Find the product or quotient, as indicated. Leave your answer in polar form.
-Find the product of $\mathrm { z } _ { 1 }$ and $\mathrm { z } _ { 2 }$.
$$z _ { 1 } = 8 \left( \cos \frac { 4 \pi } { 7 } + i \sin \frac { 4 \pi } { 7 } \right) , z _ { 2 } = \frac { 1 } { 5 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$
A) $40 \left( \cos \frac { 19 } { 21 } \pi + i \sin \frac { 19 } { 21 } \pi \right)$
B) $\frac { 41 } { 5 } \left( \cos \frac { 4 } { 21 } \pi + i \sin \frac { 4 } { 21 } \pi \right)$
C) $\frac { 8 } { 5 } \left( \cos \frac { 4 } { 21 } \pi + i \sin \frac { 4 } { 21 } \pi \right)$
D) $\frac { 8 } { 5 } \left( \cos \frac { 19 } { 21 } \pi + i \sin \frac { 19 } { 21 } \pi \right)$

Question

Quizplus · Accepted Answer

To find the product of two complex numbers in polar form, you multiply their magnitudes and add their angles. Here, the magnitudes are $8$ and $\frac{1}{5}$, and the angles are $\frac{4\pi}{7}$ and $\frac{\pi}{3}$. The product's magnitude is $8 \times \frac{1}{5} = \frac{8}{5}$, and the sum of the angles is $\frac{4\pi}{7} + \frac{\pi}{3} = \frac{12\pi + 7\pi}{21} = \frac{19\pi}{21}$. Therefore, the product in polar form is $\frac{8}{5} \left( \cos \frac{19\pi}{21} + i \sin \frac{19\pi}{21} \right)$.

Find the Product or Quotient, as Indicated z1\mathrm { z } _ { 1 }z1​

Find the Product or Quotient, as Indicated $\mathrm { z } _ { 1 }$