Use demovires theorem to write the complex number in trigonometric form. (√(3) + i)^(3)

Question

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


`(√(3) + i)^(3)`

Answer 1

`\sqrt3+i=2\left(\frac{\sqrt3}2+\frac12i\right)`


`\begin{cases}\cos\frac\pi6=\frac{\sqrt3}2\\\\\sin\frac\pi6=\frac12\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\frac\pi2+i\sin\frac\pi2\right)=8i`