# How do you divide (i+2) / (9i+4) in trigonometric form?

Jun 14, 2018

color(blue)((2 + i) / (4 + i9) = 0.175 - i 0.144

#### Explanation:

To divide $\frac{2 + i}{4 + i 9}$ using trigonometric form.

${z}_{1} = \left(2 + i\right) , {z}_{2} = \left(4 + i 9\right)$

${r}_{1} = \sqrt{{2}^{2} + {1}^{2}} = \sqrt{5}$

${r}_{2} = \sqrt{{9}^{2} + {4}^{2}} = \sqrt{97}$

${\theta}_{1} = \arctan \left(\frac{1}{2}\right) = {26.57}^{\circ}$

${\Theta}_{2} = \arctan \left(\frac{9}{4}\right) = {66.06}^{\circ}$

${r}_{1} / {r}_{2} = \sqrt{5} \frac{\cdot}{\sqrt{97}} \approx 0.227$

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

${z}_{1} / {z}_{2} = \left(0.227\right) \cdot \left(\cos \left(26.57 - 66.06\right) + i \sin \left(26.57 - 66.06\right)\right)$

${z}_{1} / {z}_{2} = 0.227 \cdot \left(\cos \left(- 39.49\right) + i \sin \left(- 39.49\right)\right) = 0.227 \left(0.772 - i 0.636\right)$

color(blue)((2 + i) / (4 + i9) = 0.175 - i 0.144