Chapter 11
Introduction to trigonometry
11.1 Trigonometry
Trigonometryisthebranchofmathematicswhichdeals
with the measurement of sides and angles of trian-
gles, and their relationship with each other. There are
many applications in engineering where a knowledge
of trigonometry is needed.
11.2 The theorem of Pythagoras
With reference to Fig. 11.1, the side opposite the right
angle(i.e.sideb)iscalledthehypotenuse.Thetheorem
of Pythagorasstates:
‘In any right-angled triangle, the square on the
hypotenuse is equal to the sum of the squares on the
other two sides.’
Hence b^2 =a^2 +c^2
B
A
a C
b
c
Figure 11.1
Problem 1. In Fig. 11.2, find the length ofEF.
E d F
D
f 5 5cm
e 5 13cm
Figure 11.2
By Pythagoras’ theorem:
e^2 =d^2 +f^2
Hence 132 =d^2 + 52
169 =d^2 + 25
d^2 = 169 − 25 = 144
Thus d=
√
144 =12cm
i.e. EF=12cm
Problem 2. Two aircraft leave an airfield at the
same time. One travels due north at an average
speed of 300km/h and the other due west at an
average speed of 220km/h. Calculate their distance
apart after 4hours.
After 4hours, the first aircraft has travelled 4× 300 =
1200km, due north, and the second aircraft has trav-
elled 4× 220 =880km due west, as shown in Fig. 11.3.
Distance apart after 4hours=BC.
A
N B
W E
S
C
1200km
880km
Figure 11.3
From Pythagoras’ theorem:
BC^2 = 12002 + 8802 = 1440000 + 774400
andBC=
√
( 2214400 )
Hence distance apart after 4 hours=1488km.