Basic Engineering Mathematics, Fifth Edition

Non-right-angled triangles and some practical applications 211

28.5 m

(^728) 34.6 m
52.4 m^758
Figure 23.14

5. Determine the length of membersBFandEB
   in the roof truss shown in Figure 23.15.

4m

2.5 m

508

4m

2.5 m

508

5m 5m

E

ABC

F D

Figure 23.15

6. A laboratory9.0m wide has a span roof which
   slopes at 36◦on one side and 44◦on the other.
   Determine the lengths of the roof slopes.


7. PQandQRare the phasors representing the
   alternating currents in two branches of a cir-
   cuit. PhasorPQis 20.0A and is horizontal.
   PhasorQR(which is joined to the end ofPQ
   to form trianglePQR)is14.0A and is at an
   angle of 35◦to the horizontal. Determine the
   resultant phasorPRand the angle it makes
   with phasorPQ.

23.6 Further practical situations

involving trigonometry

Problem 11. A vertical aerial stands on

horizontal ground. A surveyor positioned due east

of the aerial measures the elevation of the top as

48 ◦. He moves due south 30.0 m and measures the

elevation as 44◦. Determine the height of the aerial

In Figure 23.16,DCrepresents theaerial,Ais the initial
position of the surveyor andBhis final position.

From triangle ACD,tan48◦=

DC

AC

from which

AC=

DC

tan48◦

Similarly, from triangleBCD,BC=

DC

tan44◦

B

C

D

A

448 30.0m

488

Figure 23.16

For triangleABC, using Pythagoras’ theorem,

BC^2 =AB^2 +AC^2

(

DC

tan44◦

) 2

=( 30. 0 )^2 +

(

DC

tan48◦

) 2

DC^2

(

1

tan^244 ◦

−

1

tan^248 ◦

)

= 30. 02

DC^2 ( 1. 072323 − 0. 810727 )= 30. 02

DC^2 =

30. 02

0. 261596

= 3440. 4

Hence,height of aerial,DC=

√

3340. 4 =58.65m.

Problem 12. A crank mechanism of a petrol

engine is shown in Figure 23.17. ArmOAis

10 .0cm long and rotates clockwise aboutO.The

connecting rodABis 30.0cm long and endBis

constrained to move horizontally.

(a) For the position shown in Figure 23.17,

determine the angle between the connecting

rodABand the horizontal, and the length

ofOB.

(b) How far doesBmove when angleAOB

changes from 50◦to 120◦?

508

O

30.0cm 10.0cm

A

B

Figure 23.17