Find the product. Write the product in rectangular form, using exact values.
-$\left[ 8 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) \right] \left[ 6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right) \right]$
A) $12 \sqrt { 3 } + 12 \mathrm { i }$
B) $- 12 + 12 \sqrt { 3 } \mathrm { i }$
C) $24 \mathrm { i }$
D) $- 48$

Question

Quizplus · Accepted Answer

The product of two complex numbers in polar form, $r_1(\cos \theta_1 + i\sin \theta_1)$ and $r_2(\cos \theta_2 + i\sin \theta_2)$, is given by $r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$. Here, $r_1 = 8$, $\theta_1 = 210^\circ$, $r_2 = 6$, and $\theta_2 = 330^\circ$. The product is $8 \times 6(\cos(210^\circ + 330^\circ) + i\sin(210^\circ + 330^\circ)) = 48(\cos 540^\circ + i\sin 540^\circ)$. Since $\cos 540^\circ = \cos 180^\circ = -1$ and $\sin 540^\circ = \sin 180^\circ = 0$, the product in rectangular form is $-48$.

Find the Product A) 123+12i12 \sqrt { 3 } + 12 \mathrm { i }123​+12i

Find the Product A) $12 \sqrt { 3 } + 12 \mathrm { i }$