Download Elementary Set

170
New Trend Mathematics S4B — Supplement
Chapter 12
Application of Trigonometry
WARM - UP E XERCISE
1. Find the marked unknowns in the following figures. (Leave your answers in surd form if
necessary.)
(a) A
(b)
x
4
x
4
2
B
3
C
2. Find the marked unknowns in the following figures. (Correct your answers to 1 decimal place.)
(a)
(b)
x
3
x
y
8
5
18
3. Find the marked unknowns in the following figures. (Leave your answers in surd form if
necessary.)
C
(a)
(b)
x
y
4
B
x
4
A
D
8
4. Solve the following equations for 0 <  < 180. (Correct your answers to 1 decimal place.)
(a) sin   0.3
(b) cos   0.74
5. Express the following trigonometric ratios in terms of acute angles.
(a) cos(30  70)
(b) sin (180  12  46)
6. Find the value of each of the following trigonometric ratios. (Leave your answers in surd form.)
(a) sin 120 
(b) cos135 
171
Chapter 12 Application of Trigonometry
B UILD - UP E XERCISE
[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set,
Intermediate Set and Advanced Set. Yo u may choose to complete any ONE set according to your need. ]
Exercise 12A
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
  El em en tar y S et
Level 1
1. Solve ABC if B  90 and a  c  7.
 
2. Solve ABC if A  90, a  5 and B  35.
3. In the figure, BCD is a straight line. ABC  90, ACB  57,
ADB  32 and AB  10 cm.
(a) Find the length of CB.
(b) Find the lengths of AD and CD.
A
10 cm
57
32
D
A
Find the area of each of the following triangles. (6  9)
A
6.
7.
5 cm
5 cm
4 cm
A
73
6 cm
B
B
8.
A
C
C
9.
3 cm B
146
A
110
7 cm
B
C
12 cm
C
C
58
E
B
5. In the figure, ABC  ACD  90, AB  4 cm, AC  10 cm and
BAC  2CAD.
(a) Find BAC.
(b) Find the length of CD.
10 cm
B
D
17 cm
D
A
4 cm
B
10 cm
C
Ex.12A Elementary Set
4. In the figure, AD and BC intersect at E. BAC  ABD  90,
ACB  58, BD  17 cm and AC  10 cm.
(a) Find the length of AB.
(b) Find ADB.
C
172
New Trend Mathematics S4B — Supplement
10. Find  in ABC with the given area where 0 <  < 90.
B
6 cm

A
10 cm
Area  20 cm2
11. Find the value of x in ABC with the given area.
C
C
x cm
30
10 cm
A
Find the area of each of the following parallelograms. (12  14)
A
A
D
12.
13.
Ex.12A Elementary Set
16 cm
4 cm
B
Area  40 cm2
D
125
50
B
14.
20 cm
A
C
6 cm
147
D
B
5 cm
C
B
3 cm
C
Level 2
15. In the figure, ABC is a straight line. DAB  90, DBA  60,
DCB  30, BC  30 cm and AD  h cm.
(a) Find ADB and ADC.
(b) Express AB in terms of h according to ABD and ACD.
(c) Hence find the value of h.
D
h cm
60
A
16. In the figure, AB  15 cm, AC  8 cm and CAB  50.
(a) Find the area of ABC.
(b) Find the height of ABC from C to AB.
(c) Find x and B.
Ex.12A Intermediate Set
 Intermediate Set
Level 1
17. Solve ABC if B  90, a  10 and b  18.
B
30
30 cm
C
C
x
8 cm
50
A
B
15 cm

18. Solve ABC if A  50, B  90 and a  10.
19. In the figure, ABC is a straight line. DBC  90, BCD  30,
ADB  45 and AD  10 cm.
(a) Find DAB and BDC.
(b) Find the length of DB.
(c) Find the length of AC.
D
10 cm 45
30
A
B
C
173
Chapter 12 Application of Trigonometry
20. In the figure, BCD is a straight line. BAC  18, AB  12 cm and
CD  16 cm.
(a) Find the length of BC.
(b) Find ADC.
D
16 cm
C
A
18
12 cm
B
Find the area of each of the following triangles. (21  23)
C
A
21.
22.
5 cm
8 cm
117
48
A
7 cm
8 cm
B
C
B
B
23.
76
A
17 cm
C
B
24. Find  in ABC with the given area where 0 <  < 90.
22 cm

14 cm
A
Area  140 cm2
25. Find the value of x in ABC with the given area.
A
x cm
C
60
15 3 cm
B
Find the area of each of the following parallelograms. (26  28)
D
C
B
26.
27.
7 cm
1 cm
61
A
B
3 cm
135
A
D
28.
A
10 cm
B
D
16 cm 40
C
C
C
Area  85 cm2
Ex.12A Intermediate Set
13 cm
174
New Trend Mathematics S4B — Supplement
Level 2
29. In the figure, ABCD is a parallelogram. BD is a diagonal with length
20 cm. If AB  14 cm and BDC  26, find the area of parallelogram
ABCD.
14 cm
A
B
20 cm
26
D
Ex.12A Intermediate Set
30. In the figure, AB  14 cm, CD  18 cm, BC  10 cm, ABC  102 and
BCD  90.
(a) Find the length of BD.
(b) Find CBD and ABD.
(c) Hence find the area of quadrilateral ABCD.
31. In the figure, AB  10 cm, BC  17 cm and B  48.
(a) Find the area of ABC.
(b) Find the height of ABC from A to BC.
(c) Find b, A and C.
C
A
14 cm
B
102 10 cm
D
C
18 cm
B
48
17 cm
10 cm
A
32. In the figure, AB  12 cm, AC  14 cm, AD  11 cm, BAC  49 and
CAD  32.
(a) Find the area of quadrilateral ABCD.
(b) Find the area of BCD.
C
A
49 12 cm
11 cm
32
14 cm
B
D
C
 Advanced Set
Level 1

