# How do you multiply  (2+3i)(2-7i)  in trigonometric form?

Jun 17, 2018

color(blue)((2 + i 3) * (2 - i 7) = 26.25 (0.9524 - i 0.3047)

#### Explanation:

$\left(2 + i 3\right) \cdot \left(2 - i 7\right)$

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

${r}_{1} = \sqrt{{2}^{2} + {3}^{2}} = \sqrt{13}$

${r}_{2} = \sqrt{{2}^{2} + {7}^{2}} = \sqrt{53}$

${\theta}_{1} = \arctan \left(\frac{3}{2}\right) = {56.31}^{\circ}$

${\theta}_{2} = \arctan \left(- \frac{7}{2}\right) = - 74.05 = {285.95}^{\circ} , \text{ IV Quadrant}$

${z}_{1} {z}_{2} = \left(\sqrt{13} \cdot \sqrt{53}\right) \left(\cos \left(56.31 + 285.95\right) + \sin \left(74.05 + 285.95\right)\right)$

$\left(2 + i 3\right) \cdot \left(2 - i 7\right) = 26.25 \left(\cos 342.26 + i \sin 342.26\right)$

$\left(2 + i 3\right) \cdot \left(2 - i 7\right) = 26.25 \left(0.9524 - i 0.3047\right)$