# How do you divide ( 3i-2) / (i +4 ) in trigonometric form?

Jul 8, 2018

color(maroon)((-2 + 3 i) / (4 + i) ~~ -0.2931 - i 0.8231

#### Explanation:

To divide $\frac{- 2 + 3 i}{4 + i}$ using trigonometric form.

${z}_{1} = \left(- 2 + 3 i\right) , {z}_{2} = \left(4 + i\right)$

r_1 = sqrt(-2^2 + 3^2) = sqrt 13

${r}_{2} = \sqrt{{4}^{2} + {1}^{2}} = \sqrt{17}$

${\theta}_{1} = \arctan \left(- \frac{2}{3}\right) = {146.31}^{\circ} , \text{ II quadrant}$

${\Theta}_{2} = \arctan \left(\frac{4}{1}\right) = {75.96}^{\circ} , \text{ I quadrant}$

${z}_{1} / {z}_{2} = \left({r}_{1} / {r}_{2}\right) \cdot \left(\cos \left({\theta}_{1} - {\theta}_{2}\right) + i \sin \left({\theta}_{1} - {\theta}_{2}\right)\right)$

${z}_{1} / {z}_{2} = \sqrt{\frac{13}{17}} \cdot \left(\cos \left(146.31 - 75.96\right) + i \sin \left(146.31 - 75.96\right)\right)$

${z}_{1} / {z}_{2} = \sqrt{\frac{13}{17}} \cdot \left(\cos \left(70.41\right) + i \sin \left(70.41\right)\right)$

color(maroon)((-2 + 3 i) / (4 + i) ~~ -0.2931 - i 0.8231#