33. Solve ABC if B  90, b  12 and a  10 .
Ex.12A Advanced Set
34. Solve ABC if A  40, C  90 and b  15 .
35. In the figure, ABC is a straight line. CAD  90, BDA  27,
CDB  9, CD  85 cm and AB  h cm.
(a) Find the length of AD.
(b) Hence find the value of h.
C
85 cm
9
B
h cm
27
D
A
175
Chapter 12 Application of Trigonometry
Find the area of each of the following triangles. (36  37)
A
36.
37.
12.3 cm
A
B
19.7 cm
152
C
9 cm
B
83.6
C
38. Find  in ABC with the given area where 0 <  < 90.
B
4 3 cm

C
4 3 cm
A
Area  9 3 cm2
39. Find the value of x in ABC with the given area.
A
x cm
B
C
Find the area of each of the following parallelograms. (40  41)
14.8 cm
A
40. 10 cm A
41.
D
80
B
118
5.7 cm
2 13 cm
D
Area  100 cm2
C
B
C
Level 2
42. In the figure, ABC is a straight line. ADB  15, BDC  20,
AB  20 cm and BC  x cm.
(a) By considering BDC, express DC in terms of x.
(b) By considering ADC, express DC in terms of x.
(c) Hence find the length of BC.
D
15
A 20 cm B x cm C
20
Ex.12A Advanced Set
35
176
New Trend Mathematics S4B — Supplement
43. In the figure, AB // DC and DAB  90. AB  15 cm, CD  8 cm
and ABD  25.
(a) Find the length of BD.
(b) Find the area of CBD.
D
8 cm
C
25
15 cm
A
B
A
44. In the figure, ABCD is a rhombus with sides of a cm each. Prove
that the area of rhombus ABCD is the greatest if BAD  90.
a cm
B
D
C
Ex.12A Advanced Set
45. In the figure, ADC is a straight line. BC  15.2 cm, BD  9.8 cm,
AD  7 cm, ACB  29.2 and CBD  20.
(a) Find the area of BCD.
(b) Find the length of CD.
(c) Hence find the area of ABC.
46. In the figure, regular hexagon ABCDEF is inscribed in a circle
with radius 8 cm and centre O.
(a) Find EOD.
(b) Find the area of EOD.
(c) Find the area of hexagon ABCDEF.
A
7 cm
D
9.8 cm
29.2
20
15.2 cm
C
A
F
B
O
8 cm
E
C
D
47. In the figure, quadrilateral ABCD is inscribed in a circle with
centre O and diameter 8 cm. ABC  .
(a) Express COD in terms of .
(b) If AOD : COD : BOC  1 : 3 : 1, find .
(c) Find the area of quadrilateral ABCD.
B
A
D
O

C
B
177
Chapter 12 Application of Trigonometry
Exercise 12B
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
  El em en tar y S et
Level 1
In each of the following triangles, find x. (1  2)
A
1.
2.
68
B
B
x cm
107
x cm
34
A
35
8 cm
 
6 cm
C
C
In each of the following triangles, find . (3  4)
A
3.
A
4.
8 cm
10 cm
7 cm
67
B
B

35
7 cm

C
C
Ex.12B Elementary Set
5. In ABC, if A  55, a  7 cm and b  8 cm, find B.
6. Solve acute-angled triangle ABC with A  30, a  6 cm and b  11 cm.
Solve ABC under each of the following conditions. (7  12)
7. A  39, B  131 and a  5 cm
8. B  50, C  70 and b  10 cm
9. B  70, b  15 cm and c  13 cm
11. C  47, a  11 cm and c  6 cm
10. A  135, a  5 cm and b  8 cm
12. B  40, a  8 cm and b  6 cm
Level 2
13. In the figure, ABCD is a quadrilateral. BC  10 cm, ABC  45,
BAC  55, CAD  70 and ADC  80.
(a) Find the length of AC.
(b) Find the length of AD.
A
D
55 80
B
14. In ABC, if A : B : C  2 : 1 : 1,
(a) find A, B and C.
(b) find a : b : c. (Leave your answer in surd form.)
70
45
10 cm
C
178
New Trend Mathematics S4B — Supplement
 Intermediate Set
Level 1
15. In ABC, find the value of x.

