Multiply the complex numbers.Write the answer in trigonometric form. $9 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right) 4 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right)$
A) $36 \left( \cos \frac { 67 \pi } { 42 } + i \sin \frac { 67 \pi } { 42 } \right)$
B) $36 \left( \cos \frac { 10 \pi } { 13 } + i \sin \frac { 10 \pi } { 13 } \right)$
C) $36 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D) $13 \left( \cos \frac { 67 \pi } { 42 } + i \sin \frac { 67 \pi } { 42 } \right)$
E) $13 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Question

Multiply the complex numbers.Write the answer in trigonometric form.   $9 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right) 4 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right)$  
A) $36 \left( \cos \frac { 67 \pi } { 42 } + i \sin \frac { 67 \pi } { 42 } \right)$
B) $36 \left( \cos \frac { 10 \pi } { 13 } + i \sin \frac { 10 \pi } { 13 } \right)$
C)  $36 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$
D)  $13 \left( \cos \frac { 67 \pi } { 42 } + i \sin \frac { 67 \pi } { 42 } \right)$
E)  $13 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Quizplus · Accepted Answer

When multiplying complex numbers in trigonometric form, $r_1(\cos \theta_1 + i\sin \theta_1) \cdot r_2(\cos \theta_2 + i\sin \theta_2) = r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$. Here, $r_1 = 9$, $r_2 = 4$, $\theta_1 = \frac{7\pi}{6}$, and $\theta_2 = \frac{3\pi}{7}$. Multiplying $r_1$ and $r_2$ gives $36$, and adding $\theta_1$ and $\theta_2$ gives $\frac{7\pi}{6} + \frac{3\pi}{7} = \frac{67\pi}{42}$.

Multiply the Complex Numbers 9(cos⁡7π6+isin⁡7π6)4(cos⁡3π7+isin⁡3π7)9 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right) 4 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right)9(cos67π​+isin67π​)4(cos73π​+isin73π​)

Multiply the Complex Numbers $9 \left( \cos \frac { 7 \pi } { 6 } + i \sin \frac { 7 \pi } { 6 } \right) 4 \left( \cos \frac { 3 \pi } { 7 } + i \sin \frac { 3 \pi } { 7 } \right)$