210 Basic Engineering Mathematics
458
V 2 5 100 V
V 1 5 40 V
A
B
O
Figure 23.11
Angle OBA = 180
◦
− 45
◦
= 135
◦
Applying the cosine rule,
OA
2
= V
2
1 + V
2
2 − 2V 1 V 2 cos OBA
= 40
2
+ 100
2
− {2(40)(100) cos 135
◦
}
= 1600 + 10000 − {−5657}
= 1600 + 10000 + 5657 = 17257
Thus, resultant,OA =
√
17257 = 131.4 V
Applying the sine rule
131.4
sin 135 ◦ =
100
sin AOB
from which
sin AOB =
100 sin135 ◦
131.4
= 0.5381
Hence, angle AOB = sin −1 0.5381 = 32.55 ◦ (or
147.45 ◦ , which is not possible)
Hence, the resultant voltage is 131.4 volts at 32.55 ◦
to V 1
Problem 10. In Figure 23.12, PR represents the
inclined jib of a crane and is 10.0 m long. PQ is
4.0 m long. Determine the inclination of the jib to
the vertical and the length of tie QR.
P
Q
R
1208
4.0 m
10.0 m
Figure 23.12
Applying the sine rule,
PR
sin 120 ◦ =
PQ
sin R
from which sin R =
PQ sin 120 ◦
PR
=
(4.0) sin 120 ◦
10.0
= 0.3464
Hence, ∠R = sin
−1 0.3464 = 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
sin 120 ◦ =
QR
sin 39.73 ◦
from which
length of tie,QR =
10.0 sin39.73 ◦
sin 120 ◦
= 7.38 m
Now try the following Practice Exercise
Practice Exercise 92 Practical situations
involving trigonometry (answers on
page 350)
1. A ship P sails at a steady speed of 45 km/h
in a direction of W 32 ◦ N (i.e. a bearing of
302
◦ ) from a port. At the same time another
ship Q leaves the port at a steady speed of
35 km/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
52.0 m and 34.0 m, respectively. If the area
of the plot is 620 m 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 rod PR is 8.0 m long and PQ is 4.5 m long,
determine (a) the length of jib RQ and (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 quadrilateral, as shown in Figure 23.14, and its
area is 1510 m 2 . Determine the length of the
perimeter of the site.
458
V 2 5 100 V
V 1 5 40 V
A
B
O
Figure 23.11
Angle OBA = 180
◦
− 45
◦
= 135
◦
Applying the cosine rule,
OA
2
= V
2
1 + V
2
2 − 2V 1 V 2 cos OBA
= 40
2
+ 100
2
− {2(40)(100) cos 135
◦
}
= 1600 + 10000 − {−5657}
= 1600 + 10000 + 5657 = 17257
Thus, resultant,OA =
√
17257 = 131.4 V
Applying the sine rule
131.4
sin 135 ◦ =
100
sin AOB
from which
sin AOB =
100 sin135 ◦
131.4
= 0.5381
Hence, angle AOB = sin −1 0.5381 = 32.55 ◦ (or
147.45 ◦ , which is not possible)
Hence, the resultant voltage is 131.4 volts at 32.55 ◦
to V 1
Problem 10. In Figure 23.12, PR represents the
inclined jib of a crane and is 10.0 m long. PQ is
4.0 m long. Determine the inclination of the jib to
the vertical and the length of tie QR.
P
Q
R
1208
4.0 m
10.0 m
Figure 23.12
Applying the sine rule,
PR
sin 120 ◦ =
PQ
sin R
from which sin R =
PQ sin 120 ◦
PR
=
(4.0) sin 120 ◦
10.0
= 0.3464
Hence, ∠R = sin
−1 0.3464 = 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
sin 120 ◦ =
QR
sin 39.73 ◦
from which
length of tie,QR =
10.0 sin39.73 ◦
sin 120 ◦
= 7.38 m
Now try the following Practice Exercise
Practice Exercise 92 Practical situations
involving trigonometry (answers on
page 350)
1. A ship P sails at a steady speed of 45 km/h
in a direction of W 32 ◦ N (i.e. a bearing of
302
◦ ) from a port. At the same time another
ship Q leaves the port at a steady speed of
35 km/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
52.0 m and 34.0 m, respectively. If the area
of the plot is 620 m 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 rod PR is 8.0 m long and PQ is 4.5 m long,
determine (a) the length of jib RQ and (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 quadrilateral, as shown in Figure 23.14, and its
area is 1510 m 2 . Determine the length of the
perimeter of the site.