Find Quotients of Complex Numbers in Polar Form
-$$\begin{array} { l }
\mathrm { z } _ { 1 } = 6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right) \
\mathrm { z } _ { 2 } = 12 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)
\end{array}$$
A) $\frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
B) $\left. \frac { 1 } { 2 } \cos \frac { 2 \pi } { 3 } - i \sin \frac { 2 \pi } { 3 } \right)$
C) $\frac { 1 } { 2 } \left( \cos \frac { 4 \pi } { 3 } - \mathrm { i } \sin \frac { 4 \pi } { 3 } \right)$
D) $\frac { 1 } { 2 } \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$

Question

Find Quotients of Complex Numbers in Polar Form
-$$\begin{array} { l } 
\mathrm { z } _ { 1 } = 6 \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right) \
\mathrm { z } _ { 2 } = 12 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)
\end{array}$$
A) $\frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
B) $\left. \frac { 1 } { 2 } \cos \frac { 2 \pi } { 3 } - i \sin \frac { 2 \pi } { 3 } \right)$
C) $\frac { 1 } { 2 } \left( \cos \frac { 4 \pi } { 3 } - \mathrm { i } \sin \frac { 4 \pi } { 3 } \right)$
D) $\frac { 1 } { 2 } \left( \cos \frac { 4 \pi } { 3 } + i \sin \frac { 4 \pi } { 3 } \right)$

Quizplus · Accepted Answer

The quotient of two complex numbers in polar form is obtained by dividing their magnitudes and subtracting their angles. For $z_1$ and $z_2$, the magnitude of the quotient is $\frac{6}{12} = \frac{1}{2}$, and the angle is $\frac{3\pi}{2} - \frac{5\pi}{6} = \frac{9\pi - 5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3}$. Therefore, the quotient is $\frac{1}{2}(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3})$.

Find Quotients of Complex Numbers in Polar Form - A) 12(cos⁡2π3+isin⁡2π3)\frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)21​(cos32π​+isin32π​)

Find Quotients of Complex Numbers in Polar Form
- A) $\frac { 1 } { 2 } \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$