Higher Engineering Mathematics

Geometry and trigonometry

B

12

Introduction to trigonometry

12.1 Trigonometry

Trigonometryis the branch of mathematics which
deals with the measurement of sides and angles
of triangles, and their relationship with each other.
There are many applications in engineering where a
knowledge of trigonometry is needed.

12.2 The theorem of Pythagoras

With reference to Fig. 12.1, the side opposite the
right angle (i.e. sideb) is called thehypotenuse.
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.’
Henceb^2 =a^2 +c^2

Figure 12.1

Problem 1. In Fig. 12.2, find the length ofEF.

Figure 12.2

By Pythagoras’ theorem:

e^2 =d^2 +f^2

Hence 13^2 =d^2 + 52

169 =d^2 + 25

d^2 = 169 − 25 = 144

Thus d=

√

144 =12 cm

i.e. EF=12 cm

Problem 2. Two aircraft leave an airfield at the

same time. One travels due north at an aver-

age speed of 300 km/h and the other due west

at an average speed of 220 km/h. Calculate their

distance apart after 4 hours.

After 4 hours, the first aircraft has travelled

4 × 300 =1200 km, due north, and the second air-

craft has travelled 4× 220 =880 km due west, as

shown in Fig. 12.3. Distance apart after 4 hours=BC.

Figure 12.3

From Pythagoras’ theorem:

BC^2 = 12002 + 8802 =1 440 000+774 400

andBC=

√

(2 214 400)

Hence distance apart after 4 hours=1488 km.

Now try the following exercise.

Exercise 52 Further problems on the the-

orem of Pythagoras

1. In a triangleCDE,D= 90 ◦,CD= 14 .83 mm
   andCE= 28 .31 mm. Determine the length of
   DE. [24.11 mm]


2. TrianglePQRis isosceles,Qbeing a right
   angle. If the hypotenuse is 38.47 cm find
   (a) the lengths of sidesPQandQR, and