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Chapter 20

Angles and triangles
20
...
1)
...
This chapter involves the measurement of
angles and introduces types of triangle
...
2

c

Angular measurement

h e
g f

R

An angle is the amount of rotation between two straight
lines
...

If a circle is divided into 360 equal parts, then each part
is called 1 degree and is written as 1◦
i
...

or

1 revolution = 360◦
1
1 degree is
th of a revolution
360

Some angles are given special names
...

• An angle equal to 90◦ is called a right angle
...

• Any angle greater than 180◦ and less than 360◦ is
called a reflex angle
...

• If two angles add up to 90◦ they are called complementary angles
...

• Parallel lines are straight lines which are in the same
plane and never meet
...
1
...
1016/B978-1-85617-697-2
...
1

With reference to Figure 20
...
Such pairs of
angles are called vertically opposite angles
...
Such pairs of
angles are called corresponding angles
...
Such pairs of angles are called
alternate angles
...
Such pairs of
angles are called interior angles
...
2
...

i
...


1 degree = 60 minutes

which is written as

1◦ = 60
...
41◦ 29
29◦
= 41
...

1 minute further subdivides into 60 seconds,
1 minute = 60 seconds

i
...


which is written as

1 = 60
...

43◦ 29
+ 27◦ 43
71◦ 12
1◦

(i) 29 + 43 = 72
(ii) Since 60 = 1◦, 72 = 1◦12

(Notice that for minutes, 1 dash is used and for seconds,
2 dashes are used
...


20
...
2 Radians and degrees
One radian is defined as the angle subtended at the centre
of a circle by an arc equal in length to the radius
...
)
With reference to Figure 20
...

(iv) 43◦ + 27◦ + 1◦ (carried) = 71◦
...

This answer can be obtained using the
follows
...

1
...
Press ◦ ’ ’ ’
◦
4
...
Press +
6
...
Press ◦ ’ ’ ’
8
...

10
...

Problem 2
...

(ii) 1◦ or 60 is ‘borrowed’ from the degrees column,
which leaves 83◦ in that column
...

(iv) 83◦ − 56◦ = 27◦, which is placed in the degrees
column
...
2

When s is the whole circumference, i
...
when
s = 2πr,
s
2πr
θ= =
= 2π
r
r
In one revolution, θ = 360◦
...
e
...
30◦
1 rad =
π

Here are some worked examples on angular measurement
...

3
...
Enter 84
2
...
Press ’ ’ ’
5
...

◦
7
...
Enter 39
9
...
Press =
Answer = 27◦ 34

calculator as
Enter 13
Enter 56
Press ◦ ’ ’ ’

Thus, 84◦ 13 − 56◦39 = 27◦ 34
...


Evaluate 19◦ 51 47 + 63◦ 27 34
19◦ 51 47
+ 63◦ 27 34
83◦ 19 21
1◦ 1

Angles and triangles
(i) 47 + 34 = 81
(ii) Since

60

=

1 , 81

= 1 21

(iii) The 21 is placed in the seconds column and 1
is carried in the minutes column
...
Enter 63
4
...
Press =

0
...
753 × 60 = 45
...
18 = 0
...


This answer can be obtained using the calculator as
follows
...
Enter 51
1
...
Press ◦ ’ ’ ’
◦
4
...
Enter 47
6
...
Press +
8
...
Press ◦ ’ ’ ’
◦
10
...
Press ’ ’ ’
12
...
Press ◦ ’ ’ ’
14
...

Problem 4
...
45◦ by calculator
60
27◦
39◦27 = 39
= 39
...

3
...
Enter 39
2
...
Press =
6
...
Press ’ ’ ’
◦
Answer = 39
...
Convert 63◦ 26 51 to degrees in
decimal form, correct to 3 decimal places
51
= 63◦ 26
...
85◦
= 63
...
85 = 63
60
63◦ 26 51 = 63
...

This answer can be obtained using the calculator as
follows
...
Enter 26
6
...
4475◦

Problem 6
...
753◦ to degrees, minutes
and seconds

(v) Since 60 = 1◦, 79 = 1◦19

(vii) 19◦ + 63◦ + 1◦ (carried) = 83◦
...


