Download Advanced Set

9.1
Chapter 9 Applications of Trigonometry
Chapter 9
Applications of Trigonometry
WARM-UP EXERCISE
1. Find the unknown in each of the following. (Leave your answers in surd form.)
(a)
A
(b)
E
x
4
D
7
I
(c)
12
4
y
J
5
K
18
F
B
z
C
2. Simplify the following algebraic fractions.
(a)
3x
6y

x  2y x  2y
(b)
7
4

x y yx
3. Make the variable inside each pair of square brackets as the subject of the formula.
1
y
[y]
2b
b
[b]
(a) x  4 
(b) a 
4. According to each of the following triangles, find the values of cos , sin  and tan .
(Correct your answers to 2 decimal places if necessary.)
(a)
D
17
E
15
F
(c)
D
4.8
E

7.3
8

E
D
(b)
9.8

6.3
12.2
F
F
7.9
9.2
New Trend Mathematics S3B — Junior Form Supplementary Exercises
BUILD-UP EXERCISE
[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Set and
Advanced Set. You may choose to complete any ONE set according to your need. ]
Exercise 9A
[ Do not use calculators in this exercise. If necessary, leave your answers in surd form. ]
 Elementary Set
Level 1
1. Find the value of each of the following.
4
(a)
tan 45
(c) sin 60 tan 30

(b) cos 2 60
(d)
tan 30
sin 30
2. Find the value of each of the following.
(a) sin 45  cos 45
(b) 2 cos 30  tan 60
(c) tan 30  cos 60
(d) 2 sin 60  3 cos 30
3. Find the value of each of the following.
Ex.9A Elementary Set
(a)
sin 60
3  tan 45
(b)
cos 30
sin 30 tan 30
(c)
tan 45  cos 60
sin 30
(d)
cos 60 2 sin 30

sin 60 tan 45
4. Find  in each of the following.
(a)
2 sin   1
(b) 1  sin  
(c)
3 tan   3
(d)
5. Find  in each of the following.
tan 45
(a)
 2
cos 3
(c)
4 3
4
tan 2
1
2
2 cos   1  0
(b) 2 sin 2  3  0
(d)
3 tan 3  1  0
Chapter 9 Applications of Trigonometry
9.3
Level 2
6. Find the unknown in each of the following figures.
(a)
D
(b)
A
(c)
G
H
12
60
b
4.8
a
8
c
45
B
E
C
(d)
(e)
J
I
F
(f)
M
P
Q
f
d
5
60
K
Ex.9A Elementary Set
30
2 3
e
60
30
N
O
15
R
L
 Advanced Set
Level 1

1. Find the value of each of the following.
(b) cos 2 30
cos 30
(d)
sin 60
(f) sin 2 30 tan 2 60
2. Find the value of each of the following.
(a) sin 30  cos 60
(c)
sin 60  tan 30
cos 60
(b) 2 cos 30  tan 60
(d)
3 tan 30 2 cos 30

sin 45
cos 45
3. Find  in each of the following.
(a) 2 sin   1
(c) cos

2

2
2
(b) 4 cos   2 3
(d) 2 3 tan 4  6
4. Find  in each of the following.
(a) 3 tan 2  3
(c)
3
1
tan 5

1
3
1
(d) cos 2  
4
(b) 2 sin
Ex.9A Advanced Set
(a) sin 2 45
1
(c)
sin 30
tan 60
(e)
sin 30
9.4
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Level 2
5. Find the unknown in each of the following figures.
(a)
A
P
(b)
(c)
X
Y
60
4
45
3
b
R
B
a
c
2
30
Q
C
Z
Ex.9A Advanced Set
A
(d)
A
(e)
5 2
45
D
E
a
D
A
B
60
b
y
45
30
4
(f)
C
BCD is a straight line.
3
B
C
45
30
C
B
AEC and BED are straight lines.
D
P
6. In the figure, QCDR is a straight line, PQR is an
equilateral triangle, ABCD is a rectangle. If A and B are
the mid-points of PR and PQ respectively, and each side of
PQR is 6 cm long,
A
(a) find the length and width of rectangle ABCD.
(b) find the area of rectangle ABCD.
R
B
D
C
Q
Exercise 9B
[ Do not use calculators in this exercise. If necessary, leave your answers in surd form. ]
Ex.9B Elementary Set
 Elementary Set
Level 1
1. Simplify each of the following.
1
(a)
tan cos 
(c)
2 tan 
sin 
(b)

