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Chapter 21

Introduction to trigonometry
21
...

The theorem of Pythagoras and trigonometric ratios
are used with right-angled triangles only
...

In this chapter, three trigonometric ratios – i
...
sine,
cosine and tangent – are defined and then evaluated
using a calculator
...


21
...


From equation (1):
b = a2 + c2
Transposing√equation (1) for a gives a 2 = b2 − c2 , from
which a = b2 − c 2
2
2
2
Transposing
√equation (1) for c gives c = b − a , from
2
2
which c = b − a
Here are some worked problems to demonstrate the
theorem of Pythagoras
...
In Figure 21
...

In the right-angled triangle ABC shown in Figure 21
...
1
DOI: 10
...
00021-1

a

c 5 3 cm

B

Figure 21
...
e
...


A
c

A

a

C

Hence,
√
25 = ±5 but in a practical example like this an answer
of a = −5 cm has no meaning, so we take only the
positive answer
...


182 Basic Engineering Mathematics
PQR is a 3, 4, 5 triangle
...
e
...

Problem 2
...
3, find the length of EF

From Pythagoras’ theorem,
BC 2 = 12002 + 8802
= 1440000 + 774400 = 2214400
√
BC = 2214400 = 1488 km
...


D
e5 13 cm

f 5 5 cm
E

Now try the following Practice Exercise

F

d

Figure 21
...


Find the length of side x in Figure 21
...


2

41 cm

169 = d 2 + 25

x

d 2 = 169 − 25 = 144
√
d = 144 = 12 cm

Thus,

40 cm

d = EF = 12 cm

i
...


DEF is a 5, 12, 13 triangle, another right-angled
triangle which has integer values for all three sides
...
5

2
...
6(a)
...


Find the length of side x in Figure 21
...


Problem 3
...
One travels due north at an average
speed of 300 km/h and the other due west at an
average speed of 220 km/h
...
7 mm

as shown in Figure 21
...
The distance apart after
4 hours = BC
...
6

E
S

C

Figure 21
...
3 mm
(b)

B

1200 km

880 km

A

4
...
Determine the length of
AC, correct to 2 decimal places
...


A tent peg is 4
...
0 m high
tent
...


In a triangle ABC, ∠B is a right angle,
AB = 6
...
78 cm
...


14
...
8 shows a cross-section of a component that is to be made from a round bar
...

x

90◦ ,

7
...
83 mm and CE = 28
...

Determine the length of DE
...


Show that if a triangle has sides of 8, 15 and
17 cm it is right-angled
...


183

Triangle PQR is isosceles, Q being a right
angle
...
46 cm find (a)
the lengths of sides PQ and QR and (b) the
value of ∠QPR
...
A man cycles 24 km due south and then 20 km
due east
...
Find the distance
between the two men
...
A ladder 3
...
0 m from the
wall
...
8

21
...
9,
opposite side
hypotenuse

sine θ =

‘Sine’ is abbreviated to ‘sin’, thus sin θ =

BC
AC
C

12
...
One
travels due west at 18
...
6 knots
...

13
...
7 shows a bolt rounded off at one
end
...


m

4m

␾7

se

nu

te
po

Hy

Opposite

␪
A

Adjacent

B

Figure 21
...
Remembering these three equations
is very important and the mnemonic ‘SOH CAH TOA’
is one way of remembering them
...
7

184 Basic Engineering Mathematics
SOH indicates sin = opposite ÷ hypotenuse

sin C =

CAH indicates cos = adjacent ÷ hypotenuse
TOA indicates tan = opposite ÷ adjacent

cos C =

Here are some worked problems to help familiarize
ourselves with trigonometric ratios
...
In triangle PQR shown in
Figure 21
...
10

sin θ =

opposite side
PQ
5
=
=
= 0
...
9231
hypotenuse
PR
13

tanθ =

opposite side
PQ
5
=
=
= 0
...
In triangle ABC of Figure 21
...
47
= 0
...
778
4
...
7996
5
...
47
=
= 0
...
62
4
...
7996
5
...
47
=
= 0
...
778
4
...
3314
3
...
If tan B = , determine the value of
15
sin B, cos B, sin A and tan A
A right-angled triangle ABC is shown in Figure 21
...

