# How do you evaluate  e^( ( 3 pi)/8 i) - e^( ( 19 pi)/6 i) using trigonometric functions?

Apr 9, 2016

$1.249 + 1.424 i$

#### Explanation:

According to Euler's formula,

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

Using values for $x$ from the question,

$x = \frac{3 \pi}{8}$
${e}^{\frac{3 \pi}{8} i} = \cos \left(\frac{3 \pi}{8}\right) + i \sin \left(\frac{3 \pi}{8}\right)$
$= \cos 67.5 + i \sin 67.5$
$= 0.383 + 0.924 i$

$x = \frac{19 \pi}{6}$
${e}^{\frac{19 \pi}{6} i} = \cos \left(\frac{19 \pi}{6}\right) + i \sin \left(\frac{19 \pi}{6}\right)$
$= \cos 570 + i \sin 570$
$= - 0.866 - 0.500 i$

Putting these two values together,

$0.383 + 0.924 i + 0.866 + 0.500 i = 1.249 + 1.424 i$

You can check the trigonometric values, there's a margin for error with rounding (or if I've accidentally done radians rather than degrees or vice versa on a calculator) but the equation is correct. Simply use Euler's formula, enter your values for $x$ and solve with a calculator.