# How do you divide ( -i+4) / (7i +5 ) in trigonometric form?

$\frac{\sqrt{1258}}{74} \cdot \left[\cos \left({\tan}^{-} 1 \left(\frac{- 33}{13}\right)\right) + i \sin \left({\tan}^{-} 1 \left(\frac{- 33}{13}\right)\right)\right]$ OR

$\frac{\sqrt{1258}}{74} \cdot \left[\cos \left(- {68.4986}^{\circ}\right) + i \sin \left(- {68.4986}^{\circ}\right)\right]$

#### Explanation:

$\frac{- i + 4}{7 i + 5} = \frac{4 - i}{5 + 7 i}$

$\frac{4 - i}{5 + 7 i} \cdot \frac{5 - 7 i}{5 - 7 i} = \frac{20 - 5 i - 28 i - 7}{25 + 49} = \frac{13 - 33 i}{74}$

$\frac{13}{74} - \frac{33}{74} i$

Compute the magnitude $r$ ,let $x = \frac{13}{74}$ and $y = \frac{- 33}{74}$

$r = \sqrt{{x}^{2} + {y}^{2}} = \sqrt{{\left(\frac{13}{74}\right)}^{2} + {\left(\frac{- 33}{74}\right)}^{2}} = \frac{\sqrt{1258}}{74}$

Compute the Argument $\phi$

$\phi = {\tan}^{-} 1 \left(\frac{y}{x}\right) = {\tan}^{-} 1 \left(\frac{\frac{- 33}{\cancel{74}}}{\frac{13}{\cancel{74}}}\right) = {\tan}^{-} 1 \left(\frac{- 33}{13}\right)$

$\phi = - {68.4986}^{\circ}$

so that

$\frac{- i + 4}{7 i + 5} = \frac{4 - i}{5 + 7 i}$

$\frac{4 - i}{5 + 7 i} = \frac{\sqrt{1258}}{74} \cdot \left[\cos \left(- {68.4986}^{\circ}\right) + i \sin \left(- {68.4986}^{\circ}\right)\right]$