# How do you divide ( 4i+1) / (-8i +5 ) in trigonometric form?

Mar 30, 2017

$0.315 i - 0.303$

#### Explanation:

First, convert both the numerator and denominator into polar form.

Converting $4 i + 1$ to polar:

$r = \sqrt{{4}^{2} + {1}^{2}} = \sqrt{17}$
$\theta = {\tan}^{-} 1 \left(\frac{4}{1}\right) = {75.96}^{\circ}$

$4 i + 1 = \sqrt{17} \textcolor{w h i t e}{\text{-" angle color(white)".}} {75.96}^{\circ}$

Converting $- 8 i + 5$ to polar:

$r = \sqrt{{\left(- 8\right)}^{2} + {5}^{2}} = \sqrt{89}$
$\theta = {\tan}^{-} 1 \left(- \frac{8}{5}\right) = - {57.99}^{\circ}$

$- 8 i + 5 = \sqrt{89} \textcolor{w h i t e}{\text{-" angle color(white)".}} - {57.99}^{\circ}$

Dividing in polar form:

(sqrt17 color(white)"-" angle color(white)"."75.96^@) / ( sqrt89 color(white)"-" angle color(white)"."-57.99^@) = sqrt17/sqrt89 color(white)"-" angle color(white)"-" (75.96-(-57.99))^@

$= \frac{\sqrt{17}}{\sqrt{89}} \textcolor{w h i t e}{\text{-" angle color(white)".}} {133.95}^{\circ}$

Now to convert back to rectangular:

$R e = r \cos \theta = \frac{\sqrt{17}}{\sqrt{89}} \cos {133.95}^{\circ} = - 0.303$

$I m = r i \sin \theta = \frac{\sqrt{17}}{\sqrt{89}} i \textcolor{w h i t e}{\text{.}} \sin {133.95}^{\circ} = 0.315 i$

So the final answer is $0.315 i - 0.303$