Basic Engineering Mathematics, Fifth Edition

210 Basic Engineering Mathematics

458

V 25 100V

V 15 40V

A

O B

Figure 23.11

AngleOBA= 180 ◦− 45 ◦= 135 ◦

Applying the cosine rule,

OA^2 =V 12 +V 22 − 2 V 1 V 2 cosOBA

= 402 + 1002 −{ 2 ( 40 )( 100 )cos135◦}

= 1600 + 10000 −{− 5657 }

= 1600 + 10000 + 5657 = 17257

Thus, resultant,OA=

√

17257 = 131. 4 V

Applying the sine rule^131.^4

sin135◦

=

100

sinAOB

from which sinAOB=

100sin135◦

131. 4

= 0. 5381

Hence, angleAOB=sin−^10. 5381 = 32. 55 ◦(or

147. 45 ◦, which is not possible)

Hence,the resultant voltage is 131.4 volts at 32.55◦

toV 1

Problem 10. In Figure 23.12,PRrepresents the

inclined jib of a crane and is 10.0m long.PQis

4 .0m long. Determine the inclination of the jib to

the vertical and the length of tieQR.

P

Q

R

1208

4.0m 10.0m

Figure 23.12

Applying the sine rule,

PR

sin120◦

=

PQ

sinR

from which sinR=

PQsin120◦

PR

=

( 4. 0 )sin120◦

10. 0

= 0. 3464

Hence, ∠R=sin−^10. 3464 = 20. 27 ◦ (or 159. 73 ◦,

which is not possible)

∠P= 180 ◦− 120 ◦− 20. 27 ◦= 39. 73 ◦,which is the

inclination of the jib to the vertical

Applying the sine rule,

10. 0

sin120◦

=

QR

sin39. 73 ◦

from which length of tie,QR=

10 .0sin39. 73 ◦

sin120◦

= 7 .38m

Now try the following Practice Exercise

PracticeExercise 92 Practical situations

involving trigonometry (answers on

page 350)

1. A shipPsails at a steady speed of 45km/h
   in a direction of W 32◦N (i.e. a bearing of
   302 ◦) from a port. At the same time another
   shipQleaves the port at a steady speed of
   35km/h in a direction N 15◦E (i.e. a bearing
   of 015◦). Determine their distance apart after
   4 hours.


2. Two sides of a triangular plot of land are
   52 .0m and 34.0 m, respectively. If the area
   of the plot is 620m^2 , find (a) the length
   of fencing required to enclose the plot and
   (b) the angles of the triangular plot.


3. A jib crane is shown in Figure 23.13. If the
   tie rodPRis 8.0m long andPQis 4.5m long,
   determine (a) the length of jibRQand (b) the
   angle between the jib and the tie rod.
   R

1308 P

Q

Figure 23.13

4. A building site is in the form of a quadri-
   lateral, as shown in Figure 23.14, and its
   area is 1510m^2. Determine the length of the
   perimeter of the site.