Multiply the complex numbers below and leave the result in trigonometric form. $\left[ 8 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right) \right] \left[ 10 \left( \cos \frac { 5 \pi } { 9 } + i \sin \frac { 5 \pi } { 9 } \right) \right]$
A) $18 \left( \cos \frac { 53 } { 63 } \pi + i \sin \frac { 53 } { 63 } \pi \right)$
B) $80 \left( \cos \frac { 23 } { 63 } \pi + i \sin \frac { 23 } { 63 } \pi \right)$
C) $80 \left( \cos \frac { 53 } { 63 } \pi + i \sin \frac { 53 } { 63 } \pi \right)$
D) $18 \left( \cos \frac { 1 } { 9 } \pi + i \sin \frac { 1 } { 9 } \pi \right)$
E) $80 \left( \cos \frac { 1 } { 9 } \pi + i \sin \frac { 1 } { 9 } \pi \right)$

Question

Multiply the complex numbers below and leave the result in trigonometric form.   $\left[ 8 \left( \cos \frac { 2 \pi } { 7 } + i \sin \frac { 2 \pi } { 7 } \right) \right] \left[ 10 \left( \cos \frac { 5 \pi } { 9 } + i \sin \frac { 5 \pi } { 9 } \right) \right]$
A) $18 \left( \cos \frac { 53 } { 63 } \pi + i \sin \frac { 53 } { 63 } \pi \right)$
B)  $80 \left( \cos \frac { 23 } { 63 } \pi + i \sin \frac { 23 } { 63 } \pi \right)$
C)  $80 \left( \cos \frac { 53 } { 63 } \pi + i \sin \frac { 53 } { 63 } \pi \right)$
D)  $18 \left( \cos \frac { 1 } { 9 } \pi + i \sin \frac { 1 } { 9 } \pi \right)$
E) $80 \left( \cos \frac { 1 } { 9 } \pi + i \sin \frac { 1 } { 9 } \pi \right)$

Quizplus · Accepted Answer

When multiplying complex numbers in trigonometric form, you multiply the magnitudes and add the angles. Here, the magnitudes are 8 and 10, so their product is 80. The angles are $2\pi/7$ and $5\pi/9$, so their sum is $(18\pi + 35\pi)/63 = 53\pi/63$. Therefore, the correct answer is $80(\cos(53\pi/63) + i\sin(53\pi/63))$.

Multiply the Complex Numbers Below and Leave the Result in Trigonometric