Teach Yourself Trigonometry

176 Pages·2003·2.62 MB·English

by  P. Abbott

This Page Intentionally No Longer Blank
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teach
yourself
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trigonometry
p. abbott
revised by hugh neill
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Rrst published In UK 1992 by Hodder Arnold, 338 Euston Road, London, NW1 3BH.
First published in US 1992 by Contemporary Books, a Division of the McGraw-Hill
Companies, 1 Prudential Plaza, 130 East Randolph Street, Chicago, IL 60601 USA.
This edition published 2003.
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CopyrightC 1970,1991,1998,2003, Paul Abbot and Hugh Neill
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preface viii
0
01 hbdoricalbackg~nd 1
Introduction 2
what is trigonometry? 3
the origins of trigonometry 3
02 the tangent 5
Introduction 6
the idea of the tangent ratio 7
a definition of tangent 8
values of the tangent 9
notation for angles and sides 10
using tangents 10
opposite and adjacent sides 14
03 sine and cosine 18
introduction 19
definition of sine and cosine 20
using the sine and cosine 21
trigonometric ratios of 45°, 30° and 60° 25
using the calculator accurately 27
slope and gradient 27
projections 28
muHistage problems 30
04 in three dimensions 35
introduction 36
pyramid problems 36
box problems 39
wedge problems 41
05 angles of any magnitude 45
vi
introduction 46
n
;0. sine and cosine for any angle 46
i graphs of sine and cosine functions 48
the tangent of any angle 50
graph of the tangent function 51
sine, cosine and tangent 52
06 solving simple equations 53
introduction 54
solving equations involving sines 54
solving equations involving cosines 57
solving equations involving tangents 59
07 the sine and cosine fonnulae 62
notation 63
area of a triangle 63
the sine fonnula for a triangle 66
the ambiguous case 68
the cosine fonnula for a triangle 69
introduction to surveying 73
finding the height of a distant object 73
distance of an inaccessible object 75
distance between two inaccessible but visible objects 76
triangulation 76
08 radians 83
introduction 84
radians 84
length of a circular arc 84
converting from radians to degrees 86
area of a circular sector 87
09 relations between the ratios 90
introduction 91
secant, cosecant and cotangent 91
10 ratios of compound angles 96
compound angles 97
formulae for sin(A + B) and sin(A-B) 97
formulae for cos(A + B) and cos(A-B) 99
fonnulae for tan(A + B) and tan(A-B) 100
worked examples 101
muHiple angle fonnulae 103
identities 106
vii
more trigonometric equations 108
n
11 the form a sin x + b cos x 110 ;0.
introduction 111 i
the form Y= a sin X+ bcos x 111
using the alternative form 114
12 the factor formulae 118
the first set of factor formulae 119
the second set of factor formulae 122
13 circles related to a biangle 126
the circumcircle 127
the incircle 131
the ecircles 132
heron's formula: the area of a triangle 134
14 general solution of equations 138
the equation sin 8 =s in a 139
the equation cos 8 = cos a 141
the equation tan 8 = tan a 141
summary of results 142
glossary 145
summary of btgonomebic formulae 148
anSW81'8 151
index 163
I I
viii
Teach Yourself Trigonometry has been substantially revised and
rewritten to take account of modern needs and recent develop
ments in the subject.
It is anticipated that every reader will have access to a scientific
calculator which has sines, cosines and tangents, and their
inverses. It is also important that the calculator has a memory,
so that intermediate results can be stored accurately. No sup
port has been given about how to use the calculator, except in
the most general terms. Calculators vary considerably in the
keystrokes which they use, and what is appropriate for one cal
culator may be inappropriate for another.
There are many worked examples in the book, with complete,
detailed answers to all the questions. At the end of each worked
example, you will find the symbol I to indicate that the exam
ple has been completed, and what follows is text.
Some of the exercises from the original Teach Yourself
Trigonometry have been used in this revised text, but all the
answers have been reworked to take account of the greater
accuracy available with calculators.
I would like to thank Linda Moore for her help in reading and
correcting the text. But the responsibility for errors is mine.
Hugh Neill
November 1997
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In this chapter you will learn:
• what trigonometry is
• a little about its origins.
1.1 Introduction
2
i One of the earliest known examples of the practical application
i of geometry was the problem of finding the height of one of the
Egyptian pyramids. This was solved by Thales, the Greek
! philosopher and mathematician (c. 640 BC to 550 BC) using sim
ilar triangles.
i
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fig1R1.1
Thales observed the length of the shadows of the pyramid and a
stick, AB, placed vertically into the ground at the end of the
shadow of the pyramid, shown in Figure 1.1.
QB and BC represent the lengths of the shadow of the pyramid
and the stick. Thales said 'The height of the pyramid is to the
length of the stick, as the length of the shadow of the pyramid
is to the length of the shadow of the stick.'
That is, in Figure 1.1,
PQ QB
=
AB BC"
As QB, AB, and BC are known, you can calculate PQ.
We are told that the king, Amasis, was amazed at Thales' appli
cation of an abstract geometrical principle to the solution of
such a problem.
This idea is taken up in Chapter 02, in introducing the idea of
the tangent ratio.