# How do you evaluate  e^( (3 pi)/2 i) - e^( (5 pi)/3 i) using trigonometric functions?

Apr 9, 2016

$1.006 + 0.824 i$

#### Explanation:

According to Euler's formula,

${e}^{i x} = \cos x + i \sin x$.

Using values for $x$ from the equation,

$x = \frac{3 \pi}{2}$
${e}^{\frac{3 \pi}{2} i} = \cos \left(\frac{3 \pi}{2}\right) + i \sin \left(\frac{3 \pi}{2}\right)$
$= \cos 270 + i \sin 270$
$= 0.984 - 0.176 i$

$x = \frac{5 \pi}{3}$
${e}^{\frac{5 \pi}{3} i} = \cos \left(\frac{5 \pi}{3}\right) + i \sin \left(\frac{5 \pi}{3}\right)$
$= \cos 300 + i \sin 300$
$= - 0.022 - i$

Inserting these two values into the equation above,

${e}^{\frac{3 \pi}{2} i} - {e}^{\frac{5 \pi}{3} i} = 0.984 - 0.176 i + 0.022 + i$
$= 1.006 + 0.824 i$