# How do you multiply e^((11pi )/ 12 ) * e^( pi i )  in trigonometric form?

Jul 27, 2018

color(purple)(e^((11 pi)/(12) i) * e^(( pi) i) ~~ 0.9659 - 0.2588 i, IV Quadrant.

#### Explanation:

${e}^{\frac{11 \pi}{12} i} \cdot {e}^{\left(\pi\right) i}$

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

$\therefore {e}^{\frac{11 \pi}{12} i} = \left(\cos \left(\frac{11 \pi}{12}\right) + i \sin \left(\frac{11 \pi}{12}\right)\right)$

$= - 0.9659 + 0.2588 i$, II Quadrant

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

$= - 1 ,$ II Quadrant.

$\therefore {e}^{\frac{11 \pi}{12} i} \cdot {e}^{\left(\pi\right) i}$

$\approx \left(- 0.9659 + 0.2588 i\right) \cdot \left(- 1\right)$

$\approx 0.9659 - 0.2588 i$

color(purple)(e^((11 pi)/(12) i) * e^(( pi) i) ~~ 0.9659 - 0.2588 i, IV Quadrant.

Verification :

$\implies {e}^{i} \left(\left(\frac{11 \pi}{12}\right) + \left(\pi\right)\right)$

$\implies {e}^{i} \left(\frac{23 \pi}{12}\right)$

$\implies \cos \left(\frac{23 \pi}{12}\right) + i \sin \left(\frac{23 \pi}{12}\right)$

$\implies 0.9659 - 0.2588 i$, IV Quadrant.