Basic Engineering Mathematics, Fifth Edition

Introduction to trigonometry 191

3. A ladder rests against the top of the perpendi-
   cular wall of a building and makes an angle of
   73 ◦with the ground. If the foot of the ladder is
   2m from the wall, calculate the height of the
   building.


4. Determine the lengthxin Figure 21.26.

x

568 10mm

Figure 21.26

21.6 Angles of elevation and depression

If, in Figure21.27,BCrepresents horizontal groundand
ABa vertical flagpole, theangle of elevationof the top
of the flagpole,A, from the pointCis the angle that the
imaginary straight lineACmust be raised (or elevated)
from the horizontalCB; i.e., angleθ.

A

C B



Figure 21.27

P

Q R



Figure 21.28

If, in Figure 21.28,PQrepresents a vertical cliff and
Ra ship at sea, theangle of depressionof the ship
from pointPis the angle through which the imaginary
straight linePRmust be lowered (or depressed) from
the horizontal to the ship; i.e., angleφ.(Note,∠PRQis
alsoφ−alternate anglesbetween parallel lines.)

Problem 23. An electricity pylon stands on

horizontal ground. At a point 80m from the base of

the pylon, the angle of elevation of the top of the

pylon is 23◦. Calculate the height of the pylon to the

nearest metre

Figure 21.29 shows the pylon AB and the angle of

elevation ofAfrom pointCis 23◦.

80m

238

A

C B

Figure 21.29

tan23◦=

AB

BC

=

AB

80

Hence, height of pylonAB=80tan23◦

= 80 ( 0. 4245 )= 33 .96m

=34m to the nearest metre.

Problem 24. A surveyor measures the angle of

elevation of the top of a perpendicular building as

19 ◦. He moves 120m nearer to the building and

finds the angle of elevation is now 47◦. Determine

the height of the building

The buildingPQand the angles of elevation are shown

in Figure 21.30.

P

Q

h

x

R

S

120

(^478198)
Figure 21.30
In trianglePQS,tan19◦=
h
x+ 120
Hence, h=tan19◦(x+ 120 )
i.e.h= 0. 3443 (x+ 120 ) (1)
In trianglePQR,tan47◦=
h
x
Hence, h=tan47◦(x)i.e.h= 1. 0724 x (2)
Equating equations (1) and (2) gives
0. 3443 (x+ 120 )= 1. 0724 x
0. 3443 x+( 0. 3443 )( 120 )= 1. 0724 x
( 0. 3443 )( 120 )=( 1. 0724 − 0. 3443 )x
41. 316 = 0. 7281 x
x=
41. 316
0. 7281
= 56 .74m