Chapter 9, Problem 22RE

To determine

To find:Thestandard form $a+bi$ of the given expression.

Answer to Problem 22RE

The standard form $a+bi$ of the expression ${\left[3\left(\cos 20°+i\sin 20°\right)\right]}^3$ is $\frac{27}{2}+\frac{27\sqrt{3}}{2}i$ .

Explanation of Solution

Given information:

The given expressionis ${\left[3\left(\cos 20°+i\sin 20°\right)\right]}^3$ .

Calculation:

Using the De Moivre’s theorem which states that $z^n=r^n\left[\cos \left(n\theta \right)+i\sin \left(n\theta \right)\right]$

Here, $z=r\left[\cos \left(n\theta \right)+i\sin \left(n\theta \right)\right]$ is a complex number and $n$ is positive integer.

Calculate the standard form $a+bi$ .

  $\begin{array}{c}{\left[3\left(\cos 20°+i\sin 20°\right)\right]}^3=3^3\left(\cos \left(3\cdot 20°\right)+i\sin \left(3\cdot 20°\right)\right)\\ =27\left(\cos 60°+i\sin 60°\right)\\ =27\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)\\ =\frac{27}{2}+\frac{27\sqrt{3}}{2}i\end{array}$

Therefore, the standard form $a+bi$ of the expression ${\left[3\left(\cos 20°+i\sin 20°\right)\right]}^3$ is $\frac{27}{2}+\frac{27\sqrt{3}}{2}i$ .