# How do you multiply  (-2-9i)(-3-4i)  in trigonometric form?

May 19, 2018

$\left(- 30 + 35 i\right)$

#### Explanation:

Any complex equation in the form of $a + b i$ i.e. (vector form) can be written as $r {e}^{\theta i}$ (rectangular form)
Here,
$r \rightarrow$ magnitude of the vector, and
$\theta \rightarrow$ the angle between the vector form and the components

now, $r {e}^{\theta i}$ is equivalent to $r \left(\cos \theta + i \sin \theta\right)$

this tells us,
$a = r \cos \theta$
and $b = r \sin \theta$
thus, by solving the 2 above equations, $r = \sqrt{{a}^{2} + {b}^{2}}$
and $\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right)$

So, solving for $\left(- 2 - 9 i\right)$,
${r}_{1} = \sqrt{85}$
${\theta}_{1} = 77.47$ degrees

And,solving for $\left(- 3 - 4 i\right)$,
${r}_{2} = 5$
${\theta}_{2} = 53.13$ degrees

so, we get,
(−2−9i)(−3-4i)=sqrt85 e^(77.47i) x $5 {e}^{53.13 i} = 5 \sqrt{85} {e}^{130.6 i}$

:. (−2−9i)(−3-4i)=5sqrt85(cos130.6+isin130.6)

:. (−2−9i)(−3-4i)=5sqrt85(-0.651+0.759i)

:. (−2−9i)(−3-4i)=(-30+35i)