2
...
Enter 51
8
...
753◦ = 53◦ 45 11
This answer can be obtained using the calculator as
follows
...
Enter 53
...
Press =
Answer = 53◦45 10
...
Press ◦ ’ ’ ’
Now try the following Practice Exercise
Practice Exercise 76 Angular measurement
(answers on page 348)
1
...


Evaluate 76◦ 31 − 48◦37

3
...


Evaluate 41◦ 37 16 + 58◦ 29 36

5
...


Evaluate 79◦26 19 − 45◦58 56 + 53◦ 21 38

7
...


8
...


9
...
952◦ to degrees and minutes
...
Convert 58
...

Here are some further worked examples on angular
measurement
...
State the general name given to the
following angles: (a) 157◦ (b) 49◦ (c) 90◦ (d) 245◦
(a) Any angle between 90◦ and 180◦ is called an
obtuse angle
...

(b) Any angle between 0◦ and 90◦ is called an acute
angle
...

(c)

An angle equal to 90◦ is called a right angle
...

Thus, 245◦ is a reflex angle
...

48◦ 39

Find the angle complementary to

(b) An angle of 180◦ lies on a straight line
...
3(b),
180◦ = 53◦ + θ + 44◦
from which,

θ = 180◦ − 53◦ − 44◦ = 83◦

Problem 11
...
5

If two angles add up to 90◦ they are called complementary angles
...

74◦ 25

108⬚

Find the angle supplementary to

39⬚
␪

58⬚

If two angles add up to 180◦ they are called supplementary angles
...
5

180◦ − 74◦ 25 = 105◦ 35
Problem 10
...
4
...
54
...
63
...
4
...
143 m2

Exercise 111 (page 260)
1
...
59 m3

2
...
20
...
(a) 2 A (b) 50 V (c) 2
...
0
...
1 A
5
...
13 cm2 , 368
...

3
...

7
...


i − j − 4k
−i + 7j − k
−3i + 27j − 8k
i + 7
...
6i + 4
...
9k

2
...

6
...

10
...
5j − 10k
2i + 40j − 43k

Chapter 30
Exercise 118 (page 279)

2
...
5 V (b) 3 A
4
...
83 V (b) 0

1
...
5 sin(A + 63
...
(a) 20
...
62) volts
(b) 12
...
33) volts
3
...
395)

Chapter 29
Exercise 119 (page 281)
Exercise 113 (page 266)
1
...

2
...
scalar
4
...
scalar
6
...
vector 8
...
vector

Exercise 114 (page 273)
1
...

3
...

5
...
35 N at 18
...
62◦ to the 12 m/s velocity
16
...
57◦ to the 13 N force
28
...
30◦ to the 18 N force
32
...
80◦ to the 30 m displacement

1
...
5 sin(A + 63
...
(a) 20
...
62) volts
(b) 12
...
33) volts
3
...
395)

Exercise 120 (page 283)
1
...
5 sin(A + 63
...
(a) 20
...
62) volts
(b) 12
...
33) volts
3
...
395) 4
...
11 sin(ωt + 0
...
8
...
173)

Answers to practice exercises
Exercise 138 (page 324)
1
...
(a) 0
...
5 V
3
...
9 V/s
4
...
635 Pa/m

Chapter 35

5 4
x +c
4
2 3
x +c
3
...
(a) x 5 − x 2 + c
5
2

2
...
(a)
6
...

8
...


3x 2
2

− 5x + c

2
u2
ln x + c
(b)
− ln u + c
3
2
√
√
18 √ 5
14
...
(a)

Exercise 139 (page 328)
1
...
(a) −6 cos x + c
2
3 2x
12
...
(a)

355

4t 3
1
+c
(b) − + 4t +
t
3

Exercise 140 (page 330)