3 sin  cos 
tan 
Chapter 9 Applications of Trigonometry
9.5
2. Simplify each of the following.
(a) 3sin 2   3cos2 
(c)
1
cos 2 

sin  sin 
(b) 4 cos2   4 sin 2 
(d) tan 2  cos2   cos2 
3. Find  in each of the following.
4. Simplify each of the following.
3 cos 
(a)
sin( 90  )
(c)
sin( 90  )
 tan 
cos(90  )
(b) cos   sin 37
1
(d) tan 58 
tan 
(b) cos(90  ) tan(90  )
(d)
tan 
1

cos(90  ) sin( 90  )
5. Find the value of each of the following.
(a) sin 2 27  sin 2 63
(b) tan 52 
(c) tan 18 tan 72
(d)
1
tan 38
cos 69
sin 21
6. Find the value of each of the following.
(a) 5  3 tan 53 tan 37
(c) cos2 48  cos2 42
(b) sin 34 cos 56  sin 56 cos 34
sin 18
(d) cos 72 tan18 
tan 72
Level 2
3
, find the values of cos  and tan .
5
5
(b) If cos   , find the values of sin  and tan .
7
3
(c) If tan   , find the values of sin  and cos .
10
7. (a) If sin  
8. If sin  
1
24
.
, find the value of cos  
25
tan 
Ex.9B Elementary Set
(a) sin   cos 42
1
(c) tan  
tan 49
9.6
New Trend Mathematics S3B — Junior Form Supplementary Exercises
9. Simplify each of the following.
(a)
Ex.9B Elementary Set
(c)
cos 2 
 sin 
sin 
1
cos 2 
 tan 2 
(b) tan θ cos3 θ  sin 3 θ
(d)
cos 2   cos 4 
tan 
10. Find  in each of the following.
1
 tan 4
cos 4
(a) sin 3  cos 2
(b)
(c) sin   cos(  36)
(d) tan(4  10) 
 Advanced Set
Level 1
1. Simplify each of the following.
cos 
(a)
sin 
1
tan 32

(b) cos  tan   sin 
sin  cos 
tan 
(c) sin 2   cos2 
(d) 1 
(e) sin θ cos θ tan θ  1
(f) sin  cos  tan   cos2 
Ex.9B Advanced Set
2. Find  in each of the following.
(a) sin   cos 20
1
(c) tan  
tan 40
(b) cos   sin 45
1
(d) tan 37 
tan 
3. Simplify each of the following.
sin 
cos(90  )
(b)
sin 
sin( 90  )
(c) sin( 90  ) tan 
(d)
sin  tan 
cos(90  ) tan(90  )
(b)
1
 tan 55
tan 35
(a)
4. Find the value of each of the following.
(a) sin 2 50  sin 2 40
(c)
cos 60
cos 30
(d) tan 22 tan 68
Chapter 9 Applications of Trigonometry
5. Find the value of each of the following.
sin 60
(a)
1
cos 30
(c) sin 2 62  sin 2 28
9.7
(b) sin 40 cos 50  sin 50 cos 40
(d) sin 24 tan 66 
cos 66
tan 24
Level 2
6. According to each of the following trigonometric ratios, find the remaining two
trigonometric ratios.
7. Simplify each of the following.
sin  cos 
(a)

cos  sin 
(c) cos3   cos   sin 2  cos 
8. Simplify each of the following.
sin( 90  )
(a) cos(90  ) 
tan(90  )
cos 
(c) 7 cos(90  ) 
tan(90  )
9. Find  in each of the following.
1
(a) tan(  55) 
tan10
(c) sin 3  cos 63
(b) sin  
3
4
(b) 4  3cos2   3sin 2 
(d)
(b)
9  9 cos 2 
2 tan 
2
cos(90  )

tan(90  )
cos 
(b) sin(   18)  cos(  32)
(d) cos(  10)  sin( 2  40)
10. Prove that each of the following is an identity.
1
sin 
 1
(a)
(b) tan 2  
1
cos  tan 
cos 2 
1
 tan   tan(90  )
sin  cos 
(c) sin 2 (90  )  1  sin 2 
(d)
(e) 1  2 sin  cos   (cos   sin ) 2
(f) sin 2   (1  cos ) 2  2(1  cos )
Ex.9B Advanced Set
6
7
8
(c) tan  
5
(a) cos  
9.8
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Exercise 9C
[ Do not use calculators in this exercise. If necessary, leave your answers in surd form. ]
 Elementary Set
Level 1