A
10 cm
x cm
52
B
16. In ABC, A is an acute angle. Find A.
C
B
8 cm
38
C
12 cm
A
17. Solve acute-angled triangle ABC with C  35, c  4 cm and a  6 cm.
Ex.12B Intermediate Set
Solve ABC under each of the following conditions. (18  23)
18. A  79, B  43 and b  5 cm
19. C  120, b  14 cm and c  16 cm
20. C  45, a  10 cm and c  13 cm
21. A  50, a  10 cm and b  13 cm
22. A  130, a  8 cm and b  14 cm
23. B  145, b  9 cm and c  5 cm
24. In ABC, if sin A : sin B : sin C  1 : 3 : 3 and a  15, find b and c.
Level 2
25. The figure shows a triangle ABD. C is a point on BD. AB  7 cm,
ABC  40, BAC  45 and CAD  15.
(a) Find the length of AC.
(b) Hence find the length of AD.
A
45
7 cm
40
B
26. In the figure, BAC : ABC : ACB  2 : 2 : 5, AC  AE and
BC  5 cm.
(a) Find BAC, ABC and ACB.
(b) Find the length of ED.
(c) Find the total area of ACBDE.
15
C
D
C
5 cm
A
B
E
D
179
Chapter 12 Application of Trigonometry
A
12 cm
9 cm
48
D
C
28. In the figure, the radius of the circle with centre A is 5 cm. ABC is
a straight line. If CD  9 cm and CAD  40,
(a) find ACD and ADC.
(b) find the area of ACD.
(c) find the length of BC.
 Advanced Set
Level 1
29. In ABC, find the value of x.
B
D
5 cm
9 cm
40
A
C
B
Ex.12B Intermediate Set
27. In the figure, ABCD is a parallelogram with AC  12 cm,
BC  9 cm and ACD  48. ABC is an acute angle.
(a) Find CAB.
(b) Find ABC.
(c) Find the area of parallelogram ABCD.

A
4 3 cm
B
48
4 3 cm
x cm
C
30. In ABC, find .
B
x cm
2x cm
60
C
Ex.12B Advanced Set

A
Solve ABC under each of the following conditions. (31  36)
31. A  60, C  45 and b  8 3 cm
32. C  100, b  7 2 cm and c  10 3 cm
33. B  30, b  12 cm and c  20 cm
34. A  47, a  7 cm and b  8 cm
35. A  140, a  9 cm and b  7 cm
36. C  125, b  16 cm and c  13 cm
Level 2
37. In ABC, B  , A  30  , C  58 and a  13 cm.
(a) Find .
(b) Find the length of AC.
(c) Find the area of ABC.
B

13 cm
30  
58
C
A
180
New Trend Mathematics S4B — Supplement
38. In the figure, BMC is a straight line. ABM  75, AMB  75,
ACM  35 and BC  30 cm.
(a) Find the length of AM.
(b) Find the area of AMC.
B
75
75
A
M
30 cm
35
C
Ex.12B Advanced Set
39. The figure shows a triangle PQR. S is a point on PR. PQ  15 cm,
QR  QS  7 cm and QPS  25.
(a) Find PRQ.
(b) Find the length of PS.
R
S
7 cm
25
15 cm
P
Q
40. In ABC, if sin A : sin B : sin C  6 : 5 : 9 and the perimeter is 40 cm, find a.
41. In the figure, D is a point on AC. DC  4 cm and BD  10 cm.
(a) Find the length of AD.
(b) Find the area of ABC.
B
45
10 cm
60
A
42. In the figure, ABCD is a trapezium with AD // BC. AD  16 cm,
BC  7.5 cm, ABC  130 and ADB  50.
(a) Find the lengths of AB and BD.
(b) Hence find the area of trapezium ABCD.
A
D 4 cm C
16 cm
D
50
130
B 7.5 cm C
Exercise 12C
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
  El em en tar y S et
 
Ex.12C Elementary Set
Level 1
In each of the following triangles, find the marked unknown. (1  4)
B
1. A
2.
6
15
119
7
c
B
32
8
A
C
b
C
181
Chapter 12 Application of Trigonometry
3.
B
10
B
4.
4

A
6
5
9
C

A
7
C
5. Find the greatest angle of ABC if a  4, b  8 and c  10.
6. Find the smallest angle of ABC if a  12, b  10 and c  7.
Solve ABC under each of the following conditions. (7  12)
7. A  60, b  5 and c  8
8. B  50, a  8 and c  17
11. a  9, b  13 and c  3
10. a  11, b  8 and c  7
12. a  18, b  7 and c  13
Level 2
13. In the figure, ABCD is a quadrilateral with AB  12 cm,
BC  7 cm, CD  10 cm, ABC  90 and ACD  35. Find the
lengths of AC and AD.
A
D
12 cm
35
B
14. In the figure, AD  7.5 cm, CD  5.3 cm, BC  4 cm, ADB  52
and CBD  32.
(a) Find BDC and BCD.
(b) Find the lengths of BD and AB.
(c) Hence find the area of quadrilateral ABCD.
7 cm
10 cm
C
A
7.5 cm
D
52
32
4 cm
5.3 cm
B
C
15. In the figure, CDB is a straight line. AC  11 cm, CD  8 cm,
BD  7 cm and ABC  35. AB is shorter than BD.
(a) Find the length of AB.
(b) Find the length of AD and DAB.
(c) Find the area of ABD.
A
11 cm
C
8 cm
D
35
7 cm
B
Ex.12C Elementary Set
9. C  147, a  3 and b  6
182
New Trend Mathematics S4B — Supplement
 Intermediate Set

Level 1
In each of the following triangles, find the marked unknown. (16  17)
B
B
5
16.
17.
30
c
C
15
8
A

