How do you write the following in trigonometric form and perform the operation given (1+sqrt3i)/(6-3i)?

Jul 17, 2018

$\textcolor{g r e e n}{\implies 0.0178 + 0.2976 i}$

Explanation:

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

${z}_{1} = 1 + 3 i , {z}_{2} = 6 - 3 i$

${r}_{1} = \sqrt{{1}^{2} + {\sqrt{3}}^{2}} = 2$

${\theta}_{1} = {\tan}^{\sqrt{3}} / \left(1\right) = {60}^{\circ} ^ \circ , \text{ I Quadrant}$

${r}_{2} = \sqrt{{6}^{2} + {\left(- 3\right)}^{2}} = \sqrt{45}$

${\theta}_{2} = {\tan}^{-} 1 \left(- \frac{3}{6}\right) \approx {333.43}^{\circ} , \text{ IV Quadrant}$

${z}_{1} / {z}_{2} = \frac{2}{\sqrt{45}} \left(\cos \left(60 - 333.43\right) + i \sin \left(60 - 333.43\right)\right)$

$\textcolor{g r e e n}{\implies 0.0178 + 0.2976 i}$