Perform the operation shown below and leave the result in trigonometric form. $$\left[ 6 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right) \right] \left[ 8 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right) \right]$$
A) $14 \left( \cos 230 ^ { \circ } + i \sin 230 ^ { \circ } \right)$
B) $48 \left( \cos 76 ^ { \circ } + i \sin 76 ^ { \circ } \right)$
C) $14 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
D) $14 \left( \cos 8 ^ { \circ } + i \sin 8 ^ { \circ } \right)$
E) $48 \left( \cos 230 ^ { \circ } + i \sin 230 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

Multiplying two complex numbers in trigonometric form, $r_1(\cos \theta_1 + i\sin \theta_1)$ and $r_2(\cos \theta_2 + i\sin \theta_2)$, results in a new complex number $r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$. Here, $6 \times 8 = 48$ and $40^\circ + 190^\circ = 230^\circ$, leading to the answer $48(\cos 230^\circ + i\sin 230^\circ)$.

Perform the Operation Shown Below and Leave the Result in Trigonometric