# How do you write the trigonometric form in complex form 3/2(cos((5pi)/3)+isin((5pi)/3)))?

Oct 4, 2016

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

#### Explanation:

As $\cos \left(\frac{5 \pi}{3}\right) = \cos \left(2 \pi - \frac{\pi}{3}\right) = \cos \left(- \frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$

and $\sin \left(\frac{5 \pi}{3}\right) = \sin \left(2 \pi - \frac{\pi}{3}\right) = \sin \left(- \frac{\pi}{3}\right) = - \sin \left(\frac{\pi}{3}\right) = - \frac{\sqrt{3}}{2}$

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

= $\frac{3}{2} \left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right)$

= $\frac{3}{4} - i \frac{3 \sqrt{3}}{4}$