8
C
A
18. Find the greatest angle of ABC if a  9, b  9 and c  16.
19. Find the smallest angle of ABC if a  11, b  13 and c  18.
20. In PQR, p  10 cm, q  7 cm and r  4 cm.
(a) Find P.
(b) Find the area of PQR.
Ex.12C Intermediate Set
Solve ABC under each of the following conditions. (21  25)
21. A  13 , b  6 and c  14
22. B  85 , a  14 and c  21
23. C  40 , a  7 and b  12
24. a  5, b  4 and c  7
25. a  27, b  43 and c  14
Level 2
26. In ABC, the ratio of a, b and c is 4 : 5 : 2. Find cos A.
27. In the figure, ABCD is a quadrilateral with AB  12 cm,
BC  16 cm, CD  13 cm, ABC  45 and ACD  30. Find the
lengths of AC and AD.
D
A
13 cm
30
12 cm
45
16 cm
B
28. In the figure, PQRS is a parallelogram with PS  11 cm, RS  9 cm
and PSR  115.
(a) Find the length of PR.
(b) Find the length of QS.
Q
R
9 cm
115
P
29. In the figure, BCD is a straight line.
(a) Find ABC.
(b) Find the length of AD.
C
11 cm
S
A
8
B
7
13
C
15
D
183
Chapter 12 Application of Trigonometry
A
C
Ex.12C Intermediate Set
30. In the figure, ABC is a triangle where the ratio of a, b and c is
7 : 6 : 5.
(a) Find A, B and C.
(b) If the perimeter of ABC is 126 cm, find a, b and c.
(c) Find the area of ABC.
B
 Advanced Set

Level 1
In each of the following triangles, find the marked unknown. (31  32)
A
4 3
31. A
32.
B
55
10
c
9 2

C
C
5 3
B
33. Find the greatest angle of ABC if a  15 3 , b  12 3 and c  17 .
34. Find the smallest angle of ABC if a  6 2 , b  3 2 and c  3 5 .
35. C  40, a  5 5 and b  7 5
Ex.12C Advanced Set
Solve ABC under each of the following conditions. (35  37)
36. a  17, b  16 and c  25
37. a  9 3 , b  5 2 and c  4 7
Level 2
38. In the figure, CED is a straight line and ABED is a parallelogram.
AB  2 cm, AD  4 cm, BC  5 cm and CD  6 cm.
(a) Find ADC.
(b) Find the area of trapezium ABCD.
39. In the figure, ABCDE is a regular pentagon with sides of 16 cm each.
(a) Find AED and CAD.
(b) Find the length of AD.
(c) Find the area of pentagon ABCDE.
A 2 cm B
5 cm
4 cm
D
E
C
6 cm
A
16 cm
E
B
D
40. In the figure, OBC is a triangle. Find the length of BC.
C
O
4
D
9
5
B
A
10
7
C
184
New Trend Mathematics S4B — Supplement
41. In the figure, PQRS is a quadrilateral with PQ  14 cm,
QR  10 cm, RS  8 cm, PS  6 cm and SPQ  26.
(a) Find the length of diagonal QS.
(b) Hence find QRS.
P
26
6 cm
S
14 cm
8 cm
Q
10 cm
R
42. In ABC, the ratio of sin A, sin B and sin C is 7 : 4 : 5.
(a) Find a : b : c.
(b) Find cos A.
Ex.12C Advanced Set
43. In the figure, a quadrilateral ABCD is inscribed in a circle.
AB  24, BC  14, AC  32 and AD  CD.
(a) Find ABC.
(b) Find ADC.
(c) Hence find the length of AD.
D
A
32
24
14
B
44. In the figure, ABCD is a rhombus with sides of 14 cm each. If the
area of rhombus ABCD is 60 cm 2 and AC is shorter than BD,
(a) find BAD.
(b) find ABC.
(c) find the lengths of the diagonals.
45. In the figure, ABC is inscribed in a circle with centre O. Suppose
AB  10 cm and the ratio of AOB, BOC and COA is 3 : 4 : 2,
(a) find AOB, BOC and COA.
(b) find the radius of the circle.
(c) find the area of ABC.
C
B
A
C
D
A
10 cm
O
C
B
Exercise 12D
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
  El em en tar y S et
 
Ex.12D Elementary Set
Level 1
Find the area of each of the following triangles. (1  2)
B
6 cm
1.
2.
C
A
8 cm
4 cm
6 cm
12 cm
B
A
5 cm
C
185
Chapter 12 Application of Trigonometry
Find the area of each of the triangles with the sides given as follows. ( 3  6)
3. a  8, b  8 and c  10
4. a  4, b  5 and c  7
6. a  12, b  13 and c  14
Level 2
7. Find the value of x with the given area of ABC.
A
6x cm
3x cm
B
8. ABDC is a trapezium with AC // BD. The perimeter of ABC is
56 cm and a : b : c  2 : 3 : 2.
(a) Find a, b and c.
(b) Find the area of ABC.
(c) Find the area of trapezium ABDC.
 Intermediate Set
Level 1
Find the area of each of the following triangles. (9  10)
B
9.
10. A
B
C
D
A
C

