Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.

Question

Quizplus · Accepted Answer

We can use the formulae $z_1 z_2 = r_1 r_2 (\cos(	heta_1+	heta_2) + i\sin(	heta_1+	heta_2))$ to multiply the given complex numbers in trigonometric form.

For options A, B, and C, the product involves a negative value for the real part, meaning that the angle $	heta$ needs to be greater than $\pi/2$ for the trigonometric form to be valid. However, for these options, the angles are all less than $\pi/2$ which means that their real parts will be positive. Hence, these options are not viable.

For option D, we have
$$
\begin{aligned}
z_1 z_2 &= (4 \cos \frac{5\pi}{6} + 4i \sin \frac{5\pi}{6}) (-3 \cos \frac{\pi}{3} + 3i \sin \frac{\pi}{3}) \
&= -12 \cos (\frac{\pi}{3} - \frac{5\pi}{6}) - 12i \sin(\frac{\pi}{3} - \frac{5\pi}{6}) \
&= 12 ( \cos \frac{\pi}{6} - i\sin\frac{\pi}{6})\
&= 12(\frac{\sqrt{3}}{2} - \frac{i}{2}) \
&= 6\sqrt{3} - 6i.
\end{aligned}
$$

For option E, we have
$$
\begin{aligned}
z_1 z_2 &= (-3 \cos \frac{\pi}{3} + 3i \sin \frac{\pi}{3}) (8 \cos \frac{\pi}{4} + 8i \sin \frac{\pi}{4}) \
&= -24 \cos(\frac{\pi}{3} - \frac{\pi}{4}) - 24 i \sin (\frac{\pi}{3} - \frac{\pi}{4})\
&= -12\sqrt{6} - 12i\sqrt{2}.
\end{aligned}
$$
Option D gives the correct answer, so the answer is D.

Perform the Indicated Operation Using Trigonometric Form