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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
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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
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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
nearest minute
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Find the acute angle tan−1 7
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4523 means ‘the angle whose tangent is 7
...

Using a calculator,
1
...
Press tan

4
...
Press =

3
...
4523

The answer 82
...
is displayed
...
Press ◦ ”’ and 82◦ 21 26
...

Hence, tan−1 7
...
36◦ = 82◦21 correct to the
nearest minute
...
In triangle EFG in Figure 21
...
30 cm

8
...


Determine, correct to 3 decimal places,
5 cos14◦15
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Determine, correct to 4 significant figures,
7 tan 79◦9
...


Determine
2π
(a) sine
3

G

Figure 21
...
30
i
...

= 0
...

sin G =
8
...
26406429
...


Find the acute angle sin−1 0
...


6
...
9648 in degrees,
correct to 2 decimal places
...


Find the acute angle tan−1 3
...


8
...
1381 in degrees
and minutes
...


Find the acute angle cos−1 0
...


G = 15
...
e
...
31◦ or 15◦19
...
Evaluate the following expression,
correct to 3 significant figures
4
...
7 sin 66◦1
7
...
1681,
tan 49◦ 26 = tan 49
60
sin 66◦1 = 0
...
8698

Hence,

(b) cos 1
...
672

10
...
8971 in degrees
and minutes
...
In the triangle shown in Figure 21
...


5

4
...
7 sin 66◦ 1
7
...
2 × 1
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7 × 0
...
1 × 0
...
9060 − 3
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5253
=
6
...
1756
= 0
...
247,
correct to 3 significant figures
...
17

12
...
18, determine angle θ in degrees and minutes
...


Determine, correct to 4 decimal places,
3 sin 66◦41
...
18

187

188 Basic Engineering Mathematics
13
...
5 cos 67◦34 − sin 90◦
2 tan 45◦
14
...
83◦ )(2
...
48◦)
4
...
56◦

21
...
This is achieved using
(a)

cos 42◦ =

BC
6
...
2
= 8
...
5832 + 6
...
609889 = 8
...


Alternatively,

by

Problem 19
...

Determine the length of AC and hence evaluate
sin A, cos C and tan A
Triangle ABC is shown in Figure 21
...


the theorem of Pythagoras and/or

A

(b) trigonometric ratios
...
e
...
As long as at least
three facts are known, the other three can usually be
calculated
...

Problem 18
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19, find the lengths AC and AB

B

C

12 cm

Figure 21
...
9231
hypotenuse
13
(Remember SOH CAH TOA)

A

and

cos C =

adjacent side 12
=
or 0
...
400
adjacent side
5

Problem 20
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21, find the lengths of PQ and PR
428
B

6
...
19

There is usually more than one way to solve such a
triangle
...
2
(Remember SOH CAH TOA)

Transposing gives
AC

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