Find the product or quotient, as indicated. Leave your answer in polar form.
-Find the product of $\mathrm { z } _ { 1 }$ and $\mathrm { z } _ { 2 }$.
$$z _ { 1 } = 7 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right) , z _ { 2 } = 5 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$$
A) $35 \left( \cos \frac { 5 } { 6 } \pi + i \sin \frac { 5 } { 6 } \pi \right)$
B) $12 \left( \cos \frac { 23 } { 12 } \pi + i \sin \frac { 23 } { 12 } \pi \right)$
C) $12 \left( \cos \frac { 5 } { 6 } \pi + i \sin \frac { 5 } { 6 } \pi \right)$
D) $35 \left( \cos \frac { 23 } { 12 } \pi + i \sin \frac { 23 } { 12 } \pi \right)$

Question

Quizplus · Accepted Answer

The product of two complex numbers in polar form, $z_1 = r_1(\cos \theta_1 + i\sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i\sin \theta_2)$, is given by $z_1z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$. Here, $r_1 = 7$, $r_2 = 5$, $\theta_1 = \frac{5\pi}{4}$, and $\theta_2 = \frac{2\pi}{3}$. Thus, the product is $35[\cos(\frac{5\pi}{4} + \frac{2\pi}{3}) + i\sin(\frac{5\pi}{4} + \frac{2\pi}{3})]$, which simplifies to $35[\cos(\frac{23\pi}{12}) + i\sin(\frac{23\pi}{12})]$.

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

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