# How do you evaluate  e^( ( pi)/4 i) - e^( ( 11 pi)/8 i) using trigonometric functions?

Aug 13, 2018

The answer is $= \frac{\sqrt{2}}{2} - \frac{\sqrt{2 - \sqrt{2}}}{2} + i \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2}\right)$

#### Explanation:

Apply Euler's Identity

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

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

$= \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$

${e}^{i \frac{11}{8} \pi} = \cos \left(\frac{11}{8} \pi\right) + i \sin \left(\frac{11}{8} \pi\right)$

$\cos \left(2 \theta\right) = 2 {\cos}^{2} \theta - 1$

$\cos \theta = \sqrt{\frac{1 + \cos 2 \theta}{2}}$

cos(11/8pi)=sqrt((1+cos(11/4pi)/2)

$= \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}$

$= \frac{\sqrt{2 - \sqrt{2}}}{2}$

$\cos \left(2 \theta\right) = 1 - 2 {\sin}^{2} \theta$

$\sin \theta = \sqrt{\frac{1 - \cos \left(2 \theta\right)}{2}}$

$\sin \left(\frac{11}{8} \pi\right) = \sqrt{\frac{1 - \cos \left(\frac{11}{4} \pi\right)}{2}}$

$= \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$

$= \frac{\sqrt{2 + \sqrt{2}}}{2}$

Finally,

${e}^{i \frac{\pi}{4}} - {e}^{i \frac{11}{8} \pi}$

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

$= \frac{\sqrt{2}}{2} - \frac{\sqrt{2 - \sqrt{2}}}{2} + i \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2 + \sqrt{2}}}{2}\right)$