# How do you multiply  (3-5i)(1-16i)  in trigonometric form?

Jul 8, 2018

color(brown)((-3 - 5 i) * (1 - 16 i) ~~ -0.323 + i 0.1673

#### Explanation:

To divide $\left(- 3 - 5 i\right) \cdot \left(1 - 16 i\right)$ using trigonometric form.

${z}_{1} = \left(- 3 - 5 i\right) , {z}_{2} = \left(1 - 16 i\right)$

r_1 = sqrt(-3^2 + 5^2) = sqrt 34

${r}_{2} = \sqrt{{1}^{2} + - {16}^{2}} = \sqrt{257}$

${\theta}_{1} = \arctan \left(- \frac{5}{-} 3\right) = {239.04}^{\circ} , \text{ III quadrant}$

${\Theta}_{2} = \arctan \left(- \frac{16}{1}\right) = {273.58}^{\circ} , \text{ IV quadrant}$

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

${z}_{1} / {z}_{2} = \sqrt{\frac{34}{257}} \cdot \left(\cos \left(239.04 + 273.58\right) + i \sin \left(239.04 + 273.58\right)\right)$

${z}_{1} / {z}_{2} = \sqrt{\frac{34}{257}} \cdot \left(\cos \left(512.62\right) + i \sin \left(512.62\right)\right)$

color(brown)((-3 - 5 i) * (1 - 16 i) ~~ -0.323 + i 0.1673#