \\ \sin \theta = \frac{y}{| z |}\end{matrix}\right.$

Here,

$z = - 7 + 7 i$

$| z | = \sqrt{{\left(- 7\right)}^{2} + {\left(7\right)}^{2}} = \sqrt{49 + 49} = \sqrt{98} = 7 \sqrt{2}$

Therefore,

$z = 7 \sqrt{2} \left(- \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}\right)$

$\implies$, $\left\{\begin{matrix}\cos \theta = - \frac{1}{\sqrt{2}} \\ \sin \theta = \frac{1}{\sqrt{2}}\end{matrix}\right.$

$\implies$, $\theta = \frac{3}{4} \pi$

The trigonometric form is

$z = 7 \sqrt{2} \left(\cos \left(\frac{3}{4} \pi\right) + i \sin \left(\frac{3}{4} \pi\right)\right)$, $\left[\mod 2 \pi\right]$