# How do you evaluate  e^( ( 11 pi)/6 i) - e^( ( 17 pi)/8 i) using trigonometric functions?

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i} = 0.884 {e}^{- 1.505 i}$

#### Explanation:

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i} = x + i y$

${e}^{\frac{11 \pi}{6} i} = \cos \left(\frac{11 \pi}{6}\right) + i \sin \left(\frac{11 \pi}{6}\right)$

${e}^{\frac{17 \pi}{8} i} = \cos \left(\frac{17 \pi}{8}\right) + i \sin \left(\frac{17 \pi}{8}\right)$

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i} = \left(\cos \left(\frac{11 \pi}{6}\right) + i \sin \left(\frac{11 \pi}{6}\right)\right) - \left(\cos \left(\frac{17 \pi}{8}\right) + i \sin \left(\frac{17 \pi}{8}\right)\right)$

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i}$
$= \left(\cos \left(\frac{11 \pi}{6}\right) - \cos \left(\frac{17 \pi}{8}\right)\right)$
$+ i \left(\sin \left(\frac{11 \pi}{6}\right) - \sin \left(\frac{17 \pi}{8}\right)\right)$

$\cos \left(\frac{11 \pi}{6}\right) - \cos \left(\frac{17 \pi}{8}\right) = - 2 \sin \left(\frac{\frac{11 \pi}{6} + \frac{17 \pi}{8}}{2}\right) \sin \left(\frac{\frac{11 \pi}{6} - \frac{17 \pi}{8}}{2}\right)$

$\frac{11 \pi}{6} + \frac{17 \pi}{8} = \frac{8 \times 11 \pi}{8 \times 6} + \frac{6 \times 17 \pi}{6 \times 8}$

$= \frac{88 \pi}{48} + \frac{102 \pi}{48} = \frac{\left(88 + 102\right) \pi}{48}$

$\frac{11 \pi}{6} + \frac{17 \pi}{8} = \frac{190 \pi}{48}$
$\frac{11 \pi}{6} + \frac{17 \pi}{8} = \frac{95 \pi}{24}$

$\frac{11 \pi}{6} - \frac{17 \pi}{8} = \frac{8 \times 11 \pi}{8 \times 6} - \frac{6 \times 17 \pi}{6 \times 8}$

$= \frac{88 \pi}{48} - \frac{102 \pi}{48} = \frac{\left(88 - 102\right) \pi}{48}$

$\frac{11 \pi}{6} - \frac{17 \pi}{8} = \frac{- 14 \pi}{48}$
$\frac{11 \pi}{6} - \frac{17 \pi}{8} = \frac{- 7 \pi}{24}$

$\frac{1}{2} \left(\frac{11 \pi}{6} + \frac{17 \pi}{8}\right) = \frac{95 \pi}{48}$

$\frac{1}{2} \left(\frac{11 \pi}{6} - \frac{17 \pi}{8}\right) = \frac{- 7 \pi}{48}$

$- 2 \sin \left(\frac{\frac{11 \pi}{6} + \frac{17 \pi}{8}}{2}\right) \sin \left(\frac{\frac{11 \pi}{6} - \frac{17 \pi}{8}}{2}\right) = - 2 \sin \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right)$

$\cos \left(\frac{11 \pi}{6}\right) - \cos \left(\frac{17 \pi}{8}\right) = - 2 \sin \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right)$

$\sin \left(\frac{11 \pi}{6}\right) - \sin \left(\frac{17 \pi}{8}\right) = 2 \cos \left(\frac{\frac{11 \pi}{6} + \frac{17 \pi}{8}}{2}\right) \sin \left(\frac{\frac{11 \pi}{6} - \frac{17 \pi}{8}}{2}\right)$

$\frac{1}{2} \left(\frac{11 \pi}{6} + \frac{17 \pi}{8}\right) = \frac{95 \pi}{48}$

$\frac{1}{2} \left(\frac{11 \pi}{6} - \frac{17 \pi}{8}\right) = \frac{- 7 \pi}{48}$

$\sin \left(\frac{11 \pi}{6}\right) - \sin \left(\frac{17 \pi}{8}\right) = 2 \cos \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right)$

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i}$
$= \left(\cos \left(\frac{11 \pi}{6}\right) - \cos \left(\frac{17 \pi}{8}\right)\right)$
$+ i \left(\sin \left(\frac{11 \pi}{6}\right) - \sin \left(\frac{17 \pi}{8}\right)\right)$

$\cos \left(\frac{11 \pi}{6}\right) - \cos \left(\frac{17 \pi}{8}\right) = - 2 \sin \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right)$
$\sin \left(\frac{11 \pi}{6}\right) - \sin \left(\frac{17 \pi}{8}\right) = 2 \cos \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right)$

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i} = - 2 \sin \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right) + i \left(2 \cos \left(\frac{95 \pi}{48}\right) \sin \left(\frac{- 7 \pi}{48}\right)\right)$

$\frac{95 \pi}{48} = 6.218$
$\frac{7 \pi}{48} = 0.458$

$\cos \left(\frac{95 \pi}{48}\right) = 0.998$
$\sin \left(\frac{95 \pi}{48}\right) = - 0.0654$
$\cos \left(\frac{- 7 \pi}{48}\right) = 0.897$
$\sin \left(\frac{- 7 \pi}{48}\right) = - 0.442$

-2(-0.0654)xx(-0.442)+i(2(0.998)xx(-0.442)

$2 \times 0.0654 \times 0.442 - 2 \times 0.998 \times 0.442 i$

${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i} = 0.058 - 0.882 i$

r=sqrt(0.058^2+(-0.882)^2=0.884

$\theta = {\tan}^{-} 1 \frac{- 0.882}{0.058} = - 1.505$

$0.058 - 0.882 i = 0.884 \left(\cos \left(- 1.505\right) + i \sin \left(- 1.505\right)\right)$

cos(-1.505)+isin(-1.505)=e^((-1.505)i
${e}^{\frac{11 \pi}{6} i} - {e}^{\frac{17 \pi}{8} i} = 0.884 {e}^{- 1.505 i}$