9 cm
20 cm
C
C
12 cm
B
Ex.12D Intermediate Set
A
5x cm
Area  16 cm2
18 cm
18 cm
12 cm
Ex.12D Elementary Set
5. a  9, b  11 and c  15
Find the area of each of the triangles with the sides given as follows. (11  12)
11. a  8, b  5 and c  9
12. a  5.6, b  9.2 and c  6.7
Level 2
13. Find the value of x with the given area of ABC.
A
2x
cm
3
x cm
B
C
Area  25 cm2
14. In the figure, a : b : c  4 : 7 : 9 and the perimeter of ABC is 60 cm.
(a) Find a, b and c.
(b) Find the area of ABC.
C
b
a
B
c
A
186
New Trend Mathematics S4B — Supplement
Ex.12D Intermediate Set
15. In the figure, BAC  50, BCA  38, AB  5.8 cm, AD  6 cm
and CD  4 cm.
(a) Find the length of BC.
(b) Find the length of AC.
(c) Find the area of quadrilateral ABCD.
A
5.8 cm 50
6 cm
B
D
38
4 cm
C
16. In the figure, AB  AC  x cm and BC  12 cm. If the area of ABC
is 48 cm 2, find the value of x.
A
x cm
x cm
B
 Advanced Set
Level 1
17. Find the area of ABC.
C
12 cm

A
5 5
B
7 5
C
Find the area of each of the triangles with the sides given as follows. (18  19)
18. a  18.7, b  14.1 and c  9.4
19. a  3 7 , b  6 7 and c  5 7
Ex.12D Advanced Set
Level 2
20. Find the value of x in the figure.
5 x cm
B
A
x cm
2x cm
C
Area  8 cm2
21. In the figure, ABCD is a quadrilateral with AB  4 cm, BC  10 cm,
CD  17 cm, AD  15 cm and BAD  90.
(a) Find the length of BD.
(b) Find the area of ABD.
(c) Hence find the area of quadrilateral ABCD.
C
10 cm
B
4 cm
A
22. In the figure, ABC is a triangle with a : b : c  5 : 4 : 4 and
perimeter 52 cm. D and E are points on AC and BC respectively.
(a) Find a, b and c.
(b) Find the area of ABC.
(c) If the ratio of the areas of ABC and DEC is 3 : 1, find the
area of quadrilateral ABED.
17 cm
D
15 cm
A
D
B
E
C
187
Chapter 12 Application of Trigonometry
23. The figure shows a quadrilateral ABCD. BAD  75, AB  8 cm,
AD  6.2 cm, CD  10 cm, BC  9 cm and BD  x cm.
(a) Find the value of x.
(b) Find the area of CBD.
(c) Find the area of quadrilateral ABCD.
B
9 cm
8 cm
A
x cm
75
6.2 cm
C
10 cm
24. In the figure, ABCD is a square with area 4x 2 cm 2. AFB, DEA,
CHD and BGC are four identical isosceles triangles and
AF  3x cm.
(a) Find the length of AB.
F
A
B
D
C
E
G
(b) If the area of AFB is 128 cm , find the value of x.
(c) Find the perimeter of octagon AFBGCHDE.
2
Ex.12D Advanced Set
D
H
25. In the figure, AB  6 cm, AC  8 cm and BC  2x cm. If the area of
ABC is 24 cm 2, find the value of x.
A
8 cm
6 cm
B
2x cm
C
Exercise 12E
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
  El em en tar y S et
 
Ex.12E Elementary Set
Level 1
1. Change the following compass bearings into true bearings.
(a) N15E
(b) S37E
2. Change the following true bearings into compass bearings.
(a) 020
(b) 255
N
3. In the figure, find the true bearing of C from A.
C
50
N
A
188
New Trend Mathematics S4B — Supplement
In each of the following figures, find the compass bearing of C from A. (4  5)
N
4. N
5.
N
B
82
A
55
18
C
47
A
N
80
C
B
Find the marked unknowns in the following figures. (6  8)
N
N
6.
7.
15
10 m
35
B
20 m
8m

N 50
xm
10
A
30 m
ym
C
8.
D
Ex.12E Elementary Set
ym
35
C
N

A
10 m
60
B
Level 2
9. In the figure, BC  82 m and AC  67 m. The true bearings of A
from B and C are 340 and 043 respectively.
(a) Find CAB.
(b) Find ABC.
(c) Find the length of AB.
N
A
67 m
N
43
N
C
340
82 m
B
10. In the figure, two soldiers A and B stand on the same horizontal
ground. The angles of depression of A and B from observation post
C are 55 and 43 respectively. The distance between A and B is
200 m.
(a) Express AD in terms of h.
(b) Express BD in terms of h.
(c) Hence find the value of h.
C
55
43
hm
A
D
200 m
B
189
Chapter 12 Application of Trigonometry
 Intermediate Set
Level 1
11. Change the compass bearing S52E into true bearing.

