# How do you divide ( -4i+9) / (2i-12) in trigonometric form?

Feb 6, 2016

$\frac{58 - 3 i}{74}$ More calculations are given below.

#### Explanation:

Multiply the denominator, with its conjugate. It is 2i+12 here. Now multiplication would work as follows:

$\frac{\left(- 4 i + 9\right) \left(2 i + 12\right)}{\left(- 2 i + 12\right) \left(2 i + 12\right)}$

= $\frac{8 + 108 - 24 i + 18 i}{4 + 144}$

=$\frac{116 - 6 i}{148}$

=$\frac{58 - 3 i}{74}$

=$\frac{58}{74} - i \frac{3}{74}$

Now, let $r \cos \theta = \frac{58}{74}$ and $r \sin \theta = \frac{3}{74}$

On squaring and adding ${r}^{2} = \frac{3373}{5476}$ that means $r = \frac{\sqrt{3373}}{74}$ and on division it would be $\tan \theta = - \frac{3}{58}$, $\theta = {\tan}^{-} 1 \left(- \frac{3}{58}\right)$

The required trignometric form would be $r \left(\cos \theta + i \sin \theta\right)$, where r and $\theta$ would have values as worked out above.