Write the expression in the standard form a + bi.
-$\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 }$
A) $- \frac { 9 } { 2 } - \frac { 9 \sqrt { 3 } } { 2 } i$
B) $- \frac { 9 } { 2 } + \frac { 9 \sqrt { 3 } } { 2 } i$
C) $- \frac { 9 \sqrt { 3 } } { 2 } + \frac { 9 } { 2 } \mathrm { i }$
D) $- \frac { 9 \sqrt { 3 } } { 2 } - \frac { 9 } { 2 } i$

Question

Quizplus · Accepted Answer

Using De Moivre's theorem, $\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 } = 3^2(\cos(4 \cdot \frac{5\pi}{6}) + i\sin(4 \cdot \frac{5\pi}{6})) = 9(\cos\frac{10\pi}{3} + i\sin\frac{10\pi}{3})$. Simplifying the angle, $\frac{10\pi}{3}$ is equivalent to $\frac{4\pi}{3}$ (subtracting $2\pi$ for one full rotation), which corresponds to $\cos\frac{4\pi}{3} = -\frac{1}{2}$ and $\sin\frac{4\pi}{3} = -\frac{\sqrt{3}}{2}$. Thus, the expression becomes $9(-\frac{1}{2} - \frac{\sqrt{3}}{2}i) = -\frac{9}{2} - \frac{9\sqrt{3}}{2}i$.

Write the Expression in the Standard Form a + Bi [3(cos⁡5π6+isin⁡5π6)]4\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 }[3​(cos65π​+isin65π​)]4

Write the Expression in the Standard Form a + Bi $\left[ \sqrt { 3 } \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \right] ^ { 4 }$