12. Change the true bearing 157 into compass bearing.
C
13. In the figure, find the true bearing of C from A.
N
60
A
N
5
B
In each of the following figures, find the compass bearing of C from A. (14  15)
14. N
15. N
A
C
C
50
25
N
75
B
40
N
N
Ex.12E Intermediate Set
50
A
B
Find the marked unknowns in the following figures. (16  17)
N
16.
17.
N
A 160
N xm
B
40
78
B
18 m

242
C
115 m
55
ym
Level 2
18. In the figure, P and Q are two ships and A is a lighthouse. Ship Q
is due north of ship P and they are 1 km apart. The true bearings of
lighthouse A from ships P and Q are 325 and 300 respectively.
(a) Find PAQ.
(b) Find the distance between lighthouse A and ship P.
(c) Find the distance between lighthouse A and ship Q.
19. In the figure, the angles of elevation of mountain top D from
points B and C are 47 and 40 respectively. The distance between
B and C is 100 m and the height of the mountain is h m.
(a) Express AB in terms of h.
(b) Find the distance between C and D.
(c) Hence find the value of h.
N
A
300
Q
1 km
P
325
D
hm
40
47
C
B
100 m
A
190
New Trend Mathematics S4B — Supplement
Ex.12E Intermediate Set
20. In the figure, a pole PQ is standing on a slope AB with inclination
20. The inclination of PQ is 82. The distance between A and B is
25 m. The angles of depression of A and B from Q are 43 and 76
respectively.
(a) Find ABQ.
(b) Find AQ.
(c) Find PQ.
43
B
P
25 m
82
20
A
 Advanced Set
Level 1
21. In the figure, find the compass bearing of C from A.
Q
76
C

N
N
B
192
A
48
C
22. In the figure, find the true bearing of C from A.
N
B
48
C
Ex.12E Advanced Set
N
30
A
Find the marked unknowns in the following figures. (23  26)
N
N
23.
24.
A
130
xm
6 3m
xm
B
25.
20
58
C
7m
26.
N
A
xm
C
230
38
600 2 m
C
20 3 km
N
30
B
N
B
305
N

55
y km
260
A
191
Chapter 12 Application of Trigonometry
Level 2
27. In the figure, AB is a building with height h m and CD is a
tower with height 100 m. The angle of elevation of D from A
is 32 and the angle of depression of C from A is 40.
(a) Find ADC.
(b) Find the distance between A and C.
(c) Hence find the height of the building AB.
D
32
A
100 m
40
hm
B
C
A
28. In the figure, the height of two buildings AD and BC are h m
and 40 m respectively. The angle of depression of B from A is
25 and the angle of elevation of B from D is 15.
(a) Find the distance between B and D.
(b) Find DAB and ABD.
(c) Hence find the value of h.
25
E
B
40 m
15
D
29. In the figure, post AB with length 6 m inclines to the east and
makes an angle of 70 with the horizontal line AP. PQ is a
wall which inclines to the west and makes an angle of 85
with the horizontal. The sun shines from the west and the
shadow of post AB is AP and PR. If the angle of elevation of
the sun ray is 20 and AP  4 m, find PR.
C
Q
Sun ray
B
20
R
6m
70
85
A
30. In the figure, a man B was due east to a man A at 12:00 p.m.
and the distance between them was 200 m. B was walking at
0.5 ms 1 along the direction 032.
(a) Find the true bearing of B from A at 12:10 p.m.
(b) The walking speed of B had changed since 12:10 p.m.
The difference between the bearing of B from A at
12:20 p.m. and that at 12:10 p.m. was 5. Find the new
walking speed of B.
Ex.12E Advanced Set
hm
E
P
B (at 12:20 p.m.)
B (at 12:10 p.m.)
5
N
A
N
200 m
32
0.5 ms1
B (at 12:00 p.m.)
Exercise 12F
  El em en tar y S et
Level 1
1. The figure shows a cuboid. Name the projection of
(a) BE on plane ABGF.
(b) BE on plane AFED.
(c) BE on plane EFGH.
 
E
H
C
D
F
A
G
B
Ex.12F Elementary Set
[ In this exercise, unless otherwise stated, correct your answers to 3 significant fi gures if necessary. ]
192
New Trend Mathematics S4B — Supplement
2. The figure shows a cuboid.
(a) Name the angle between
(i) BH and BG.
(ii) AE and BE.
(iii) AB and BH.
(b) Which of the above is a right angle?
8 cm
E
H
2 cm
C
D
G
4 cm
F
A
B
E
3. The figure shows a cuboid.
(a) Name the angle between planes
(i) ABHE and ABGF.
(ii) ABHE and ABCD.
(iii) BFE and ABCD.
(iv) BFE and BGH.
(b) Name five planes which are perpendicular to plane AFED.
H
C
D
G
F
A
In the figure, ABCDEFGH is a rectangular block. AB  19 cm,
BC  16 cm and CH  15 cm. (4  9)
4. Find the angle between AF and AD.
B
E
H
15 cm
G
F
C
Ex.12F Elementary Set
D
5. Find the angle between EG and EB.
16 cm
A
19 cm
B
6. Find the angle between BH and plane ABCD.
7. Find the angle between BE and plane ADEF.
8. Find the angle between planes BCEF and ABCD.
9. Find the angle between planes ABHE and EFGH.
Level 2
10. In the figure, VABC is a triangular pyramid with height 10 cm. Its
base ABC is a right-angled isosceles triangle where
AB  AC  7 cm and BAC  90.
(a) Find the length of the altitude from A to BC.
(b) Hence find the angle between planes VBC and ABC.
V
10 cm
C
A
D
7 cm
B
11. In the figure, ABEF is a horizontal ground. ABCD is a hillside with
the greatest inclination of 15. AC is a track making an angle of
50 with the line of greatest slope AD of the hill and the height of
point D on the hill is 52 m.
(a) Find the length of AD.
(b) Find the length of AC.
(c) Find the angle between AC and plane ABEF.
C
D
52 m
E
B
50
A
F
15
193
Chapter 12 Application of Trigonometry
12. The figure shows a right pyramid VABCDEF with height 24 cm. Its
base is a regular hexagon with sides of 18 cm each and the length
of each slant edge is 30 cm.
(a) Find the angle between lines VB and VC.
(b) Find the angle between VC and plane ABCDEF.
(c) Find the angle between planes VAF and ABCDEF.
V
24 cm
E
F
13. In the figure, ABC lies on a horizontal plane where ACB  90.
CD is a vertical line with length h cm, DAC  35 and
AD  DB  15 cm.
(a) Find the value of h.
(b) Find the length of AB.
(c) Find the area of ABD.
Ex.12F Elementary Set
30 cm
D
C
O
A 18 cm B
D
15 cm
15 cm
h cm
35
B
C
A
 Intermediate Set

