Question

Divide.

6√3 cis(7π/6) ÷ 3√5 cis(π/3)

Answer 1

`\frac{6\sqrt 3~ \text{cis} \left( \frac{7 \pi} 6 \right)}{3 \sqrt 5 ~\text{cis} \left( (\pi)/(3)\right)}\\\\\\=\frac{2\sqrt 3 \cdot e^{i\tfrac{7\pi}{6}}}{\sqrt 5\cdot e^{i \tfrac{\pi}{3}}}\\\\\\=(2 \sqrt 3)/(\sqrt 5) \cdot \left(e^i \right)^{\tfrac{7\pi}{6} - \tfrac{\pi }{3}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[\text{cis}~ \theta = e^(i \theta)\right]\\\\\\=(2 √(3))/(\sqrt 5) \cdot \left(e^i\right)^{\tfrac{5\pi}6}\\\\\\=(2√(3))/(\sqrt 5)\cdot e^{i\tfrac{5 \pi}6}\\\\\\`

`=(2√(3))/(\sqrt 5) \left[ \cos \left((5\pi)/(6)\right) + i \sin \left((5\pi)/(6)\right) \right]~~~~~~~~~~~~~~~~~~;\left[e^(i\theta) = \cos \theta + i \sin \theta} \right]\\\\\\=(2√(3))/(\sqrt 5)\left[ \cos \left(2 * \frac{\pi}2 -(\pi)/(6) \right) + i\sin \left(2 * \frac{\pi}2 -(\pi)/(6) \right) \right]\\\\\\=(2√(3))/(\sqrt 5) \left[ -\cos \left((\pi)/(6)\right) + i \sin \left((\pi)/(6)\right) \right]\\\\\\`

`=(2√(3))/(\sqrt 5)\left( -\frac {√(3)}{2} + i \cdot \frac 12 \right)\\\\\\=-(3)/(\sqrt 5) + i (\sqrt 3)/(\sqrt 5)`

Answer 2

Explanation:

Just took the test:

6sqrt3 cis (7pie/6)/ 3sqrt5 cis (pie/3)=

2sqrt15/5 cis (5pie/6)