Question

For questions 16 - 19, write each expression in the standard form for the complex number a + bi. 1/2(cos(72)+isin(72))^5

Answer 1

Given:

The expression is:

$\dfrac{1}{2}(\cos (72)+i\sin (72))^5$

To find:

The $a+bi$ form for the given expression.

Solution:

According to De Moivre's theorem,

$(\cos \theta+i\sin \theta)^n=\cos (n\theta )+i\sin (n\theta )$

We have,

$\dfrac{1}{2}(\cos (72)+i\sin (72))^5$

Using De Moivre's theorem, we get

$=\dfrac{1}{2}(\cos (72\times 5)+i\sin (72\times 5))$

$=\dfrac{1}{2}(\cos (360)+i\sin (360))$

$=\dfrac{1}{2}(1+i(0))$

$=\dfrac{1}{2}+0i$

It is the $a+bi$ form of the given expression. Here, $a=\dfrac{1}{2},\ b=0$.

Therefore, the required expression is $\dfrac{1}{2}+0i$.