Question

Use demovires theorem to write the complex number in trigonometric form.

Answer 1

$\sqrt3+i=2\left(\dfrac{\sqrt3}2+\dfrac12i\right)$

$\begin{cases}\cos\dfrac\pi6=\dfrac{\sqrt3}2\\\\\sin\dfrac\pi6=\dfrac12\end{cases}$

$\implies\sqrt3+i=2e^{i\pi/6}$

By DeMoivre's theorem,

$(\sqrt3+i)^3=(2e^{i\pi/6})^3=8e^{i\pi/2}=8\left(\cos\dfrac\pi2+i\sin\dfrac\pi2\right)=8i$