8
If tan B = , then AC = 8 and BC = 15
...
12

i
...

from which

AB 2 = 82 + 152

AB = 82 + 152 = 17
AC
AB
BC
cos B =
AB
BC
sinA =
AB
BC
tanA =
AC
sin B =

3
...
62 cm

C

Figure 21
...
e
...
472 + 4
...
472 + 4
...
778 cm

from which

AB
AC
BC
AC
AB
BC
BC
AC
AB
AC
BC
AB

By Pythagoras, AB 2 = AC 2 + BC 2

A

B

opposite side
=
hypotenuse
adjacent side
=
hypotenuse
opposite side
=
adjacent side
opposite side
=
hypotenuse
adjacent side
=
hypotenuse
opposite side
=
adjacent side

8
17
15
=
17
15
=
17
15
=
8
=

or 0
...
8824
or 0
...
8750

Problem 7
...
Determine (a) the distance AB and
(b) the gradient of the straight line AB

Introduction to trigonometry
f (x)
8
7
6

f(x)
8
7
B
6

4
3
2

4
3
2

0

A

2

4
(a)

6

B

␪

0

8

C

A

2

4
(b)

6

15
, find sin X and cos X , in frac4
...

5
...
15, find (a) sin α (b) cos θ (c) tan θ
...
13

185

␣

17
␪
15

Figure 21
...
13(a)
...
13(b), the horizontal and vertical
lines AC and BC are constructed
...
5 cm

R

Figure 21
...
5

PQ = 7
...
5(0
...
860 cm

Introduction to trigonometry
cos 38◦ =

7
...
5
7
...
518 cm
◦
cos 38
0
...
5)2 + (5
...
59 = (9
...

Problem 21
...
22
A

35 mm

B

37 mm

XZ
, hence XZ = 20
...
0
= 20
...
3953) = 7
...
0 cos 23◦17
20
...
0(0
...
37 mm
Check: using Pythagoras’ theorem,
(18
...
906)2 = 400
...
0)2 ,
cos 23◦17 =

Now try the following Practice Exercise
Practice Exercise 85 Solving right-angled
triangles (answers on page 349)

C

Figure 21
...

sin C =

sin 23◦17 =

189

1
...
24(a) to (f), each correct to 4
significant figures
...
94595, hence
37
C = sin−1 0
...
08◦

708
13
...
08◦ = 18
...
92◦ = 37(0
...
0 mm

sin B =

(a)
15
...
0 mm
...
Solve triangle XYZ given
∠X = 90◦, ∠Y = 23◦17 and YZ = 20
...
Such a sketch is shown in
Figure 21
...

17

...
0 mm
(c)

238179
X

Figure 21
...
24

190 Basic Engineering Mathematics
E

4
...
0

x

D
8
...
0

J

(c)

L

6
...


438

(e)

278

K

M
(d)

7
...
0

538
P

(f)

Figure 21
...
Find the unknown sides and angles in the rightangled triangles shown in Figure 21
...
The
dimensions shown are in centimetres
...
0
5
...
0
(a)

Figure 21
...
25

Introduction to trigonometry

3
...
If the foot of the ladder is
2 m from the wall, calculate the height of the
building
...
Calculate the height of the pylon to the
nearest metre
Figure 21
...

A

4
...
26
...
29

tan 23◦ =

x

AB
AB
=
BC
80

Hence, height of pylon AB = 80 tan 23◦
= 80(0
...
96 m
= 34 m to the nearest metre
...
26

21
...
27, BC represents horizontal ground and
AB a vertical flagpole, the angle of elevation of the top
of the flagpole, A, from the point C is the angle that the
imaginary straight line AC must be raised (or elevated)
from the horizontal CB; i
...
, angle θ
...
A surveyor measures the angle of
elevation of the top of a perpendicular building as
19◦
...
Determine
the height of the building
The building PQ and the angles of elevation are shown
in Figure 21
...

P

A

h
478

␪

C

B

Q

R

198

S

120

x

Figure 21
...
30
P

␾

Hence,
Q

h
x + 120
h = tan 19◦(x + 120)

In triangle PQS, tan 19◦ =
R

i
...
h = 0
...
28

If, in Figure 21
...
e
...
(Note, ∠PRQ is
also φ − alternate angles between parallel lines
...
An electricity pylon stands on
horizontal ground
...
e
...
0724x

(1)

In triangle PQR, tan 47◦ =
Hence,

Equating equations (1) and

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