1. Find the area of each of the following triangles.
A
(a)
P
(b)
L
(c)
60
12 cm
7 cm
30
B
N
5 cm
Q
C
60
R
M
2. Find the area of each of the following quadrilaterals.
(a)
A
B
(b)
P
4 cm
(c)
Q
D
4 3 cm
2 5 cm
120
2 3 cm
G
E
Ex.9C Elementary Set
60
60
D
S
C
R
F
3. Find the perimeter of each of the following figures.
(a)
A
5 cm
(b)
B
D
(c)
P
8 cm
30
45
D
9 cm
C
Q
E
45
R
QRS is a straight line.
6 cm
G
60
S
F
4. In the figure, ABCD is a trapezium, ADC  60 and
BC  CD. Find the area of trapezium ABCD.
A
12 cm
B
10 cm
60
D
5. The length of a diagonal of a square is 18 cm. Find the area of the square.
C
9.9
Chapter 9 Applications of Trigonometry
Level 2
D
6. In the figure, DEFG, AFC and AGB are straight lines.
(a) Find the lengths of AC, AD and CD.
(b) Find DCE.
(c) Find the length of DE.
E
C
F
1
Ex.9C Elementary Set
(d) Hence find the value of sin 75 by considering ADG.
30
45
A
7. In the figure, ABCDEFGH is a regular octagon, where
DE  2 cm. PFE and HGP are straight lines and
GPF  90.
G
B
A
(a) Find .
B
H
C
G
D
(b) Find the length of PF.
(c) Find the area of FGP.
2 cm

(d) Hence find the area of ABCDEFGH.
P
 Advanced Set
Level 1
E
F

1. Find the area of each of the following figures.
B
(a)
(b)
3 2 cm
D
(c)
E
G
45
5 cm
5 cm
30
A
(d)
F
C
7 cm
J
K
I
(e)
P
10 cm
W
(f)
Q
30
60
12 cm
M
4 cm
10 cm
Y
60
L
S
X
R
Z
Ex.9C Advanced Set
30
H
9.10
New Trend Mathematics S3B — Junior Form Supplementary Exercises
2. Find the perimeter of each of the following figures.
A
(a)
15 cm
D
(b)
Q
P
(c)
E
30
8 3 cm
B
E
C
G
10 3 cm
60
60
H
S
F
R
60
D
4 3 cm
3. A slide is 10 m long. The angle between the slide and the ground is 45. Find the height of
the top of the slide from the ground.
Level 2
Ex.9C Advanced Set
4. In the figure, APQ is a right-angled triangle. PQR is a
straight line. QAP  60, QRB  90, ABR  90 and
AB  2 cm.
(a) Find the length of RB.
P
(b) Find the length of PR.
Q
60
A
2 cm
R
B
D
5. In the figure, AEC and DCFB are straight lines.
BDE  BAC  30, CFE  FBA  90 and CE  60 cm.
Find the length of DE.
30
60 cm
C
E
A
F
30
B
6. In the figure, AED and BCD are straight lines.
D
(a) Express AC, BC and CD in terms of a.
E
(b) Find CDE.
C
(c) Hence find the value of sin 15 by considering ACE.
15
30
A
a
B
D
7. In the figure, ABCDEFGHIJKL is a regular 12-sided
polygon with the sides 1 cm each, and ACEGIK is a regular
hexagon.
C
E
F
B
(a) Find BAC.
(b) Use ABC and ALK to form a rhombus. Then find the
area of the rhombus.
G
A
H
L
(c) Hence find the total area of the shaded regions.
Ex.9C Advanced Set
9.11
Chapter 9 Applications of Trigonometry
I
K
J
Exercise 9D
[ In this exercise, correct your answers to 3 significant figures if necessary. ]
 Elementary Set
Level 1

