# How do you multiply  (1+7i)(-3+2i)  in trigonometric form?

Nov 19, 2017

$\left(1 + 7 i\right) \left(- 3 + 2 i\right) \approx 25.495 \left(\cos 228.18 + i \sin 228.18\right)$

#### Explanation:

$Z = \left(1 + 7 i\right) \left(- 3 + 2 i\right) = \left(- 3 + 2 i - 21 i + 14 {i}^{2}\right)$

$= - 3 - 19 i - 14 = - 17 - 19 i \left[{i}^{2} = - 1\right]$

$Z$ lies on $3 r d$ quadrant

Modulus $| Z | = r = \sqrt{{\left(- 17\right)}^{2} + {\left(- 19\right)}^{2}} \approx 25.495$ ;

$\tan \alpha = \frac{b}{a} = \frac{- 19}{-} 17 \approx 1.1176 \therefore \alpha = {\tan}^{-} 1 \left(1.1176\right) \approx {48.18}^{0}$

Since $Z$ is on $3 r d$ quadrant $\therefore \theta = \pi + \alpha$ or

$\theta = 180 + 48.18 = {228.18}^{0}$. Argument : $\theta = {228.18}^{0}$

In trigonometric form expressed as $r \left(\cos \theta + i \sin \theta\right)$

$= 25.495 \left(\cos 228.18 + i \sin 228.18\right)$

$\therefore \left(1 + 7 i\right) \left(- 3 + 2 i\right) \approx 25.495 \left(\cos 228.18 + i \sin 228.18\right)$ [Ans]