Find the product. Write the product in rectangular form, using exact values.
-$\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right]$
A) $- 24 \sqrt { 3 } - 24 \mathrm { i }$
B) $- 12 + 12 \sqrt { 3 } i$
C) $24 \mathrm { i }$
D) 24

Question

Quizplus · Accepted Answer

To find the product of two complex numbers in polar form, you multiply their magnitudes and add their angles. Here, the magnitudes are 8 and 6, and the angles are 240° and 330°. The product's magnitude is $8 \times 6 = 48$, and the sum of the angles is $240° + 330° = 570°$, which is equivalent to $210°$ (since $570° - 360° = 210°$). The rectangular form of a complex number is given by $r(\cos \theta + i\sin \theta)$, where $r$ is the magnitude and $\theta$ is the angle. For $210°$, $\cos 210° = -\sqrt{3}/2$ and $\sin 210° = -1/2$. Thus, the product in rectangular form is $48[-\sqrt{3}/2 - (1/2)i] = -24\sqrt{3} - 24i$.

Find the Product [8cis⁡240∘][6cis⁡330∘]\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right][8cis240∘][6cis330∘]

Find the Product $\left[ 8 \operatorname { cis } 240 ^ { \circ } \right] \left[ 6 \operatorname { cis } 330 ^ { \circ } \right]$