Level 1
E
H
In the figure, ABCDEFGH is a rectangular block. M and N are the
M
mid-points of EF and AB respectively. It is known that AB  12 cm,
G
F
6 cm
BC  5 cm and CH  6 cm. (14  19)
14. Find the angle between AE and AD.
C
5 cm
D
15. Find the angle between FB and plane ABCD.
A
B
N
12 cm
Ex.12F Intermediate Set
16. Find the angle between BM and plane ADEF.
17. Find the angle between planes AGHD and BCHG.
18. Find the angle between planes AMN and ABGF.
19. (a) Find the lengths of MN, MH and NH.
(b) Find the angle between MN and NH.
Level 2
20. In the figure, VABCD is a right pyramid with a square base.
Suppose AB  BC  8 cm, VB  VC  9 cm and M is the mid-point
of BC,
(a) find the angle between the two diagonals of square ABCD.
(b) find the angle between VB and plane ABCD.
(c) find the angle between VM and plane ABCD.
V
D
C
M
N
A
B
194
New Trend Mathematics S4B — Supplement
21. In the figure, ABCDEF is a triangular prism with height 13 cm. Its
base DEF is a triangle, where DF  15 cm, DE  20 cm and
EDF  110.
(a) Find the angle between CE and plane DEF.
(b) Find the angle between CE and CF.
B
A
C
F
15 cm
13 cm
E
110
20 cm
D
22. In the figure, a flag pole AP is fixed vertically by two wires PB
and PC. PC  40 m, BC  25 m, BAC  56 and ACB  28.
(a) Find the length of AC.
(b) Find the height of the flag pole.
(c) Find the length of wire PB.
P
40 m
A
C
28
25 m
56
B
Ex.12F Intermediate Set
23. In the figure, ABCD and BCEF are two planes. M is the mid-point
of BC. ED  6 m, CE  15 m and AD  30 m. Find the angle
between
(a) line DC and plane BCEF.
(b) line DM and plane BCEF.
(c) line DB and plane BCEF.
A
30 m
D
6m
F
E
B
15 m
M
C
E
24. The figure shows a slope ABFE. ABCD, CDEF and ABFE are
rectangles and ABCD is perpendicular to CDEF. The greatest
inclination of the slope is 30. H is a point on BC such that
GH  BC. AB  350 m, BC  600 m and GAH  20. Let
GH  h m.
(a) Find AH and BH in terms of h.
(b) Find the length of AG.
(c) Find the lengths of HC, HD and ED.
(d) Find the total length of path AGE.
F
G
D
C
20
A 350 m
25. In the figure, a rectangular wall ABCD with AB  3 m and
BC  8 m stands vertically on the horizontal ground along the
east-west direction. The sun shines from N15E with an angle of
elevation 35 and the shadow of the wall on the ground is EBCF.
(a) Find the length of CF.
(b) Find the area of the shadow.
H
600 m
B
30
Sun rays
D
A
N
B
35
E
F
C
15
E
26. In the figure, a triangular landmark ABC stands vertically on the
horizontal ground along the east-west direction. AB  2 km,
BC  1.75 km and ABC  36. The sun shines from N42W with
an angle of elevation 30 and the shadow of the landmark on the
ground is ABT.
(a) Find the length of CD.
(b) Find the length of TD.
(c) Find the area of the shadow.
N
C
D
36
A
E
B
30
42
T
Ex.12F Intermediate Set
195
Chapter 12 Application of Trigonometry
 Advanced Set

