Question

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

Answer 1

Answer:

Option C.

Step-by-step explanation:

The given expression is

$\left(\sqrt{2}cis \dfrac{9\pi}{20}\right)^5$

It can be rewritten as

$\left(\sqrt{2}\cos \dfrac{9\pi}{20}+i\sin\dfrac{9\pi}{20}\right)^5$

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

$(\sqrt{2})^5\left(\cos \dfrac{9\pi\times 5}{20}+i\sin\dfrac{9\pi\times 5}{20}\right)$

$=4\sqrt{2}\left(\cos \dfrac{9\pi}{4}+i\sin\dfrac{9\pi}{4}\right)$

$=4\sqrt{2}\left(\cos (2\pi +\dfrac{\pi}{4})+i\sin (2\pi+\dfrac{\pi}{4})\right)$

$=4\sqrt{2}\left(\cos \dfrac{\pi}{4}+i\sin \dfrac{\pi}{4}\right)$

$=4\sqrt{2}\left(\dfrac{1}{\sqrt{2}}+i\dfrac{1}{\sqrt{2}}\right)$

$=4+4i$

Therefore, the correct option is C.