Chapter 21

Introduction to trigonometry

21.1 Introduction

Trigonometry is a subject that involves the measurement
of sides and angles of triangles and their relationship to
each other.
The theorem of Pythagoras and trigonometric ratios
are used with right-angled triangles only. However,
there are many practical examples in engineering
where knowledge of right-angled triangles is very
important.
In this chapter, three trigonometric ratios – i.e. sine,
cosine and tangent – are defined and then evaluated
using a calculator. Finally, solving right-angled trian-
gle problems using Pythagoras and trigonometric ratios
is demonstrated, together with some practical examples
involving angles of elevation and depression.

21.2 The theorem of Pythagoras

The theorem of Pythagoras states:
In any right-angled triangle, the square of the
hypotenuse is equal to the sum of the squares of the
other two sides.
In the right-angled triangleABCshown in Figure 21.1,
this means

b^2 =a^2 +c^2 (1)

B

A

a C

c b

Figure 21.1

If the lengths of any two sides of a right-angled triangle

are known, the lengthof the thirdside may be calculated

by Pythagoras’ theorem.

From equation (1): b=

√

a^2 +c^2

Transposingequation (1) foragivesa^2 =b^2 −c^2 , from

whicha=

√

b^2 −c^2

Transposing equation (1) forcgivesc^2 =b^2 −a^2 , from

whichc=

√

b^2 −a^2

Here are some worked problems to demonstrate the

theorem of Pythagoras.

Problem 1. In Figure 21.2, find the length ofBC

A B

C

b 5 4cm a

c 5 3cm

Figure 21.2

From Pythagoras, a^2 =b^2 +c^2

i.e. a^2 = 42 + 32

= 16 + 9 = 25

Hence, a=

√

25 =5cm.

√

25 =±5 but ina practical example likethisan answer

ofa=−5cm has no meaning, so we take only the

positive answer.

Thus a=BC=5cm.

DOI: 10.1016/B978-1-85617-697-2.00021-1