θ3
3

+c

1
...
5 (b) 0
...
(a) 105 (b) −0
...
(a) 6 (b) −1
...
(a) −0
...
833

5
...
67 (b) 0
...
(a) 0 (b) 4

7
...
248

8
...
2352 (b) 2
...
(a) 19
...
457

10
...
2703 (b) 9
...
proof
5
...
5
8
...
67

2
...
7
...
2
...
32
7
...
140 m

4
...
33 Nm

Index
Acute angle, 165
Acute angled triangle, 171
Adding waveforms, 278
Addition in algebra, 62
Addition law of probability, 307
Addition of fractions, 10
numbers, 1, 18
two periodic functions, 278
vectors, 267
by calculation, 270
Algebra, 61, 68
Algebraic equation, 61, 73
expression, 73
Alternate angles, 165, 191
Ambiguous case, 207
Amplitude, 199
Angle, 165
Angle, lagging and leading, 200
types and properties of, 165
Angles of any magnitude, 196
depression, 191
elevation, 191
Angular measurement, 165
velocity, 202
Annulus, 226
Arbitrary constant of integration, 325
Arc, 231
Arc length, 233
Area, 219
Area of common shapes, 219, 221
under a curve, 330
Area of circle, 222, 233
common shapes, 219
irregular figures, 257
sector, 222, 233
similar shapes, 229
triangles, 205
Arithmetic, basic, 1
Average, 299
value of waveform, 260
Axes, 130
Bar charts, 289
Base, 47
Basic algebraic operations, 61
BODMAS with algebra, 71
fractions, 13
numbers, 6

Boyle’s law, 46
Brackets, 6, 68
Calculation of resultant phasors, 281,
283
Calculations, 22, 28
Calculator, 22
addition, subtraction, multiplication
and division, 22
fractions, 26
π and e x functions, 28, 118
reciprocal and power functions, 24
roots and ×10 x functions, 25
square and cube functions, 23
trigonometric functions, 27
Calculus, 313
Cancelling, 10
Cartesian axes, 131
co-ordinates, 214
Charles’s law, 42, 142
Chord, 230
Circle, 222, 230, 233
equation of, 236
properties of, 230
Circumference, 230
Classes, 293
Class interval, 293
limits, 295
mid-point, 293, 295
Coefficient of proportionality, 45
Combination of two periodic functions,
278
Common factors, 69
logarithms, 111
prefixes, 53
shapes, 219
Complementary angles, 165
Completing the square, 105
Cone, 245
frustum of, 252
Congruent triangles, 175
Construction of triangles, 179
Continuous data, 288
Co-ordinates, 130, 131
Corresponding angles, 165
Cosine, 27, 183
graph of, 195
Cosine rule, 205, 281
wave, 195

Cross-multiplication, 75
Cube root, 23
Cubic equation, 161
graphs, 161
units, 240
Cuboid, 240
Cumulative frequency distribution,
293, 297
curve, 293
Cycle, 199
Cylinder, 241
Deciles, 304
Decimal fraction, 216
places, 13, 18
Decimals, 16
addition and subtraction, 19
multiplication and division, 19
Definite integrals, 328
Degrees, 27, 165, 166, 232
Denominator, 9
Dependent event, 307
Depression, angle of, 191
Derivatives, 315
standard list, 321
Derived units, 53
Determination of law, 147
involving logarithms, 150
Diameter, 230
Difference of two squares, 103
Differential calculus, 313
coefficient, 315
Differentiation, 313, 315
from first principles, 315
of ax n , 315
of e ax and ln ax, 320
of sine and cosine functions, 318
successive, 322
Direct proportion, 40, 42
Discrete data, 288
standard deviation, 302
Dividend, 63
Division in algebra, 62
Division of fractions, 12
numbers, 3, 4, 19
Divisor, 63
Drawing vectors, 266

358 Index
Parabola, 156
Parallel lines, 165
Parallelogram, 219
method, 267
Peak value, 199
Pentagon, 219
Percentage component bar chart, 289
error, 36
relative frequency, 289
Percentages, 33
Percentile, 304
Perfect square, 105
Perimeter, 171
Period, 199
Periodic function, 200
plotting, 238
Periodic time, 200
Phasor, 280
Pictograms, 289
Pie diagram, 289
Planimeter, 257
Plotting periodic functions, 238
Polar co-ordinates, 214
Pol/Rec function on calculator, 217
Polygon, 210
frequency, 293, 296
Population, 289
Power, 47
series for e x , 119
Powers and roots, 47
Practical problems
quadratic equations, 108
simple equations, 77
simultaneous equations, 96
straight line graphs, 141
trigonometry, 209
Precedence, 6, 71
Prefixes, 53
Presentation of grouped data, 292
statistical data, 288
Prism, 240, 242
Probability, 306
laws of, 307
Production of sine and cosine waves,
198
Proper fraction, 9
Properties of circles, 230
triangles, 171
Proportion, 40
Pyramid, 244
volumes and surface area of frustum
of, 252
Pythagoras’ theorem, 181
Quadrant, 230
Quadratic equations, 102

