De Moivre’s Theorem

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 $sin 3 x,$ $cos 7 x$ ) 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 $(z)$ 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 ${cos}^{2} θ + {sin}^{2} θ \equiv 1$
know how to expand ${(x + y)}^{n}$ when $n$ is a positive integer

Learning Outcomes

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

1 De Moivre’s theorem

2 De Moivre’s theorem and root finding

2.1 Arg(

z

) revisited