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:

`((1)/(2)(cos((\pi)/(5))+isin((\pi)/(5))))^5`

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

Apply exponential property to simplify the expression.

We have:

`((1)/(2)(cos((\pi)/(5))+\imaginaryI s\imaginaryI n((\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} (((1)/(2))^5(cos((\pi)/(5)*5))+isin((\pi)/(5)*5)) \\ \\ (1)/(32)(cos(\pi)+isin(\pi)) \end{gathered}`

ANSWER:

`(1)/(32)(cos(\pi)+\imaginaryI s\imaginaryI n(\pi))`