### Introduction

In this Section we introduce De Moivre’s theorem and examine some of its consequences. We shall see that one of its uses is in obtaining relationships between trigonometric functions of multiple angles (like $sin3x,$ $cos7x$ ) and powers of trigonometric functions (like ${sin}^{2}x$ , ${cos}^{4}x$ ). Another important use of De Moivre’s theorem is in obtaining complex roots of polynomial equations. In this application we re-examine our definition of the argument arg $\left(z\right)$ of a complex number.

#### Prerequisites

- be familiar with the polar form of a complex number
- be familiar with the Argand diagram
- be familiar with the trigonometric identity ${cos}^{2}\theta +{sin}^{2}\theta \equiv 1$
- know how to expand ${\left(x+y\right)}^{n}$ when $n$ is a positive integer

#### Learning Outcomes

- employ De Moivre’s theorem in a number of applications
- fully define the argument arg $\left(z\right)$ of a complex number
- obtain complex roots of complex numbers