Chapter 6.6, Problem 4E

To determine

To fill in the blanks:

Theorem states, if $n$ is the positive integer and $z=r\left(\cos \theta +i\sin \theta \right)$ is the complex number, then $z^n=r^n\left(\cos n\theta +i\sin n\theta \right)$ .

Answer to Problem 4E

De Moivre’s Theorem states, if $n$ is the positive integer and $z=r\left(\cos \theta +i\sin \theta \right)$ is the complex number, then $z^n=r^n\left(\cos n\theta +i\sin n\theta \right)$ .

Explanation of Solution

Given information:

The incomplete sentence is:

‘ Theorem states that if $n$ is the positive integer and $z=r\left(\cos \theta +i\sin \theta \right)$ is the complex number, then $z^n=r^n\left(\cos n\theta +i\sin n\theta \right)$ .’

This theorem gives the formula for computing the powers of the complex numbers. Firstly, gain the intuition for theorem of de Moivre’s considering what’s happen when multiply the complex numbers itself.

Therefore, the complete sentence is:

‘De Moivre’s Theorem states, if $n$ is the positive integer and $z=r\left(\cos \theta +i\sin \theta \right)$ is the complex number, then $z^n=r^n\left(\cos n\theta +i\sin n\theta \right)$ .’