1. Arrange the following gradients in descending order of their angles of inclination.
1
I. 1 : 10
II. 1 : 5
III. 1 : 9
2
IV. 1 : 0.8
V. 1 : 20
VI. 1 : 1.4
1 : 10
3. The horizontal distance and vertical distance of a slope are 40 m and 8 m respectively. Find
the gradient of the slope.
4. A person walks 80 m down a road with the gradient of
1
. Find the vertical distance
3
travelled.
Level 2
5. The figure shows a wooden platform ABCD. It is given
that AB // DC, CE  AB, AE  30 m, DC  14 m, EB  8 m
and CBE  50.
(a) Find the gradients of AD and CB. Express your
answers in the form of 1 : n.
(b) Which side of the platform is steeper?
D
14 m
C
50
A
30 m
E 8m B
Ex.9D Elementary Set
2. The road sign in the figure shows that the gradient of a
road is 1 : 10. If the angle of inclination of the road is ,
find .
9.12
New Trend Mathematics S3B — Junior Form Supplementary Exercises
6. The figure shows a part of a map with the scale of
1 cm : 0.25 km. The length of XY in the map is
1.8 cm.
(a) Find the actual distance of XY.
550 m
600 m
650 m
(b) Find the angle of inclination of XY.
X
Y
Scale 1 cm : 0.25 km
Ex.9D Elementary Set
7. The figure shows a map with the scale of
1 : 10 000. O denotes the location of a signpost, and
OA and OB denote two paths to there. On the map,
OA  2.8 cm and OB  1.6 cm.
260 m
A
240 m
O
(a) Find the gradient of OA.
(b) Find the gradient of OB.
220 m
200 m
(c) Which path is flatter?
180 m
B
160 m
Scale 1 : 10 000
D
8. The figure shows a slope with three sections where
their horizontal distances are 244 m, 206 m and
148 m. The gradients of AB, BC and CD are 1 : 2.2,
1 : 1.9 and 1 : 0.84 respectively. If a man travels
from A to D, how far does he travel?
C
B
A
244 m
Ex.9D Advanced Set
 Advanced Set
Level 1
206 m
148 m

1. Find the angle of inclination corresponding to each of the following gradients.
(a) 1 : 3
(b) 1 : 0.5
(c) 1 : 22.5
2. Find the gradient corresponding to each of the following angles of inclination. Express
your answers in the form of 1 : n.
(a) 10
(b) 40.5
(c) 67.25
9.13
Chapter 9 Applications of Trigonometry
3. A car travels 3.2 km up a slope with the gradient of 1 : 15. Find the vertical rise of the car.
4. The horizontal and vertical distances of a road are 420 m and 35 m respectively. Find the
gradient of the road.
1
5. A man walks 240 m down a path with the gradient of . Find the horizontal and vertical
8
distances he walks.
Level 2
6. The figure shows a cross section of a hill. The
lengths of two sides AB and BC of the hill are
460 m and 530 m respectively, and the height of
the hill is 360 m. Find the gradients of AB and BC,
and the horizontal distance of AC.
B
A
Horizontal ground
7. The figure shows a part of a map. The actual
horizontal distance of AB is 270 m. Find the angle
of inclination of AB.
B
360 m
340 m
A
320 m
300 m
8. The figure shows a part of a map with the scale of
1 : 5 000. The line segment AB measures 3.2 cm.
450 m
A
(a) Find the actual horizontal distance of AB.
(b) Find the gradient of AB.
425 m
400 m
375 m
B
350 m
Scale 1 : 5 000
325 m
Ex.9D Advanced Set
C
9.14
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Exercise 9E
[ In this exercise, correct your answers to 3 significant figures if necessary. ]
 Elementary Set
Level 1

1. The figure shows a basketball stand. Two players look at A from B and C. Find the
respective angles of elevation from B and C to A.
A
68
42
C
B
2. According to the figure, find
A
Ex.9E Elementary Set
20
45
B
C
D
(a) the angle of elevation to A from C.
(b) the angle of depression to D from A.
3. The figure shows that a person in a boat looks at point A of a lighthouse at the angle of
elevation 19.5. If the distances of the eye level and point A from the sea level are 2.4 m
and 20 m respectively, find the horizontal distance between the boat and point A.
A
20 m
19.5
2.4 m
9.15
Chapter 9 Applications of Trigonometry
4. A person looks at the top Y of building B
from the top X of building A at the angle
of elevation 28. If the height of building
A is 40 m and these two buildings are
240 m apart, find the height of building
B.
Y
X
28
40 m
A
B
240 m
5. A person looks at peak B from peak A at the angle of depression 7.4. It is given that the
heights of peaks A and B are 627 m and 508 m respectively. Find the horizontal distance
between the two peaks.
A
7.4
B
627 m
6. A person looks at the foot B of a church
from the top A of a building at the angle
of depression . Given that the height of
the building is 82 m, and it is 245 m away
from the church, find .
Ex.9E Elementary Set
508 m
A

82 m
245 m
7. A person sits at the front A of the upper
deck of a bus and looks at the top B of a
building. Given that A and B are 3.5 m
and 120 m from the ground respectively,
and the front of the bus is 164 m away
from the building, find the angle of
elevation to B from the eyes of the
person.
B
B
120 m

