Question

Represent the following complex number in trigonometric form:
$(-\sqrt{3}+i{)}^3$

Answer 1

$z=-\sqrt{3}+i=r{e}^{i\theta }$(let)

$r=\sqrt{3+1}=4$

$\theta ={\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)=5\pi /6$

$z=2e^{i5\pi /6}$

$z^3=2e^{3i\pi /2}$

$⟹2(2\cos 5\pi /2+i\sin 5\pi /2)$

$⟹2i$