by completing the square, 105
factorization, 102
formula, 106
graphically, 156
practical problems, 108
Quadratic formula, 106
graphs, 156
Quadrilaterals, 219
properties of, 219
Quartiles, 303
Radians, 27, 165, 166, 232
Radius, 230
Range, 295
Ranking, 299
Rates of change, 323
Ratio and proportion, 40
Ratios, 40
Reciprocal, 24
Rectangle, 219
Rectangular axes, 131
co-ordinates, 131
prism, 240
Reduction of non-linear laws to linear
form, 147
Reflex angle, 165
Relative frequency, 289
velocity, 276
Resolution of vectors, 269
Resultant phasors, by drawing, 280
horizontal and vertical components,
283
plotting, 278
sine and cosine rules, 281
Rhombus, 219
Right angle, 165
Right angled triangle, 171
solution of, 188
Sample, 289
Scalar quantities, 266
Scalene triangle, 171
Scales, 131
Sector, 222, 230
area of, 233
Segment, 230
Semicircle, 230
Semi-interquartile range, 304
Set, 289
Short division, 4
Significant figures, 17, 18
Similar shapes, 229, 256
triangles, 176
Simple equations, 73
practical problems, 77
Simpson’s rule, 258

Simultaneous equations, 90
graphical solution, 155
in three unknowns, 99
in two unknowns, 90
practical problems, 96
Sine, 27, 183
graph of, 195
Sine rule, 205, 281
wave, 198, 260
mean value, 260
Sinusoidal form A sin(ωt ± α), 202
SI units, 53
Slope, 134
Solution of linear and quadratic
equations simultaneously, 110
Solving right-angled triangles, 188
simple equations, 73
Space diagram, 276
Sphere, 246
Square, 23, 219
numbers, 23
root, 25, 48
units, 219
Standard deviation, 302
discrete data, 302
grouped data, 303
Standard differentials, 321
form, 56
integrals, 326
Statistical data, presentation of, 288
terminology, 288
Straight line, 165
equation of, 135
Straight line graphs, 132
practical problems, 141
Subject of formulae, 83
Subtraction in algebra, 62
Subtraction of fractions, 10
numbers, 1, 18
vectors, 274
Successive differentiation, 322
Supplementary angles, 165
Surface areas of frusta of pyramids and
cones, 252
of solids, 247
Symbols, 28
Tally diagram, 293, 296
Tangent, 27, 183, 230
graph of, 195
Terminating decimal, 17
Theorem of Pythagoras, 181
Transposition of formulae, 83
Transversal, 165
Trapezium, 220
Trapezoidal rule, 257

Index
Triangle, 171, 219
Triangles, area of, 205
congruent, 175
construction of, 179
properties of, 171
similar, 176
Trigonometric functions, 27
Trigonometric ratios, 183
evaluation of, 185
graphs of, 195
waveforms, 195
Trigonometry, 181
practical situations, 209
Turning points, 156

Ungrouped data, 289
Units, 53
Upper class boundary, 293
Use of calculator, 22
Vector addition, 267
subtraction, 274
Vectors, 266
addition of, 267
by calculation, 267
by horizontal and vertical
components, 269
drawing, 266
subtraction of, 274
Velocity, relative, 276

Vertical axis intercept, 133
bar chart, 289
component, 269, 283
Vertically opposite angles, 165
Vertices of triangle, 172
Volumes of common solids, 240
frusta of pyramids and cones, 252
irregular solids, 259
pyramids, 244
similar shapes, 256
Waveform addition, 278
y-axis intercept, 135
Young’s modulus of elasticity, 143

359

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