Download Trigonometry --- Bearing Problems, Angle of Elevation and

TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
Q1.
The diagram shows four points, A, B, C and D, on a piece of horizontal land.
It is given that AB = 45 meters, AD = 25 meters and
BD = 28 meters.
(a)
Calculate angle ADB.
(b)
Given also that CD = 22 meters and that angle
ACD = 33o, calculate angle ADC.
(c)
The line BD is produced beyond D.
Calculate the shortest distance from C to this
extended line.
(d)
D is the foot of a vertical mast, DT.
The angle of elevation of the top of the mast,
T, from A is 40o.
Calculate the angle of elevation of T from B.
Ans.
(a)
252  282  452
cos ADB 
2(25)(28)
ADB 116.1o
(b)
sin 33o  22
25
DAC  28.64o
sin DAC 
DAC  180o  33o  28.64o  118.4o
(c)
Diagram
E is the foot of the perpendicular from C to BD produced.
EDC  ADC  ADE
 ADC  (180o  ADB)
118.4o 180o  116.1o  54.5o
The shortest distance from C to the extended
Line = CE = 22 sin 54.5o = 17.9 m
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
(d)
DT = 25 tan 40o
DT 25 tan 40o
tan TDB =

DB
28
Angle of elevation of T from B = 36.8o
Q2.
A vertical flagpole, 18 m high, stands on horizontal ground.
Calculate the angle of elevation of the top of the flagpole from a point, on the ground 25 m
from its base.
Ans.
35.8o
Q3.
The diagram shows the position of a harbor, H, and three island A, B and C.
C is due North of H.
ˆ = 128o.
The bearing of A from H is 062o and HAB
HA = 54 km and AB = 31 km.
(a)
Calculate the distance HB.
(b)
Find the bearing of B from A.
(c)
The bearing of A from C is 133o.
Calculate the distance AC.
(d)
A lightship, L, is positioned due North
of H and equidistant from A and H.
Calculate the distance HL.
(c)
Q4.
65.2 km
(d)
57.5 km
An aircraft waiting to land is flying around a triangular
Circuit ABC.
A, B and C are vertically above three beacons, X, Y and Z.
T is the control tower at the airport, and T, X, Y and Z
lie in a horizontal plane.
BC = 18 km, CA = 22 km and AB =b24 km.
(a)
(i)
The plane is flying at 200 km/h.
Calculate the time, in minutes and seconds,
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
that
The aircraft takes to complete one circuit.
(ii)
(b)
Calculate the largest angle of triangle ABC.
Z is due West of T.
The bearing of X from Z is 042o and the bearing of X from T is 338o.
(c)
(i)
Find the angles of triangle TXZ.
(ii)
Calculate TX.
The aircraft is flying at a constant height of 2600 meters.
Calculate the angle of depression of the tower, T, from the aircraft when it is at A.
Ans.
Q5.
(a)
(i)
19 min 12 sec
(b)
64o
(c)
Angle of depression = 8.4o
(ii)
C = 73.0o
(ii)
TX = 17.6 km
The diagram shows the positions of A and B.
Find the bearing of
(a)
A from B,
(b)
B from A.
Ans.
(a)
300o
Q6.
Three paths, AB, BC and CA, run along the edges of a horizontal triangular field ABC.
(b)
120o
BC = 51 m, AC = 72 m and angle ACB = 81o.
(a)
Calculate the length of AB.
(b)
Calculate the area of the field ABC.
(c)
Calculate the shortest distance from
C to AB.
(d)
A vertical tree, CT, has its base at C.
The angle of elevation of the top of the
tree from A is 15o.
(e)
John measured the largest angle of
elevation of the top of the tree as seen
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
form the path AB.
Calculate this angle of elevation.
Ans.
(a)
81.5 m
(b)
1810 m2 correct to 3 figures
(c)
44.5 m
(d)
CT = 19.3 m
(e)
23.4o
A is due North of O.
Q7.
(a)
ˆ = 12o.
A ship sailed from O to B, where AOB
Write down the bearing of B from O.
(b)
At B, the ship turned and sailed to C,
ˆ = 50o.
Where OBC
Calculate the bearing of C from B.
Ans.
(a)
348o
(b)
218o
Q8.
Three points, A, B and C, lie on a horizontal field. Angle BAC = 75o and the bearing of C from A
is 217o.
AB = 72 m and AC = 60 m.
(a)
(b)
Calculate
(i)
