# How do you evaluate e^( ( pi)/12 i) - e^( ( 5 pi)/8 i) using trigonometric functions?

$\left\{\cos \left(\frac{\pi}{12}\right) - \cos \left(\frac{5 \pi}{8}\right)\right\} + i \left\{\sin \left(\frac{\pi}{12}\right) - \sin \left(\frac{5 \pi}{8}\right)\right\}$
Use Euler's formula ${e}^{\left(\theta\right) i} = \cos \left(\theta\right) + i \sin \left(\theta\right)$ on both terms:
${e}^{\left(\frac{\pi}{12}\right) i} - {e}^{\left(\frac{5 \pi}{8}\right) i} =$
$\left\{\cos \left(\frac{\pi}{12}\right) - \cos \left(\frac{5 \pi}{8}\right)\right\} + i \left\{\sin \left(\frac{\pi}{12}\right) - \sin \left(\frac{5 \pi}{8}\right)\right\}$