3.5 m
164 m
A
9.16
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Level 2
8. The figure shows a kite which is 30 m above
the ground. The angles of elevation to the
kite from A and B on the ground are 45 and
62 respectively. Given that the projection of
the kite on the ground forms a horizontal
line with A and B, find the distance between
A and B.
30 m
45
62
A
9. In the figure, the angles of elevation to the
top of a tree from A and B on the ground are
40 and  respectively. The horizontal
distances of A and B from the bottom C of
the tree are 18 m and 12 m respectively. Find
.
B

40
A
Ex.9E Elementary Set
10. In the figure, the shorter flagpole is 5 m
high, and its shadow BC is 6 m long. The
longer flagpole is 8 m high and its shadow is
AC.
(a) Find .
18 m
C
12 m
B

8m
(b) Find the distance between the two
flagpoles.
5m
A
C
B
6m
11. In the figure, a person looks at point A on
the ground from the top C of building X at
the angle of depression 70. Another one
looks at A from the top D of building Y at
the angle of depression 40. Given that the
two buildings stand on the same horizontal
line as point A, find the distance between the
two buildings.
12. In the figure, someone looks at points A and
B on the ground from the top X of a building
at the angles of depression 45 and 38
respectively. Given that the building stands
on the same horizontal line as A and B, find
the distance between A and B.
C
70
D
40
60 m
45 m
X
Y
A
X
38
45
90 m
A
B
9.17
Chapter 9 Applications of Trigonometry
13. From the top Y of building B, someone looks at the top X and foot Z of building A at the
angles of depression 15 and 30 respectively as shown. Find the height of building A.
Y
15
X
30
65 m
Building B
Ex.9E Elementary Set
Building A
Z
14. In the figure, two people look at a helicopter from
points A and B on the ground at the angles of
elevation 69 and 52 respectively. The distance
between A and B is 200 m. If A, B and the
projection of the helicopter lie on the same
horizontal line, find the height of the helicopter
above the ground.
C
69
52
A
B
200 m
 Advanced Set
Level 1
1. According to the figure,

B
(a) find the angle of elevation from C to B.
(b) find the angle of depression from B to A.
A
2. In the figure, the height SP of a tree and the length
of PQ are equal. PQR is a horizontal line.
(a) Find the angle of depression from S to Q.
38
Ex.9E Advanced Set
15
C
S
20
(b) Find the angle of elevation from R to S.
P
Q
R
9.18
New Trend Mathematics S3B — Junior Form Supplementary Exercises
3. In the figure, the height of a school is 20 m
and a spot on the ground is 30 m away from
the school. If the angle of elevation to the top
of the school from the spot is , find .
School 20 m

30 m
4. A person looks at an aeroplane from B at the
angle of elevation of 25. Given that the
aeroplane is flying at a height of 2 000 m
above the ground at that time, find the
horizontal distance between the person and the
aeroplane.
A
2 000 m
25
C
B
5. On a horizontal ground, Howard stands 25 m away from a flagpole of 15 m high. What is
the angle of depression from the top of the flagpole to the position of Howard?
Ex.9E Advanced Set
6. According to the figure, find the value of h.
hm
50
1.6 m
10 m
Level 2
7. In the figure, a helicopter is 160 m above the
ground. The angles of elevation to the
helicopter from points A and B on the ground
are 48 and 69 respectively. Given that the
projection of the helicopter on the ground lies
on the same horizontal level as A and B, find
the distance between A and B.
160 m
69
B
48
A
9.19
Chapter 9 Applications of Trigonometry
8. In the figure, someone looks at the top of a lighthouse from X and Y on the ground at the
angles of elevation 24 and 32 respectively. If the horizontal distance between X and the
top of the lighthouse is 148 m, find the horizontal distance between Y and the top of the
lighthouse.
32
24
X
Y
148 m
Q
70
P
50
54 m
38 m
X
Building A
33
10. In the figure, the angles of depression from the
top of flagpole B to the top and foot of flagpole A
are 22 and 33 respectively. Given that the
height of flagpole B is 7.2 m, find the height of
flagpole A.
22
7.2 m
A
11. Simon is going to find out the height of a
bookshelf. He stands in front of it and measures
the angle of elevation of the top as 31 and the
angle of depression of the foot as 39. If his eye
level is 1.76 m above the ground, find the height
of the bookshelf.
12. Henry looks at the top and bottom of a basketball
backboard from the playground at the angles of
elevation of 18.3 and 12 respectively. If his eye
level is 1.6 m above the ground, and the
horizontal distance between him and the
backboard is 6.2 m, find the height AB of the
backboard.
Building B
B
31
39
1.76 m
A
18.3
B
12
1.6 m
6.2 m
Ex.9E Advanced Set
9. In the figure, two people look at point X on the
ground from the top P of building A and the top
Q of building B at the angles of depression 50
and 70 respectively. Given that the two
buildings stand on the same horizontal line as X,
find the distance between the two buildings.
9.20
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Ex.9E Advanced Set
13. In the figure, the angle of elevation to a kite from
X on the ground is 64, and the angle of elevation
to the kite from Y on the ground is 38. The
distance between X and Y is 74 m. If X, Y and the
projection of the kite on the ground lie on the same
horizontal line, find the height of the kite.
64
38
X
14. Two people look at point C on the ground from A
and B of a building at the angles of depression 42
and 30 respectively. If A is 14.2 m above B, find
the height of A from the ground.
Y
74 m
A
42
14.2 m
B
30
C
Exercise 9F
 Elementary Set
Level 1

