Use DeMoivre's Theorem to find the indicated power of the following complex number.
$$\left[ 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) \right] ^ { 12 }$$
A) 2048
B) 4,096
C) $4,096 + 4096 \sqrt { 3 }$
D) $2,048 + 2048 \sqrt { 3 }$
E) $- 4,096$

Question

Quizplus · Accepted Answer

DeMoivre's Theorem states that $(r(\cos \theta + i\sin \theta))^n = r^n (\cos n\theta + i\sin n\theta)$. Here, $r = 2$, $n = 12$, and $\theta = \frac{\pi}{3}$. Thus, $2^{12} = 4096$ and $\cos(12 \cdot \frac{\pi}{3}) = \cos(4\pi) = 1$, $\sin(12 \cdot \frac{\pi}{3}) = \sin(4\pi) = 0$, leading to a real number $4096$.

Use DeMoivre's Theorem to Find the Indicated Power of the Following