Question

Use DeMoivre's theorem to evaluate the expression

Answer 1

Answer:

B. $\dfrac{9}{2}-i\dfrac{9\sqrt{3}}{2}$

Step-by-step explanation:

DeMoivre's theorem:

If the complex number $z=r(\cos \alpha+i\sin\alpha),$ then for natural number n

$z^n=r^n(\cos n\alpha+i\sin n\alpha).$

In your case,

$z=\sqrt{3}\left(\cos\dfrac{5\pi}{12}+i\sin\dfrac{5\pi}{12}\right)$

and you have to evaluate $z^4.$

By DeMoivre's theorem,

$z^4=(\sqrt{3})^4\left(\cos 4\cdot \dfrac{5\pi}{12}+i\sin 4\cdot \dfrac{5\pi}{12}\right)=9\left(\cos \dfrac{5\pi}{3}+i\sin\dfrac{5\pi}{3}\right)=9\left(\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}\right)=\dfrac{9}{2}-i\dfrac{9\sqrt{3}}{2}.$