How can you use trigonometric functions to simplify  8 e^( ( 2 pi)/3 i )  into a non-exponential complex number?

May 26, 2016

$8 \left(\cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)\right) = - 4 + 4 \sqrt{3} i$

Explanation:

Using $\textcolor{b l u e}{\text{Euler's identity}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{r {e}^{\theta i} = r \left(\cos \theta + i \sin \theta\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

here r = 8 and $\theta = \frac{2 \pi}{3}$

$\Rightarrow 8 {e}^{\frac{2 \pi}{3} i} = 8 \left(\cos \left(\frac{2 \pi}{3}\right) + i \sin \left(\frac{2 \pi}{3}\right)\right)$

$= 8 \left(- \frac{1}{2} + \frac{\sqrt{3}}{2} i\right) = - 4 + 4 \sqrt{3} i$