# How do you write -3-2i in trigonometric form?

Apr 23, 2017

$\sqrt{13} \left(\cos \left(2.55\right) - i \sin \left(2.55\right)\right)$

#### Explanation:

$\text{to convert into "color(blue)"trigonometric form}$

$\text{that is } x + y i \to r \left(\cos \theta + i \sin \theta\right)$ where

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{r = \sqrt{{x}^{2} + {y}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

"and " color(red)(bar(ul(|color(white)(2/2)color(black)(theta=tan^-1(y/x);-pi< theta<=pi)color(white)(2/2)|)))

$\text{here " x=-3" and } y = - 2$

$\Rightarrow r = \sqrt{{\left(- 3\right)}^{2} + {\left(- 2\right)}^{2}} = \sqrt{13}$

$\text{now " -3-2i" is in the 3rd quadrant so must ensure that}$

$\theta \text{ is in the 3rd quadrant}$

$\theta = {\tan}^{-} 1 \left(\frac{2}{3}\right) = 0.588 \leftarrow \textcolor{red}{\text{ related acute angle}}$

$\Rightarrow \theta = - \pi + 0.588 \approx - 2.55 \leftarrow \textcolor{red}{\text{ in 3rd quadrant}}$

$\Rightarrow - 3 - 2 i = \sqrt{13} \left(\cos \left(- 2.55\right) + i \sin \left(- 2.55\right)\right)$

• color(orange)" Reminder" cos(-theta)=costheta ; sin(-theta)=-sintheta

$\Rightarrow - 3 - 2 i = \sqrt{13} \left(\cos \left(2.55\right) - i \sin \left(2.55\right)\right)$