Multiply the complex numbers below and leave the result in trigonometric form. $$\left[ 0.1 \left( \cos 260 ^ { \circ } + i \sin 260 ^ { \circ } \right) \right] \left[ 0.9 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right) \right]$$
A) $0.09 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$
B) $0.09 \left( \cos 120 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
C) $\cos 40 ^ { \circ } + i \sin 40 ^ { \circ }$
D) $0.09 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
E) $\cos 120 ^ { \circ } + i \sin 120 ^ { \circ }$

Question

Multiply the complex numbers below and leave the result in trigonometric form. $$\left[ 0.1 \left( \cos 260 ^ { \circ } + i \sin 260 ^ { \circ } \right) \right] \left[ 0.9 \left( \cos 220 ^ { \circ } + i \sin 220 ^ { \circ } \right) \right]$$ ​
A) $0.09 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$
B)​ $0.09 \left( \cos 120 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
C)​ $\cos 40 ^ { \circ } + i \sin 40 ^ { \circ }$
D)​ $0.09 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right)$
E)​ $\cos 120 ^ { \circ } + i \sin 120 ^ { \circ }$

Quizplus · Accepted Answer

When multiplying complex numbers in trigonometric form, $\left[ r_1(\cos \theta_1 + i\sin \theta_1) \right] \left[ r_2(\cos \theta_2 + i\sin \theta_2) \right]$, the result is $r_1r_2(\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2))$. Here, $r_1 = 0.1$, $r_2 = 0.9$, $\theta_1 = 260^\circ$, and $\theta_2 = 220^\circ$. Multiplying $r_1$ and $r_2$ gives $0.09$, and adding $\theta_1$ and $\theta_2$ gives $480^\circ$, which is equivalent to $120^\circ$ (since $480^\circ - 360^\circ = 120^\circ$). Therefore, the result is $0.09(\cos 120^\circ + i\sin 120^\circ)$.

Multiply the Complex Numbers Below and Leave the Result in Trigonometric