# How do you simplify 5(cospi+isinpi)*2(cos((3pi)/4)+isin((3pi)/4)) and express the result in rectangular form?

May 17, 2017

$5 \sqrt{2} \left(1 - i\right) .$

#### Explanation:

$\left\{5 \left(\cos \pi + i \sin \pi\right)\right\} \left\{2 \left(\cos \left(3 \frac{\pi}{4}\right) + i \sin \left(3 \frac{\pi}{4}\right)\right)\right\}$

$= \left\{5 \left(- 1 + 0 i\right)\right\} \left\{2 \left(- \cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right)\right\}$

$= - 5 \left\{2 \left(- \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot i\right)\right\}$

$= 5 \sqrt{2} \left(1 - i\right) ,$

Otherwise, knowing that,

$r \left(\cos \alpha + i \sin \alpha\right) \cdot R \left(\cos \beta + i \sin \beta\right) = \left(r R\right) \left(\cos \left(\alpha + \beta\right) + i \sin \left(\alpha + \beta\right)\right) ,$

we have, the Reqd. Value

$= \left(5 \times 2\right) \left\{\cos \left(\pi + 3 \frac{\pi}{4}\right) + i \sin \left(\pi + 3 \frac{\pi}{4}\right)\right\}$

$= 10 \left\{\cos \left(2 \pi - \frac{\pi}{4}\right) + i \sin \left(2 \pi - \frac{\pi}{4}\right)\right\}$

$= 10 \left\{\cos \frac{\pi}{4} - i \sin \frac{\pi}{4}\right\}$

$= 10 \left(\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right)$

$= 5 \sqrt{2} \left(1 - i\right) ,$ as derived earlier!

Enjoy Maths.!