Perform the operation shown below and leave the result in trigonometric form. $\left[ 4 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) \right] \left[ 3 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right) \right]$
A) $7 \left( \cos \frac { 23 \pi } { 21 } + i \sin \frac { 23 \pi } { 21 } \right)$
B) $12 \left( \cos \frac { 23 \pi } { 21 } + i \sin \frac { 23 \pi } { 21 } \right)$
C) $12 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
D) $12 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right)$
E) $7 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right)$

Question

Perform the operation shown below and leave the result in trigonometric form. $\left[ 4 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right) \right] \left[ 3 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right) \right]$
A) $7 \left( \cos \frac { 23 \pi } { 21 } + i \sin \frac { 23 \pi } { 21 } \right)$
B)  $12 \left( \cos \frac { 23 \pi } { 21 } + i \sin \frac { 23 \pi } { 21 } \right)$
C)  $12 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$
D)  $12 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right)$
E)  $7 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right)$

Quizplus · Accepted Answer

The product of two complex numbers in trigonometric form, $r_1(\cos \theta_1 + i\sin \theta_1)$ and $r_2(\cos \theta_2 + i\sin \theta_2)$, is given by $r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$. Here, $r_1 = 4$, $r_2 = 3$, $\theta_1 = \frac{2\pi}{3}$, and $\theta_2 = \frac{3\pi}{7}$. Multiplying $r_1$ and $r_2$ gives $12$, and adding $\theta_1$ and $\theta_2$ gives $\frac{2\pi}{3} + \frac{3\pi}{7} = \frac{14\pi + 9\pi}{21} = \frac{23\pi}{21}$. Therefore, the result is $12(\cos \frac{23\pi}{21} + i\sin \frac{23\pi}{21})$.

Perform the Operation Shown Below and Leave the Result in Trigonometric