the bearing of B from A,
(ii)
BC,
(iii)
Angle ABC,
(iv)
the bearing of C from B.
A girl standing at B is flying a kite.
The kite, K, is vertically above A.
The string, BK, attached to the kite
is at 24o to the horizontal.
Calculate the angle of elevation of
the kite when viewed form C.
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
Ans.
Q9.
(a)
(i)
292o
(ii)
(iii)
ABC = 45.7o
80.9 m
(iv).
157.7o
(b)
28.1o
(b)
Diagram I represents part of the framework of the ride.
The points A, B, C, D, E and F are on the framework.
The points H, C, G, E and F lie on a horizontal lien.
The lines BH and DG are vertical
BC = 80 cm,
HC = 60 m,
DG = 40 m
GE = 35 m
and
ˆ = 32o
DCG
Calculate
(i)
ˆ ,
HCB
(ii)
CD,
(iii)
the angle of depression of E from D.
(c)
Diagram II shows part of the ride.
The carriage that carried the family was 4.6 m long.
It was travelling at a constant speed of 15 m/s as it passed the point F.
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
(i)
Calculate, correct to the nearest hundredth of a second, the time taken for
the
carriage to pass the point F.
Ans.
Q10.
(ii)
Express 15 m/s in kilometers per hour.
(b)
(i)
HCB  41.4o
(ii)
CD = 75.5 m
(c)
(i)
0.31 s
(ii)
54 km/h
(iii)
48.8o
In the diagram, A and B are two points on a straight coastline.
B is due east of A and AB = 7 km.
The position of a boat at different times
was noted.
(a)
At 8 a.m, the boat was at C, where
ˆ = 66o and ABC
ˆ = 48o
ACB
Calculate
(b)
(i)
the bearing of B from C,
(ii)
the distance AC.
At 9 a.m, the boat was at D, where
ˆ = 41o.
AD = 6.3 km and DAB
Calculate
(i)
the area of triangle ADB,
(ii)
the shortest distance from the
boat to the coastline.
Compiled By : Sir Rashid Qureshi
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
(c)
At 11 a.m, the boat was at E, where
AE = 9 km and BE = 5 km.
Calculate the bearing of E from A.
Ans.
(a)
(i)
138o
(ii)
5.69 km
(b)
(i)
14.5 km2
(ii)
4.13 km
[
Q11.
A vertical flagpole, BF, stands at the top of a hill.
AB is the steepest path up the hill.
ˆ = 90o.
N lies vertically below B and ANB
AN = 100 m and AB = 104 m.
(a)
Show that BN = 28.6 m.
(b)
ˆ = 25o.
It is given that FAN
(i)
Write down the size of the angle of
depression of A from F.
(ii)
(c)
Calculate the height, BF, of the flagpole.
The diagram shows three other straight paths
(CB, DB and ACD) on the hill.
ˆ  NAC
ˆ  90o .
The path ACD is horizontal and BAC
CN and DN are horizontal lines.
(i)
Given that AC = 60 m,
ˆ .
Calculate BCN
(ii)
ˆ  10o ,
Given that BDN
ˆ .
Calculate DBA
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
Ans.
(a)
BN 2  AB 2  AN 2
Pythagoras’
1042 1002
 816 m2
BN  816  28.6 m
Q12.
(b)
(i)
25o
(ii)
BF = 18.1 m
(c)
(i)
BCN 13.8o
(ii)
DBA  50.8o
Diagram I
In Diagram I, the point D lies on AC and N is the
foot of the perpendicular from C to BD.
AB = 61m, AD = 30 m and DC = 45m.
Angle BAC = 41o.
(a)
(b)
Calculate BD.
Show that, correct to the nearest square meter.
The area of triangle BDA is 600 m2.
(f)
area of BCD 3
 .
area of BDA 2
(c)
Explain why
(d)
Calculate the area of triangle BCD.
(e)
Hence calculate CN.
Diagram II
The same points B, C, D and N lie on a sloping plane.
The point E is 15 m vertically below C.
The points B, E, D and N lie on a horizontal plane.
Diagram II represents this information.
Calculate the angle of elevation of C from N.
Ans.
(a)
BD = 43.1 m
(b)
Consider BDA
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TOPIC#10
Trigonometry --- Bearing Problems, Angle of Elevation and Depression,
3 Dimensional Problems.
Area  12 6130sin 41
 600.29  600 m2
Area of BCD 2 CD h

area of BDA 12  AD h
1
(c)


CD
AD
45
3

30
2
(d)
area of BCD  32  600  900m
(e)
area of BCD  12  BD  CN
1
900   43.113 CN
2
CN  41.8m2
(f)
angle of elevation = 21.1o
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
9