# How do you evaluate  e^( ( 7 pi)/4 i) - e^( ( 5 pi)/3 i) using trigonometric functions?

Aug 19, 2017

The answer is $= \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}\right) - i \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right)$

#### Explanation:

We apply Euler's formula

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

So,

${e}^{\frac{7}{4} \pi i} - {e}^{\frac{5}{3} \pi i} =$

$\cos \left(\frac{7}{4} \pi\right) + i \sin \left(\frac{7}{4} \pi\right) - \cos \left(\frac{5}{3} \pi\right) - i \sin \left(\frac{5}{3} \pi\right)$

We calculate separately

$\cos \left(\frac{7}{4} \pi\right) = \cos \left(\pi + \frac{3}{4} \pi\right) = \cos \left(\pi\right) \cos \left(\frac{3}{4} \pi\right) - \sin \left(\pi\right) \sin \left(\frac{3}{4} \pi\right)$

$= - 1 \cdot - \frac{\sqrt{2}}{2} - 0 = \frac{\sqrt{2}}{2}$

$\sin \left(\frac{7}{4} \pi\right) = \sin \left(\pi + \frac{3}{4} \pi\right) = \sin \left(\pi\right) \cos \left(\frac{3}{4} \pi\right) + \sin \left(\frac{3}{4} \pi\right) \cos \left(\pi\right)$

$= 0 + \frac{\sqrt{2}}{2} \cdot - 1 = - \frac{\sqrt{2}}{2}$

$\cos \left(\frac{5}{3} \pi\right) = \cos \left(\pi + \frac{2}{3} \pi\right) = \cos \left(\pi\right) \cos \left(\frac{2}{3} \pi\right) - \sin \left(\pi\right) \sin \left(\frac{2}{3} \pi\right)$

$= - 1 \cdot - \frac{\sqrt{3}}{2} - 0 = \frac{\sqrt{3}}{2}$

$\sin \left(\frac{5}{3} \pi\right) = \sin \left(\pi + \frac{2}{3} \pi\right) = \sin \left(\pi\right) \cos \left(\frac{2}{3} \pi\right) + \sin \left(\frac{2}{3} \pi\right) \cos \left(\pi\right)$

$= 0 + \frac{\sqrt{3}}{2} \cdot - 1 = - \frac{\sqrt{3}}{2}$

${e}^{\frac{7}{4} \pi i} - {e}^{\frac{5}{3} \pi i} = \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}\right) - i \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right)$