# How do you evaluate  e^( ( 19 pi)/12 i) - e^( (3 pi)/4 i) using trigonometric functions?

Apr 9, 2016

$0.363 + 0.686 i$

#### Explanation:

According to Euler's formula,

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

Substituting the different values for $x$ in the question,

${e}^{\frac{19 \pi}{12} i} = \cos \left(\frac{19 \pi}{12}\right) + i \sin \left(\frac{19 \pi}{12}\right)$
$= \cos 285 + i \sin 285$
$= - 0.633 + 0.774 i$

${e}^{\frac{3 \pi}{4} i} = \cos \left(\frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi}{4}\right)$
$= \cos 135 + i \sin 135$
$= - 0.996 + 0.088 i$

Using these values, the final answer is

e^((19pi)/12i) - e^((3pi)/4i) `= - 0.633 + 0.774i + 0.996 - 0.088i
$= 0.363 + 0.686 i$