vi Contents
11 Introduction to trigonometry
97
11.1 Trigonometry
97
11.2 The theorem of Pythagoras
97
11.3 Trigonometric ratios of acute angles
98
11.4 Evaluating trigonometric ratios
100
11.5 Solution of right-angled triangles
105
11.6 Angles of elevation and depression
106
11.7 Sine and cosine rules
108
11.8 Area of any triangle
108
11.9 Worked problems on the solution of
triangles and finding their areas
109
11.10 Further worked problems on solving
triangles and finding their areas
110
11.11 Practical situations involving
trigonometry
111
11.12 Further practical situations involving
trigonometry
113
12 Cartesian and polar co-ordinates
117
12.1 Introduction
117
12.2 Changing from Cartesian into polar
co-ordinates
117
12.3 Changing from polar into Cartesian
co-ordinates
119
12.4 Use of Pol/Rec functions on calculators
120
13 The circle and its properties
122
13.1 Introduction
122
13.2 Properties of circles
122
13.3 Radians and degrees
123
13.4 Arc length and area of circles and sectors 124
13.5 The equation of a circle
127
13.6 Linear and angular velocity
129
13.7 Centripetal force
130
Revision Test 4
133
14 Trigonometric waveforms
134
14.1 Graphs of trigonometric functions
134
14.2 Angles of any magnitude
135
14.3 The production of a sine and cosine wave 137
14.4 Sine and cosine curves
138
14.5 Sinusoidal form A sin(ωt ± α)
143
14.6 Harmonic synthesis with complex
waveforms
146
15 Trigonometric identities and equations
152
15.1 Trigonometric identities
152
15.2 Worked problems on trigonometric
identities
152
15.3 Trigonometric equations
154
15.4 Worked problems (i) on trigonometric
equations
154
15.5 Worked problems (ii) on trigonometric
equations
156
15.6 Worked problems (iii) on trigonometric
equations
157
15.7 Worked problems (iv) on trigonometric
equations
157
16 The relationship between trigonometric and
hyperbolic functions
159
16.1 The relationship between trigonometric
and hyperbolic functions
159
16.2 Hyperbolic identities
160
17 Compound angles
163
17.1 Compound angle formulae
163
17.2 Conversion of a sinωt + b cosωt into
R sin(ωt + α)
165
17.3 Double angles
169
17.4 Changing products of sines and cosines
into sums or differences
170
17.5 Changing sums or differences of sines and
cosines into products
171
17.6 Power waveforms in a.c. circuits
173
Revision Test 5
177
18 Functions and their curves
178
18.1 Standard curves
178
18.2 Simple transformations
181
18.3 Periodic functions
186
18.4 Continuous and discontinuous functions
186
18.5 Even and odd functions
186
18.6 Inverse functions
188
18.7 Asymptotes
190
18.8 Brief guide to curve sketching
196
18.9 Worked problems on curve sketching
197
19 Irregular areas, volumes and mean values of
waveforms
203
19.1 Areas of irregular figures
203
19.2 Volumes of irregular solids
205
19.3 The mean or average value of a waveform 206
Revision Test 6
212
20 Complex numbers
213
20.1 Cartesian complex numbers
213
20.2 The Argand diagram
214
20.3 Addition and subtraction of complex
numbers
214
20.4 Multiplication and division of complex
numbers
216
11 Introduction to trigonometry
97
11.1 Trigonometry
97
11.2 The theorem of Pythagoras
97
11.3 Trigonometric ratios of acute angles
98
11.4 Evaluating trigonometric ratios
100
11.5 Solution of right-angled triangles
105
11.6 Angles of elevation and depression
106
11.7 Sine and cosine rules
108
11.8 Area of any triangle
108
11.9 Worked problems on the solution of
triangles and finding their areas
109
11.10 Further worked problems on solving
triangles and finding their areas
110
11.11 Practical situations involving
trigonometry
111
11.12 Further practical situations involving
trigonometry
113
12 Cartesian and polar co-ordinates
117
12.1 Introduction
117
12.2 Changing from Cartesian into polar
co-ordinates
117
12.3 Changing from polar into Cartesian
co-ordinates
119
12.4 Use of Pol/Rec functions on calculators
120
13 The circle and its properties
122
13.1 Introduction
122
13.2 Properties of circles
122
13.3 Radians and degrees
123
13.4 Arc length and area of circles and sectors 124
13.5 The equation of a circle
127
13.6 Linear and angular velocity
129
13.7 Centripetal force
130
Revision Test 4
133
14 Trigonometric waveforms
134
14.1 Graphs of trigonometric functions
134
14.2 Angles of any magnitude
135
14.3 The production of a sine and cosine wave 137
14.4 Sine and cosine curves
138
14.5 Sinusoidal form A sin(ωt ± α)
143
14.6 Harmonic synthesis with complex
waveforms
146
15 Trigonometric identities and equations
152
15.1 Trigonometric identities
152
15.2 Worked problems on trigonometric
identities
152
15.3 Trigonometric equations
154
15.4 Worked problems (i) on trigonometric
equations
154
15.5 Worked problems (ii) on trigonometric
equations
156
15.6 Worked problems (iii) on trigonometric
equations
157
15.7 Worked problems (iv) on trigonometric
equations
157
16 The relationship between trigonometric and
hyperbolic functions
159
16.1 The relationship between trigonometric
and hyperbolic functions
159
16.2 Hyperbolic identities
160
17 Compound angles
163
17.1 Compound angle formulae
163
17.2 Conversion of a sinωt + b cosωt into
R sin(ωt + α)
165
17.3 Double angles
169
17.4 Changing products of sines and cosines
into sums or differences
170
17.5 Changing sums or differences of sines and
cosines into products
171
17.6 Power waveforms in a.c. circuits
173
Revision Test 5
177
18 Functions and their curves
178
18.1 Standard curves
178
18.2 Simple transformations
181
18.3 Periodic functions
186
18.4 Continuous and discontinuous functions
186
18.5 Even and odd functions
186
18.6 Inverse functions
188
18.7 Asymptotes
190
18.8 Brief guide to curve sketching
196
18.9 Worked problems on curve sketching
197
19 Irregular areas, volumes and mean values of
waveforms
203
19.1 Areas of irregular figures
203
19.2 Volumes of irregular solids
205
19.3 The mean or average value of a waveform 206
Revision Test 6
212
20 Complex numbers
213
20.1 Cartesian complex numbers
213
20.2 The Argand diagram
214
20.3 Addition and subtraction of complex
numbers
214
20.4 Multiplication and division of complex
numbers
216