Chapter 6.6, Problem 94E

To determine

To calculate:

The power of following complex number in the standard form:

  ${\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5$

Answer to Problem 94E

The power of ${\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5$ in the standard form is $32i$ .

Explanation of Solution

Given information:

  ${\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5$

Formula used:

The DeMoivre’s theorem for complex numbers is given as:

If $z=r\left(\cos +i\sin \theta \right)$ is complex number it has positive integer $n$ then

  $\begin{array}{l}{z}^n={\left[r\left(\cos \theta +i\sin \theta \right)\right]}^n\\ z^n=r^n\left(\cos n\theta +i\sin n\theta \right)\end{array}$

Calculation:

The complex number in the trigonometric form is given as:

  ${\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5$

By using DeMoivre’s theorem:

  $\begin{array}{l}{z}^n=r^n\left(\cos n\theta +i\sin n\theta \right)\\ {\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5={\left(2\right)}^5\left[\cos \left(\frac{5\pi }{10}\right)+i\sin \left(\frac{5\pi }{10}\right)\right]\\ {\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5=32\left[\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right]\\ {\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5=32\left(0+1i\right)\\ {\left[2\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)\right]}^5=32i\end{array}$