Multiply the complex numbers below and leave the result in trigonometric form. $\left[ 7 \left( \cos \frac { 6 \pi } { 7 } + i \sin \frac { 6 \pi } { 7 } \right) \right] \left[ 2 \left( \cos \frac { 4 \pi } { 9 } + i \sin \frac { 4 \pi } { 9 } \right) \right]$
A) $9 \left( \cos \frac { 82 \pi } { 63 } + i \sin \frac { 82 \pi } { 63 } \right)$
B) $9 \left( \cos \frac { 10 \pi } { 63 } + i \sin \frac { 10 \pi } { 63 } \right)$
C) $14 \left( \cos \frac { 10 \pi } { 63 } + i \sin \frac { 10 \pi } { 63 } \right)$
D) $14 \left( \cos \frac { 82 \pi } { 63 } + i \sin \frac { 82 \pi } { 63 } \right)$
E) $14 \left( \cos \frac { 58 \pi } { 63 } + i \sin \frac { 58 \pi } { 63 } \right)$

Question

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, $r_1 = 7$, $r_2 = 2$, $\theta_1 = \frac{6\pi}{7}$, and $\theta_2 = \frac{4\pi}{9}$. Multiplying the magnitudes gives $7 \cdot 2 = 14$, and adding the angles gives $\frac{6\pi}{7} + \frac{4\pi}{9} = \frac{54\pi + 28\pi}{63} = \frac{82\pi}{63}$. Therefore, the result is $14 \left( \cos \frac{82\pi}{63} + i\sin \frac{82\pi}{63} \right)$.

Multiply the Complex Numbers Below and Leave the Result in Trigonometric