# How do you evaluate  e^( ( 7 pi)/4 i) - e^( ( 8 pi)/3 i) using trigonometric functions?

May 3, 2016

$0.881 - 1.692 i$

#### Explanation:

According to Euler's formula,

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

If we substitute in values for $x$ from the question, we get

${e}^{\frac{7 \pi}{4} i} = \cos \left(\frac{7 \pi}{4}\right) + i \sin \left(\frac{7 \pi}{4}\right)$
$= \cos 315 + i \sin 315$
$= 0.707 - 0.707 i$

${e}^{\frac{8 \pi}{3} i} = \cos \left(\frac{8 \pi}{3}\right) + i \sin \left(\frac{8 \pi}{3}\right)$
$= \cos 100 + i \sin 100$
$= - 0.174 + 0.985 i$

Now you have the two parts of the question, you put them together and solve arithmetically:

${e}^{\frac{7 \pi}{4} i} - {e}^{\frac{8 \pi}{3} i}$
$= \left(0.707 - 0.707 i\right) - \left(- 0.174 + 0.985 i\right)$
$= 0.707 + 0.174 - 0.707 i - 0.985 i$

$= 0.881 - 1.692 i$