1. Complete the following table.
Compass bearing
True bearing
N30E
S42W
Ex.9F Elementary Set
S19E
010
123
270
310
N
2. According to the figure, find the compass bearing and
true bearing of B from A.
A
30
B
9.21
Chapter 9 Applications of Trigonometry
N
3. According to the figure, find
(a) the true bearing of Y from X.
Y
N
(b) the true bearing of X from Y.
X
N
Ex.9F Elementary Set
4. According to the figure, find
15
B
(a) the compass bearing of C from A.
(b) the compass bearing of B from O.
60
A
70
D
(c) the compass bearing of B from A.
60
(d) the compass bearing of C from B.
65
O
E
C
Level 2
5. Victor drives a car from city A at a bearing of S47W to city B. If he has to drive from city
B back to city A, in what direction should he drive?
 Advanced Set
Level 1

1. If the compass bearing of B from A is N54W, what is the compass bearing of A from B?
2. If the true bearing of Y from X is 230, what is the true bearing of X from Y?
N
A
(b) the compass bearing of A from C.
36
30
(c) the true bearing of B from C.
B
(d) the true bearing of O from B.
25
30
O
C
E
Level 2
4. A missile P is launched from A at a bearing of 217 to position B. If a missile Q is launched
from position B to hit missile P, in what direction should missile Q fly?
5. Patrick runs from location A for 100 m at a bearing of 045 to location B, then he runs for
100 2 m at a bearing of 270 to location C. If he then runs from B for 100 m at bearing of
135, what is his final location?
Ex.9F Advanced Set
3. According to the figure, find
(a) the compass bearing of B from A.
9.22
New Trend Mathematics S3B — Junior Form Supplementary Exercises
Exercise 9G
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
 Elementary Set
Level 1

1. Jane walks 500 m due south and then 750 m due west. Find the compass bearing of her
final position from the starting point.
2. If Alfred walks 250 m at a bearing of 297 and then walks to the west of the starting point
at a bearing of 180, find the distance between the final position and the starting point.
N
3. Vivian walks 5 km from A at a bearing of N62W to B,
and then walks 3 km at a bearing of S28W to C. Find
the true bearing of C from A.
B
N
5 km
3 km
62
28
A
C
Ex.9G Elementary Set
4. The figure shows three locations P, Q and R on a map.
P is at a bearing of N28W from Q, Q is at a bearing of
S62W from R, and P is at a bearing of S95W from R.
It is given that PQ  45 km.
N
95 R
P
62
N
28
(a) Find the distance between P and R.
Q
(b) Find the distance between Q and R.
5. The bearing of a lighthouse from a ship at A is N72E.
The ship then sails eastward to B and the bearing of the
lighthouse from B is N29W. If the distance between A
and B is 1 500 m, what will be the shortest distance
between the ship and the lighthouse during the journey?
N
N
29
72
A
1 500 m
B
Level 2
6. In the figure, a ship sails 10 km from A to B. The
bearing of B from A is N45W. Then the ship sails
15 km to C from B at the direction N20W.
(a) Find the true bearing of C from A.
(b) Find the distance of AC.
C
N
15 km
20
45
N
B
10 km
A
9.23
Chapter 9 Applications of Trigonometry
N
C
N
40
60
A
10 km
Ex.9G Elementary Set
7. The bearing of C from a car at A is N60E. The car
travels 10 km eastward to B and the bearing of C
from B is N40E. If the car continues to travel
eastward, what will be its shortest distance from C?
B
N
8. May is at a bearing of S40E and 150 m away from
Winnie. They start travelling at the same time and
May travels to the north at 5 m/s. Winnie decides to
meet May by taking the shortest path.
5 m/s
40
(a) At which direction should Winnie go?
150 m
N
(b) How far should May travel to meet Winnie?
(c) How far should Winnie travel to meet May?
(d) Find the lowest speed taken by Winnie to meet
May.
 Advanced Set
Level 1
1. In the figure, the distance between two buoys A and
B is 150 m. Boat C is due north of A, and the
compass bearing of boat C from B is N72W. Find
the distances of C from buoys A and B.

