# How do you evaluate  2 e^( ( 3 pi)/8 i) - 5 e^( ( 19 pi)/8 i) using trigonometric functions?

Apr 26, 2018

$\therefore 2 {e}^{\frac{3 \pi}{8} i} - 5 {e}^{\frac{19 \pi}{8} i} \approx - 1.15 - 2.77 i$

#### Explanation:

$2 {e}^{\frac{3 \pi}{8} i} - 5 {e}^{\frac{19 \pi}{8} i}$

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

$\frac{3 \pi}{8} \approx 1.178097 , \frac{19 \pi}{8} \approx 7.461823$

$\therefore 2 {e}^{\frac{3 \pi}{8} i} = 2 \left(\cos \left(\frac{3 \pi}{8}\right) + i \sin \left(\frac{3 \pi}{8}\right)\right)$

$= 0.765367 + 1.847759 i$

$\therefore 5 {e}^{\frac{19 \pi}{8} i} = 5 \left(\cos \left(\frac{19 \pi}{8}\right) + i \sin \left(\frac{19 \pi}{8}\right)\right)$

$\approx 1.913417 + 4.619397 i$

$\therefore 2 {e}^{\frac{3 \pi}{8} i} - 5 {e}^{\frac{19 \pi}{8} i}$

$= \left(0.765367 + 1.847759 i\right) - \left(1.913417 + 4.619397 i\right)$

$= \left(- 1.148050 - 2.771638 i\right)$

$\therefore 2 {e}^{\frac{3 \pi}{8} i} - 5 {e}^{\frac{19 \pi}{8} i} \approx - 1.15 - 2.77 i \left(2 \mathrm{dp}\right)$ [Ans]