Question

Use DeMoivre's theorem to evaluate the expression

[√3( cos 5pi/12 + i sin 5pi/12)]^4 ? write the answer in rectangular form

Answer 1

B.
`(9)/(2)-i(9√(3))/(2)`

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=√(3)\left(\cos(5\pi)/(12)+i\sin(5\pi)/(12)\right)`

and you have to evaluate
`z^4.`

By DeMoivre's theorem,

`z^4=(√(3))^4\left(\cos 4\cdot (5\pi)/(12)+i\sin 4\cdot (5\pi)/(12)\right)=9\left(\cos (5\pi)/(3)+i\sin(5\pi)/(3)\right)=9\left((1)/(2)-i(√(3))/(2)\right)=(9)/(2)-i(9√(3))/(2).`