# How do you simplify 4(cos((9pi)/4)+ising((9pi)/4))div2[cos(-pi/2)+isin(-pi/2)] and express the result in rectangular form?

## How do you simplify $4 \left(\cos \left(\frac{9 \pi}{4}\right) + i \sin \left(\frac{9 \pi}{4}\right)\right) \div 2 \left[\cos \left(- \frac{\pi}{2}\right) + i \sin \left(- \frac{\pi}{2}\right)\right]$ and express the result in rectangular form?

Mar 24, 2017

$- \sqrt{2} \left(1 - i\right) .$

#### Explanation:

Recall that, $r \left(\cos \theta + i \sin \theta\right)$ is also denoted, as $r c i s \theta ,$ &,

${r}_{1} c i s \alpha \div {r}_{2} c i s \beta = \left({r}_{1} / {r}_{2}\right) c i s \left(\alpha - \beta\right)$.

Using these, we find, that,

$\text{The given Exp.=} 4 c i s \left(9 \frac{\pi}{4}\right) \div 2 c i s \left(- \frac{\pi}{2}\right)$

$= \left(\frac{4}{2}\right) \left[c i s \left\{\left(9 \frac{\pi}{4}\right) - \left(- \frac{\pi}{2}\right)\right\}\right] .$

$= 2 \left\{c i s \left(9 \frac{\pi}{4} + \frac{\pi}{2}\right)\right\} .$

$= 2 c i s \left(11 \frac{\pi}{4}\right) .$

$= 2 \left\{\cos \left(11 \frac{\pi}{4}\right) + i \sin \left(11 \frac{\pi}{4}\right)\right\} .$

$= 2 \left\{\cos \left(3 \pi - \frac{\pi}{4}\right) + i \sin \left(3 \pi - \frac{\pi}{4}\right)\right\} .$

$= 2 \left\{- \cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right\} .$

$= 2 \left(- \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\right) .$

$\therefore \text{ The Exp.=} - \sqrt{2} \left(1 - i\right) .$

Enjoy Maths.!