How do you divide ( -i+4) / (2i +7 ) in trigonometric form?

Jun 28, 2018

color(violet)((4 -i) / (7 + 2i) ~~ 0.4905 + i 0.2831

Explanation:

To divide $\frac{4 - i}{7 + 2 i}$ using trigonometric form.

${z}_{1} = \left(4 - i\right) , {z}_{2} = \left(7 + 2 i\right)$

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

${r}_{2} = \sqrt{{7}^{2} + - {2}^{2}} = \sqrt{53}$

${\theta}_{1} = \arctan \left(- \frac{1}{4}\right) = {345.96}^{\circ} , \text{ IV quadrant}$

${\Theta}_{2} = \arctan \left(\frac{2}{7}\right) = {15.95}^{\circ} , \text{ I quadrant}$

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

${z}_{1} / {z}_{2} = \sqrt{\frac{17}{53}} \cdot \left(\cos \left(345.96 - 15.95\right) + i \sin \left(345.96 - 15.95\right)\right)$

${z}_{1} / {z}_{2} = 0.5664 \cdot \left(\cos \left(- 330.01\right) + i \sin \left(- 330.01\right)\right)$

color(violet)((4 -i) / (7 + 2i) ~~ 0.4905 + i 0.2831#