Perform the indicated operation. Write the answer in the form a + bi.
-4(cos 315° + i sin 315°) · 6(cos 225° + i sin 225°)
A) -24i
B) 24i
C) -24
D) 24

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, $r_1 = 4$, $r_2 = 6$, $\theta_1 = 315^\circ$, and $\theta_2 = 225^\circ$. Thus, the product is $4 \cdot 6(\cos(315^\circ + 225^\circ) + i\sin(315^\circ + 225^\circ)) = 24(\cos(540^\circ) + i\sin(540^\circ))$. Since $\cos(540^\circ) = \cos(180^\circ) = -1$ and $\sin(540^\circ) = \sin(180^\circ) = 0$, the result is $24(-1 + 0i) = -24$.

Perform the Indicated Operation