Question

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

A) 32+32root3i
B) 32root3 + 32i
C) 6+6root3i
D) 6root3 +6i

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( (\pi )/(18) \right)+i~sin\left( (\pi )/(18) \right) \right]^3\implies 4^3\left[ cos\left(3\cdot (\pi )/(18) \right)+i~sin\left( 3\cdot (\pi )/(18) \right) \right] \\\\\\ 64\left[ cos\left( (\pi )/(6) \right)+i~sin\left( (\pi )/(6) \right) \right]\implies 64\left[(√(3))/(2)+i~(1)/(2) \right] \\\\\\ 32√(3)+32i`