# How do you divide ( 4i+1) / (-8i +3 ) in trigonometric form?

Jun 28, 2018

color(blue)((1 + 4i) / (3 - 8i) ~~ 0.4388 + i 0.2385

#### Explanation:

To divide $\frac{1 + 4 i}{3 - 8 i}$ using trigonometric form.

${z}_{1} = \left(1 + 4 i\right) , {z}_{2} = \left(3 - 8 i\right)$

r_1 = sqrt(4^2 + 1^2) = sqrt 17

${r}_{2} = \sqrt{{3}^{2} + - {8}^{2}} = \sqrt{73}$

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

${\Theta}_{2} = \arctan \left(- \frac{8}{3}\right) = {290.56}^{\circ} , \text{ IV quadrant}$

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

${z}_{1} / {z}_{2} = \sqrt{\frac{17}{73}} \cdot \left(\cos \left(75.96 - 290.56\right) + i \sin \left(75.96 - 290.56\right)\right)$

${z}_{1} / {z}_{2} = 0.4826 \cdot \left(\cos \left(- 204.6\right) + i \sin \left(- 204.6\right)\right)$

color(blue)((1 + 4i) / (3 - 8i) ~~ 0.4388 + i 0.2385#