Trigonometric Identities Peggy Adamson

Mathematics Learning Centre

Trigonometric Identities Peggy Adamson

c 1986

University of Sydney

Contents
1 Introduction 1.1 1.2 1.3 1.4 How to use this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 2 2 4 7 11 12 13 14

2 Relations between the trigonometric functions 3 The Pythagorean identities 4 Sums and differences of angles 5 Double angle formulae 6 Applications of the sum, difference, and double angle formulae 7 Self assessment 8 Solutions to exercises

Mathematics Learning Centre, University ofSydney

1

1
1.1

Introduction
How to use this book

You will not gain much by just reading this booklet. Have pencil and paper ready to work through the examples before reading their solutions. Do all the exercises. It is important that you try hard to complete the exercises on your own, rather than refer to the solutions as soon as you are stuck.

1.2

Introduction

This unit isdesigned to help you learn, or revise, trigonometric identities. You need to know these identities, and be able to use them confidently. They are used in many different branches of mathematics, including integration, complex numbers and mechanics. The best way to learn these identities is to have lots of practice in using them. So we remind you of what they are, then ask you to work through examplesand exercises. We’ve tried to select exercises that might be useful to you later, in your calculus unit of study.

1.3

Objectives

By the time you have worked through this workbook you should • be familiar with the trigonometric functions sin, cos, tan, sec, csc and cot, and with the relationships between them, • know the identities associated with sin2 θ + cos2 θ = 1, • know theexpressions for sin, cos, tan of sums and differences of angles, • be able to simplify expressions and verify identities involving the trigonometric functions, • know how to differentiate all the trigonometric functions, • know expressions for sin 2θ, cos 2θ, tan 2θ and use them in simplifying trigonometric functions, • know how to reduce expressions involving powers and products of trigonometric functions tosimple forms which can be integrated.

Mathematics Learning Centre, University of Sydney

2

1.4

Pretest

We shall assume that you are familiar with radian measure for angles, and with the definitions and properties of the trigonometric functions sin, cos, tan. This test is included to help you check how well you remember these. 1. Express in radians angles of i. 60◦ ii. 135◦ iii. 270◦2. Express in degrees angles of π 3π i. ii. − 4 2 3. What are the values of π i. sin ii. 2 iv. sin 7π 6 V.

iii.

2π

cos cos

3π 2 5π 3

iii. vi.

tan

3π 4

tan 2π

4. Sketch the graph of y = cos x.

2

Relations between the trigonometric functions

Recall the definitions of the trigonometric functions by means of the unit circle, x2 + y 2 = 1.

sin θ = y cos θ = xθ

(x, y)

tan θ =

y x

Three more functions are defined in terms of these, secant (sec), cosecant (cosec or csc) and cotangent (cot). 1 cos θ 1 sin θ 1 tan θ

sec θ = csc θ = cot θ =

(1) (2) (3)

Mathematics Learning Centre, University of Sydney

3

The functions cos and sin are the basic ones. Each of the others can be expressed in terms of these. In particular

tan θ = cotθ =

sin θ cos θ cos θ sin θ

(4) (5)

These relationships are identities, not equations. An equation is a relation between functions that is true only for some particular values of the variable. π For example, the relation sin θ = cos θ is an equation, since it is satisfied when θ = , but 4 not for other values of θ between 0 and π. On the other hand, tan θ = sin θ is true for all values...