Question

Use DeMoivre's Theorem to find (4cis (pi/18))^3

Answer 1

$\bf \qquad \textit{power of two complex numbers} \\\\\ [\quad r[cos(\theta)+isin(\theta)]\quad ]^n\implies r^n[cos(n\cdot \theta)+isin(n\cdot \theta)]\\\\ -------------------------------$

$\bf 4\left[ cos\left( \frac{\pi }{18} \right)+i~sin\left( \frac{\pi }{18} \right) \right]^3\implies 4^3\left[ cos\left(3\cdot \frac{\pi }{18} \right)+i~sin\left( 3\cdot \frac{\pi }{18} \right) \right] \\\\\\ 64\left[ cos\left( \frac{\pi }{6} \right)+i~sin\left( \frac{\pi }{6} \right) \right]\implies 64\left[\frac{\sqrt{3}}{2}+i~\frac{1}{2} \right] \\\\\\ 32\sqrt{3}+32i$