How can you use trigonometric functions to simplify  3 e^( ( 5 pi)/3 i )  into a non-exponential complex number?

Oct 9, 2016

$3 {e}^{\left(\frac{5 \pi}{3}\right) i} = 3 \cos \left(\frac{5 \pi}{3}\right) + 3 \sin \left(\frac{5 \pi}{3}\right) i$

Explanation:

Begin with Euler's formula :

${e}^{i x} = \cos \left(x\right) + \sin \left(x\right) i$

Multiply by any magnitude, A:

$A {e}^{i x} = A \cos \left(x\right) + A \sin \left(x\right) i$

Oct 9, 2016

$\frac{- \sqrt{3} + i}{2}$

Explanation:

Using Euler's formula ${e}^{i x} = \sin x + i \cos x$...

$3 {e}^{\frac{5 \pi}{3} i} = \sin \left(\frac{5 \pi}{3}\right) + i \cos \left(\frac{5 \pi}{3}\right)$

$\textcolor{w h i t e}{a a a a} = - \frac{\sqrt{3}}{2} + i \ast \frac{1}{2} \textcolor{w h i t e}{a a a}$ from the unit circle

$\textcolor{w h i t e}{a a a a} = \frac{- \sqrt{3} + i}{2}$