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# How do you divide ( 7i+5) / ( -3i +8 ) in trigonometric form?

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#### Explanation

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Jun 25, 2018

color(blue)((5 + 7i) / (8 - 3i) ~~ 0.2603 - i 0.9728

#### Explanation:

To divide $\frac{5 + 7 i}{8 - 3 i}$ using trigonometric form.

${z}_{1} = \left(5 + 7 i\right) , {z}_{2} = \left(8 - 3 i\right)$

r_1 = sqrt(5^2 + 7^2) = sqrt 74

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

${\theta}_{1} = \arctan \left(\frac{7}{5}\right) = {54.46}^{\circ} , \text{ I quadrant}$

${\Theta}_{2} = \arctan \left(- \frac{3}{8}\right) = {339.44}^{\circ} , \text{ IV quadrant}$

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

${z}_{1} / {z}_{2} = \sqrt{\frac{74}{73}} \cdot \left(\cos \left(54.46 - 339.44\right) + i \sin \left(54.46 - 339.44\right)\right)$

${z}_{1} / {z}_{2} = 1.007 \cdot \left(\cos \left(- 284.98\right) + i \sin \left(- 284.98\right)\right) = 1.007 \left(0.2585 - i 0.966\right)$

color(blue)((5 + 7i) / (8 - 3i) ~~ 0.2603 - i 0.9728#

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