# How do you write the trigonometric form in complex form 3/4(cos((7pi)/4)+isin((7pi)/4)))?

Aug 10, 2016

$z = \frac{3 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{8} i$

#### Explanation:

Starting with the trig. part

That is $\cos \left(\frac{7 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4}\right)$

Simplifying this as follows.

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$

color(red)(|bar(ul(color(white)(a/a)color(black)(cos((7pi)/4)=cos(2pi-(7pi)/4)=cos(pi/4)color(white)(a/a)|

and color(red)(|bar(ul(color(white)(a/a)color(black)(sin((7pi)/4)=-sin(2pi-(7pi)/4)=-sin(pi/4)color(white)(a/a)|)))

$\Rightarrow \cos \left(\frac{7 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) - i \sin \left(\frac{\pi}{4}\right)$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\cos \left(\frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow \cos \left(\frac{\pi}{4}\right) - i \sin \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} i = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i$

Finally the whole expression becomes.

$\frac{3}{4} \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i\right) = \frac{3 \sqrt{2}}{8} - \frac{3 \sqrt{2}}{8} i \text{ in complex form}$