Complex Numbers - de Moivre's Theorem

De Moivre's Theorem gives a formula for calculating complex numbers. It enables us to connect complex numbers and trigonometry. Most importantly, it is incredibly useful for finding powers and roots of complex numbers. It can be stated in a number of ways:

\([r(cos\theta+isin\theta)]^n=r^n(cos\ n\theta+isin\ n\theta)\)

Whilst we can prove the formula using proof by induction, it becomes clear when we write it in Euler Form: \([re^{i\theta}]^n=r^ne^{in\theta}\)


Key Concepts

On this page, you should learn to

  • Find powers and roots of complex numbers using de Moivre's theorem
  • Carry out a proof by induction of de Moivre's theorem

Essentials

The following videos will help you understand all the concepts from this page

Proof of de Moivre's Theorem

In the following video, we look at an example of a proof by induction question applied to complex numbers. In it, we prove De Moivre's Theorem:

Let \(z=r(cosθ+isinθ)\)

Prove that \(z^{ n }≡r^{ n }[cos⁡(nθ)+isin(nθ)]\ ,\ n\in \mathbb{Z^+}\)

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

Find the roots of the equation \(z^3=8i \)

Express your answers in Cartesian Form

Hint

Full Solution

Question 2

\(z=-2+2\sqrt{3}i\)

a) Find |z| and arg(z)

b) Find \(z^6\) and simplify your answer

c) Given that \(w^4=z^3\) , find the values of \(w\) giving your answers in the form \(a+bi\)

Hint

Full Solution

 

Question 3

Find the values of n such that \((\sqrt{3}-i)^n\) is a real number

Hint

Full Solution

 

Question 4

a) By writing \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\), show that \(sin(\frac{\pi}{12})=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)

b) Work out \(cos(\frac{\pi}{12})\)

c) Hence, find the roots of the equation \(z^4=2+2\sqrt{3}i\), giving answers in the form \(z=a+ib\)

Hint

Full Solution

Question 5

a) Find the roots of the equation \(z^4-1=0\)

b) Find the roots of the equation \(z^4+1=0\)

c) Show that roots of \(z^4-1=0\) and \(z^4+1=0\) together make the roots of \(z^8-1=0\)

d) Hence, find all the roots to \(z^6+z^4+z^2+1=0\)

Hint

Full Solution

MY PROGRESS

How much of Complex Numbers - de Moivre's Theorem have you understood?