Question

Use DeMoivre's Theorem to find the following. Write your answer in
standard form.

Answer 1

Option C.

Explanation:

The given expression is

`\left(√(2)cis (9\pi)/(20)\right)^5`

It can be rewritten as

`\left(√(2)\cos (9\pi)/(20)+i\sin(9\pi)/(20)\right)^5`

According to De moivre's Theorem:

`[r(\cos\theta+i\sin \theta)]^n=r^n(\cos n\theta +i\sin n\theta)`

Using De moivre's Theorem, we get

`(√(2))^5\left(\cos (9\pi* 5)/(20)+i\sin(9\pi* 5)/(20)\right)`

`=4√(2)\left(\cos (9\pi)/(4)+i\sin(9\pi)/(4)\right)`

`=4√(2)\left(\cos (2\pi +(\pi)/(4))+i\sin (2\pi+(\pi)/(4))\right)`

`=4√(2)\left(\cos (\pi)/(4)+i\sin (\pi)/(4)\right)`

`=4√(2)\left((1)/(√(2))+i(1)/(√(2))\right)`

`=4+4i`

Therefore, the correct option is C.