Basic Engineering Mathematics, Fifth Edition

212 Basic Engineering Mathematics

(a) Applying the sine rule,

AB

sin50◦

=

AO

sinB

from which sinB=

AOsin50◦

AB

=

10 .0sin50◦

30. 0

= 0. 2553

Hence,B=sin−^10. 2553 = 14. 79 ◦(or 165. 21 ◦,

which is not possible)

Hence,the connecting rodABmakes an angle

of 14.79◦with the horizontal.

AngleOAB= 180 ◦− 50 ◦− 14. 79 ◦= 115. 21 ◦

Applying the sine rule:

30. 0

sin50◦

=

OB

sin115. 21 ◦

from which OB=

30 .0sin115. 21 ◦

sin50◦

= 35 .43cm

(b) Figure 23.18 shows the initial and final positions

of the crank mechanism.

508

O

30.0 cm 1208

BB 9

A A^9

10.0 cm

Figure 23.18

In triangleOA′B′, applying the sine rule,

30. 0

sin120◦

=

10. 0

sinA′B′O

from which sinA′B′O=

10 .0sin120◦

30. 0

= 0. 28868

Hence, A′B′O=sin−^10. 28868 = 16. 78 ◦ (or

163. 22 ◦, which is not possible)

AngleOA′B′= 180 ◦− 120 ◦− 16. 78 ◦= 43. 22 ◦

Applying the sine rule,

30. 0

sin120◦

=

OB′

sin43. 22 ◦

from whichOB′=

30 .0sin43. 22 ◦

sin120◦

= 23 .72cm

Since OB= 35 .43cm and OB′= 23 .72cm,

BB′= 35. 43 − 23. 72 = 11 .71cm

Hence,B moves 11.71cm when angle AOB

changes from 50◦to 120◦.

Problem 13. The area of a field is in the form of a

quadrilateralABCDas shown in Figure 23.19.

Determine its area

62.3m

21.4m

39.8m

1148

568

B

D

C

A

42.5m

Figure 23.19

A diagonal drawn fromBtoDdivides the quadrilateral

into two triangles.

Area of quadrilateralABCD

=area of triangleABD

+area of triangleBCD

=

1

2

( 39. 8 )( 21. 4 )sin114◦

+

1

2

( 42. 5 )( 62. 3 )sin 56◦

= 389. 04 + 1097. 5

=1487m^2

Now try the following Practice Exercise

PracticeExercise 93 More practical

situations involving trigonometry (answers

on page 350)

1. Three forces acting on a fixed point are repre-
   sented by the sides of a triangle of dimensions
   7 .2cm, 9 .6cm and 11.0cm. Determine the
   angles between thelines of action and thethree
   forces.


2. A vertical aerialAB,9.60m high, stands on
   groundwhich is inclined12◦to the horizontal.
   AstayconnectsthetopoftheaerialAtoapoint
   Con the ground 10.0m downhill fromB,the
   foot of the aerial. Determine (a) the length of
   the stay and (b) the angle the stay makes with
   the ground.


3. A reciprocating engine mechanism is shown
   in Figure 23.20. The crankABis 12.0cm long
   and the connecting rodBCis 32.0cm long.
   For the position shown determine the length
   ofACand the angle between the crank and the
   connecting rod.