Question

Whichexpression is equivalent to (=[cos(™) + i sin(™)])³ ?[cos()+sin()[cos(m) + sin(m)][cos()+sin()][cos(m) + sin(m)]10

Answer 1

Given the expression:

$(\frac{1}{2}(cos(\frac{\pi}{5})+isin(\frac{\pi}{5})))^5$

Let's select the expresssion which is equivalent to the given expression.

Apply exponential property to simplify the expression.

We have:

$(\frac{1}{2}(cos(\frac{\pi}{5})+\imaginaryI s\imaginaryI n(\frac{\pi}{5})))^5$

Apply De Moivre's Theorem:

$(r(cos\theta+isin\theta))^n=r^n(cos(\theta n)+isin(\theta n))$

Thus, we have:

$\begin{gathered} ((\frac{1}{2})^5(cos(\frac{\pi}{5}*5))+isin(\frac{\pi}{5}*5)) \\ \\ \frac{1}{32}(cos(\pi)+isin(\pi)) \end{gathered}$

ANSWER:

$\frac{1}{32}(cos(\pi)+\imaginaryI s\imaginaryI n(\pi))$