# How do you divide  (3+4i) / (9-4i)  in trigonometric form?

Feb 26, 2016

$0.51 \left(\cos \left(1.35\right) + i \sin \left(1.35\right)\right)$

#### Explanation:

Let the quotient be $q = 3 + 4 i$
and the divisor be $d = 9 - 4 i$

In trigonometric form:
$\textcolor{w h i t e}{\text{XXX}} q = {r}_{q} \left(\cos \left({\theta}_{q}\right) + i \sin \left({\theta}_{q}\right)\right)$
with
$\textcolor{w h i t e}{\text{XXX}} {r}_{q} = \sqrt{{3}^{2} + {4}^{2}} = 5$
and
$\textcolor{w h i t e}{\text{XXX}} {\theta}_{q} = \arctan \left(\frac{4}{3}\right) \approx 0.927295$

Similaryly, in trigonometric form:
$\textcolor{w h i t e}{\text{XXX}} d = {r}_{d} \left(\cos \left({\theta}_{d}\right) + i \sin \left({\theta}_{d}\right)\right)$
with
$\textcolor{w h i t e}{\text{XXX}} {r}_{d} = \sqrt{{9}^{2} + {\left(- 4\right)}^{2}} = \sqrt{97} \approx 9.848858$
and
$\textcolor{w h i t e}{\text{XXX}} {\theta}_{d} = \arctan \left(\frac{9}{- 4}\right) \approx - 0.41822$

Using trigonometric form:
$\textcolor{w h i t e}{\text{XXX}} \frac{q}{d} = \frac{{r}_{q}}{{r}_{d}} \left(\cos \left({\theta}_{q} - {\theta}_{d}\right) + i \sin \left({\theta}_{q} - {\theta}_{d}\right)\right)$

color(white)("XXX")=5/sqrt(97)(cos(0.927295-(-041822))+isin(0.927295-(-0.41822))

$\textcolor{w h i t e}{\text{XXX}} \approx 0.51 \left(\cos \left(1.35\right) + i \sin \left(1.35\right)\right)$