Find the given power. Write answer in rectangular form.
-$\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.$$^{18}$
A) $- 1$
B) $\mathrm { i }$
C) $- \mathrm { i }$

Question

Quizplus · Accepted Answer

Using De Moivre's Theorem, $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$. Here, $n = 18$ and $\theta = 30^\circ$, so $n\theta = 540^\circ$. Since $540^\circ$ is equivalent to $180^\circ$ in the unit circle, $\cos(540^\circ) = \cos(180^\circ) = -1$ and $\sin(540^\circ) = \sin(180^\circ) = 0$, resulting in $-1 + 0i = -1$.

Find the Given Power (cos⁡30∘+isin⁡30∘)\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.(cos30∘+isin30∘)

Find the Given Power $\left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ }) \right.$