N
C
N
72
A
B
Ex.9G Advanced Set
150 m
N
2. In the figure, two ships B and C sail from A. Ship B
sails 23 km to the east and ship C sails 18 km to the
south.
A
23 km
B
B
(a) Find the distance between the two ships.
(b) Find the true bearing of ship C from ship B.
18 km
B
3. Sam and Raymond sail for 3 hours on windsurfers
from the starting point A to B and C at bearings
S28W and S62E respectively. The speed of Sam is
2 km/h, and the speed of Raymond is 3 km/h. Find
the true bearing of C from B.
C
N
A
62
28
B
C
9.24
New Trend Mathematics S3B — Junior Form Supplementary Exercises
N
4. The figure shows the locations of three ships A, B and
C. Ship B is due east of ship A, and ships C and B are
240 m apart. The bearings of ship C from ships A and
B are S25W and S62W respectively. Find the
distance between ships A and C.
N
A
B
62
25
240 m
C
Level 2
N
5. The figure shows the locations of three lighthouses X,
Y and Z. Ben sails 19 km in a boat from lighthouse X
to lighthouse Y at a bearing of 203, and then sails
22 km to lighthouse Z at a bearing of 102.
(a) Find the distance between lighthouses X and Z.
203
X
19 km
N
(b) Find the true bearing of X from Z.
N
102
Y
22 km
Z
Ex.9G Advanced Set
6. In the figure, cars A and B travel from O at the same
time at bearings N30W and N64E respectively. Car A
travels at 82 km/h and car B travels at 100 km/h.
(a) Find the distance between cars A and B after
2 hours.
(b) Find the compass bearing of car B from car A
after 2 hours.
7. The bearing of location B from location A is N54E,
and locations A and B are 154 km apart. Helicopters P
and Q fly from A and B respectively, and arrive at C
at the same time. It is given that helicopter P flies at
the speed of 160 km/h at a bearing of S12E for
1.8 hours to reach C.
(a) Find the distance between B and C.
A
B
N
30
64
O
N
154 km
B
54
A
12
(b) Find the speed of helicopter Q.
(c) Find the compass bearing of C from B.
C
8. A typhoon (at location B ) is observed 300 km away
from a city (at location A) at the bearing of S75E.
This typhoon moves at a bearing of N40W at
18 km/h. It is known that the area within 200 km from
the centre of the typhoon will be affected by the
typhoon. Will this city be affected by the typhoon? If
it does, how long will it be affected?
CHAPTER TEST
N
N
A
75
300 km
40
B
(Time allowed: 1 hour)
[ In this test, unless otherwise stated, correct your answers to 3 significant figures if necessary. ]
Section A (1) [ 3 marks each ]
1. Find the value of the expression tan 30 tan 60  sin 60 cos 30 without using a calculator.
2. Find the value of sin 2 75  sin 2 15 .
3. Lawrence walks 200 m up a slope with the gradient of
1
. What is his vertical rise?
5
4. The angle of depression from the top of a building of 100 m high to a car on the ground is
35. What is the distance between the car and the building?
5. A ship sails 15 km to the east from A to B, and then sails 10 km to the north to C. Find the
bearing of A from C. (Correct your answer to 1 decimal place.)
6. If sin  
5
, find the values of cos  and tan  .
13
Section A (2) [ 6 marks each ]
7. Find the unknowns in the figure and leave your
answers in surd form.
b cm
135
30
10 cm
a cm
Ex.9G Advanced Set
9.25
Chapter 9 Applications of Trigonometry
9.26
New Trend Mathematics S3B — Junior Form Supplementary Exercises
8. Prove that sin  cos (1  tan 2 )  tan  is an identity.
9. The figure shows a part of a map with the scale of
