# How can you use trigonometric functions to simplify  22 e^( ( 3 pi)/4 i )  into a non-exponential complex number?

Feb 21, 2016

$22 \left[\cos \left(\frac{3 \pi}{4}\right) + j \sin \left(\frac{3 \pi}{4}\right)\right]$ or
$- 15.5562 + 15.5562 j$
See details below.

#### Explanation:

Following expressions have the same value

$x + y j$, Cartesian or Rectangular form

r(cos θ+j sin θ) =r cis θ =r/_θ, Polar form

where $| r | = \sqrt{{x}^{2} + {y}^{2}}$, and
$\theta = {\tan}^{- 1} \left(\frac{y}{x}\right)$
(The angle of the point on the complex plane obtained by taking inverse tangent of the complex portion over the real portion)

re^(jθ), Exponential from.

The question asks us to change from exponential form using trigonometric functions. Therefore, let's chose to change from exponential form to polar form.

re^(jθ)=r(cos θ+j sin θ)
$22 {e}^{j \frac{3 \pi}{4}} ,$ Inspection reveals that $r = 22 \mathmr{and} \theta = \frac{3 \pi}{4}$
Polar* form becomes $22 \left[\cos \left(\frac{3 \pi}{4}\right) + j \sin \left(\frac{3 \pi}{4}\right)\right]$
Inserting the value of cosine and sine of the angle $\frac{3 \pi}{4} = {135}^{\circ}$we obtain
$22 \left(- 0.7071 + 0.7071 j\right)$

Expressing in rectangular* form

$- 15.5562 + 15.5562 j$

*Both Polar Form and Rectangular Form the are required non-exponential complex numbers