How do you divide ( -7i-5) / ( 9 i -2 ) in trigonometric form?

$- 0.623 - 0.694 i$

Explanation:

$\setminus \frac{- 7 i - 5}{9 i - 2}$

$= \setminus \frac{- 5 - 7 i}{- 2 + 9 i}$

$= \setminus \frac{\setminus \sqrt{{\left(- 5\right)}^{2} + {\left(- 7\right)}^{2}} \setminus {e}^{i \left(- \setminus \pi + \setminus {\tan}^{- 1} \left(\frac{7}{5}\right)\right)}}{\setminus \sqrt{{\left(- 2\right)}^{2} + {\left(9\right)}^{2}} \setminus {e}^{i \left(\setminus \pi - \setminus {\tan}^{- 1} \left(\frac{9}{2}\right)\right)}}$

$= \setminus \frac{\setminus \sqrt{74} \setminus {e}^{i \left(- \setminus \pi + \setminus {\tan}^{- 1} \left(\frac{7}{5}\right)\right)}}{\setminus \sqrt{85} \setminus {e}^{i \left(\setminus \pi - \setminus {\tan}^{- 1} \left(\frac{9}{2}\right)\right)}}$

$= \setminus \sqrt{\frac{74}{85}} {e}^{i \left(- \setminus \pi + \setminus {\tan}^{- 1} \left(\frac{7}{5}\right)\right) - i \left(\setminus \pi - \setminus {\tan}^{- 1} \left(\frac{9}{2}\right)\right)}$

$= \setminus \sqrt{\frac{74}{85}} {e}^{i \left(- 2 \setminus \pi + \setminus {\tan}^{- 1} \left(\frac{7}{5}\right) + \setminus {\tan}^{- 1} \left(\frac{9}{2}\right)\right)}$

$= \setminus \sqrt{\frac{74}{85}} {e}^{- 3.9805 i}$

$= \setminus \sqrt{\frac{74}{85}} \left(\setminus \cos \left(- 3.9805\right) + i \setminus \sin \left(- 3.9805\right)\right)$

$= \setminus \sqrt{\frac{74}{85}} \left(\setminus \cos \left(3.9805\right) - i \setminus \sin \left(3.9805\right)\right)$

$= - 0.623 - 0.694 i$

Jul 8, 2018

color(chocolate)((-5 - 7 i) / (-2 +9 i) ~~ 0.6235 + i 0.6942

Explanation:

To divide $\frac{- 5 - 7 i}{- 2 + 9 i}$ using trigonometric form.

${z}_{1} = \left(- 5 - 7 i\right) , {z}_{2} = \left(- 2 + 9 i\right)$

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

${r}_{2} = \sqrt{- {2}^{2} + {9}^{2}} = \sqrt{85}$

${\theta}_{1} = \arctan \left(- \frac{5}{-} 7\right) = {215.54}^{\circ} , \text{ III quadrant}$

${\Theta}_{2} = \arctan \left(- \frac{2}{9}\right) = {167.47}^{\circ} , \text{ II quadrant}$

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

${z}_{1} / {z}_{2} = \sqrt{\frac{74}{85}} \cdot \left(\cos \left(215.54 - 167.47\right) + i \sin \left(215.54 - 167.47\right)\right)$

${z}_{1} / {z}_{2} = \sqrt{\frac{74}{85}} \cdot \left(\cos \left(48.07\right) + i \sin \left(48.07\right)\right)$

color(chocolate)((-5 - 7 i) / (-2 +9 i) ~~ 0.6235 + i 0.6942#