1 : 10 000. The length of AB is 4 cm.
A
(a) Find the actual horizontal distance of AB.
(b) Find the average angle of inclination of AB.
250 m
200 m
150 m
B
100 m
50 m
Scale 1 : 10 000
10. In the figure, the compass bearing of B from A is S38E,
the compass bearing of B from C is S52W, and the
compass bearing of C from A is N85E. A car travels at
the speed of 40 km/h from A to B for 2.5 hours.
(a) Find ABC.
(b) Find the distance of AC.
N
N
A
C
85
52
38
B
Section B
11. S3A students are having body height measurements
with the ruler on the wall. Alfred is 156 cm tall. But
when Patrick stands facing him 60 cm apart, he reads
Alfred’s height as 159 cm.
(a) According to the figure, explain why there is a
difference in Alfred’s height.
(3 marks)
(b) If the distance between the top of Alfred’s head and
the ruler on the wall is 10 cm, what is the angle of
elevation from Patrick’s eyes to the top of Alfred’s
head?
(3 marks)
(c) What is the distance between Patrick’s eye level and
the ground?
(3 marks)
(d) If Patrick walks 60 cm backwards, what will he read
about Alfred’s height? (Correct your answer to the
nearest cm.)
(4 marks)
60 cm
10 cm
Chapter 9 Applications of Trigonometry
9.27
Multiple Choice Questions [ 3 marks each ]
12. Find the area of the following
parallelogram, correct to the nearest
1 cm 2.
12 cm
16. Simplify
sin 
1  cos 2 
.
A. 0
B. 1
120
C. sin 
9 cm
□
D. tan 
A. 50 cm 2
17. William walks 500 m up a path with the
gradient of 1 : 12. Find his vertical rise.
B. 72 cm 2
C. 94 cm 2
□
D. 108 cm 2
13. Find the value of
3
A.
4
1
B.
4
C.
D.
A. 498 m
B. 497 m
C. 41.7 m
cos 30 sin 30
.
tan 30
□
D. 41.5 m
18. If the angle of elevation from A to B is
57, then the angle of depression from B
to A is
3
2
3
4
A. 33.
□
B. 43.
C. 57.
□
D. 137.
6
14. Given that cos   , find the values of
7
sin  and tan .
13
6 13
A. sin  
, tan  
7
13
19. What is the bearing of D from A?
N
A
13
13
B. sin  
, tan  
7
6
34
32
13
13
C. sin  
, tan  
6
6
D. sin  
6 13
13
, tan  
13
6
64
□
B
42
C
A. S34E
15. Given that sin 28.7  cos  , find .
B. S66E
A. 27.8
C. N42E
B. 28.7
D. N64E
C. 61.3
D. 65.2
D
□
□
9.28
New Trend Mathematics S3B — Junior Form Supplementary Exercises
20. Find the area of the following ABC.
24. The following shows a map of the scale
1 : 5 000. If the length of MN on the map
is 3.2 cm, find the average angle of
inclination of the path, correct to
1 decimal place.
A
8 cm
45
30
B
C
A. 42 2 cm 2
B. 54 2 cm 2
M
N
C. (8 3  24) c m 2
□
D. (8 3  8) c m 2
300 m
250 m
200 m
150 m
21. Simplify
1
tan (90  )
2
 tan 2  .
Scale 1 : 5 000
A. 43.2
A. 2 cos 2 
B. 46.8
B. 2 tan 2 
C. 58.4
C. 1
□
D. 0
22. Islands A and B are 350 m apart. The true
bearing of B from A is 130. A ship sails
from the west of B along a straight line
passing through B. What is the shortest
distance between the ship and A? (Correct
your answer to the nearest m.)
A. 225 m
25. Find the angle of elevation from S to P,
correct to 1 decimal place.
P
2.2 m
Q
2m
R
1.6 m
S
B. 268 m
2.8 m
3.6 m
1.5 m
C. 392 m
□
D. 417 m
A. 53.7
B. 51.6
C. 38.4
23. If sin   cos   1 , find .
2
□
D. 69.6
2
D. 36.3
□
A. 0
B. 30
C. 60
D. 90
□
26. The bearing of Wendy at location B from
Desmond at location A is 140, and they
are 20 m apart. Find the distance, correct
to 1 decimal place, that Wendy should
walk to the north such that the bearing of
Wendy from Desmond is 110.
Chapter 9 Applications of Trigonometry
N
110
A
140
20 m
N
B
A. 4.7 m
B. 7.3 m
C. 10.6 m
D. 35.3 m
□
9.29