Level 1
E
H
In the figure, ABCDEFGH is a rectangular block. M divides EF in the
M
ratio 1 : 3 and N divides AB in the ratio 4 : 3. It is known that
G
9 cm
AB  14 cm, BC  8 cm and CH  9 cm. (27  29)
F
27. Find the angle between BM and plane ABCD.
C
D
8 cm
28. Find the angle between planes EFN and NGH.
A
29. (a) Find the lengths of MN, NC and MC.
(b) Find the angle between MC and NC.
31. In the figure, VABC is a triangular pyramid with height VD where
D is the mid-point of BC. Its base ABC is an isosceles triangle
where AB  AC  18 cm and ACB  45. If VB  VC  24 cm,
(a) find VD.
(b) find the angle between VA and plane ABC.
(c) find the angle between VC and plane ABC.
V
15 cm
8 cm
B
C
6 cm
A
M
E
D
V
24 cm
B
A
18 cm
D
45
C
A
32. In the figure, ABCD is a regular tetrahedron. E is the mid-point of
BC. Find the angle between planes ABC and BCD.
D
B
E
C
Ex.12F Advanced Set
Level 2
30. In the figure, VABCD is a right pyramid with height 15 cm. Its
base ABCD is a rectangle where CD  6 cm and BC  8 cm. If M is
the mid-point of AD,
(a) find the length of slant edge VA.
(b) find the angle between VA and plane ABCD.
(c) find the angle between VA and plane VBD.
(d) find the angle between planes VAD and ABCD.
(e) find the angle between planes VAD and VBC.
B
N
14 cm
196
New Trend Mathematics S4B — Supplement
33. In the figure, the angle between two identical rectangles ABCD
and CDEF is 25. CD  10 m and DE  7 m.
(a) Find the distance between A and E.
(b) Find ACF.
(c) Find the angle between BD and plane CDEF.
B
C
F
10 m
A
25
7m E
D
34. A door with dimensions 1.5 m  2.5 m rotates from the position
ABCD to the position ABEF through 40 as shown in the figure.
(a) Find the length of AC.
(b) Find the distance between E and C.
(c) Find the angle between lines AC and AE.
E
40
B
C
2.5 m
F
A 1.5 m D
Ex.12F Advanced Set
35. In the figure, A, B and D are points on the same horizontal plane.
CD is a building and the bearing of the foot D of the building CD
from A and B are N38W and N55E respectively. The angle of
elevation of C from A is 65. A and B are 48 m apart and A is due
east of B.
(a) Find DBA and BDA.
(b) Find the distance between A and D.
(c) Find the height of building CD.
36. The figure shows a rectangular box with dimensions
20 cm  12 cm  8 cm. The cover of the box is opened and makes
an angle of 45 with the horizontal.
(a) Find the distance between A and K.
(b) Find the distance between A and G.
(c) Find the angle between AG and plane GHIJ.
(d) Find the distance between A and E.
(e) Find the angle between AG and AE.
37. In the figure, the angle of depression of A from an aeroplane H
right above C is 30 and the angle of elevation of H from B is 25.
The bearings of A and B from H are S20W and S50E
respectively. The distance between A and B is 500 m. Let the
altitude of the aeroplane be h m.
(a) Express AC and BC in terms of h.
(b) Find the value of h.
(c) Find the compass bearing of B from A.
C
65
N
B
38
D
55
N
A
48 m
B
A
45
12 cm
F
D
C
8 cm
E
I
J
G
20 cm
H
K
H
30
hm
N
C
20
A
50
500 m
E
25
B
E
197
Chapter 12 Application of Trigonometry
F
H
E
5 cm
10 cm
D
7 cm
C
R
39. In the figure, an aeroplane climbs from P to R with the inclination
of 30. P is due north of A and the angle of elevation of P from A
is 65. The bearing of R from A is N35E. Q and S are the
projections of P and R on the ground respectively. AQ  150 m and
AS  200 m.
(a) Find the original height PQ of the aeroplane.
(b) Find the height RS of the aeroplane.
(c) Find the true bearing of the course of the aeroplane.
30
P
N
150 m
65
Q
S
35 200 m
E
A
40. In the figure, AB is a pole with length 4 m. It inclines to the north
and makes an angle of 40 with the ground. C is the projection of
B on the ground. When the sun is due west of the pole, the shadow
of AB on the ground is AD and the angle of elevation of the sun
from D is 75.
(a) Find AC and BC.
(b) Find the length of shadow AD.
(c) Two hours later, the pole and its shadow are equal in length.
Let the new shadow be AD'. Prove that BC  CD'. What is the
angle of elevation of the sun from D' at this moment?
C HAPTER T EST
B
4 cm
A
G
Ex.12F Advanced Set
38. The figure shows a prism ABCDEFGH with length 10 cm. ABCD
is a trapezium where AB  4 cm, AD  5 cm, CD  7 cm and
BAD  ADC  90.
(a) Find the angle between line FC and plane DCHE.
(b) Find the length of FC.
(c) Find the angle between lines FC and FG.
B
N
4m
75
40
C
D
E
A
(Time allowed: 1 hour)
[ In this test, correct your answers to 1 decimal place if necessary. ]
Section A
1. In the figure, c  8, b  12 and A  92. Find the area of ABC.
(3 marks)
A
8
92
12
B
2. In the figure, ABC is a right-angled triangle. ADB  50 and
ACB  30. Find the length of CD.
(4 marks)
C
A
50
30
B 4 cm D
C
198
New Trend Mathematics S4B — Supplement
3. In the figure, find the value of x.
A
(4 marks)
80
x cm
B
4. In the figure, AB  8 cm, AC  15 cm and B  110. Find the
marked unknowns.
(4 marks)
10 cm
C
B
8 cm
110
A
y

15 cm
C
5. Solve ABC if A  50, b  8 cm and c  11 cm.
(5 marks)
6. In the figure, ABCD is a quadrilateral with AB  20 cm,
BC  19 cm, AD  22 cm, CD  25 cm and ABC  110.
(a) Find the length of AC.
(2 marks)
(b) Find the area of ACD.
(3 marks)
A
20 cm
110
22 cm
19 cm
D
Section B
7. A ship sails 18 km from C to B in the direction N70E. After
reaching B, it changes its course and sails to A which is 12 km due
south of C.
(a) Find the distance between A and B.
(3 marks)
(b) Find BAC.
(3 marks)
(c) Find the true bearing of A from B.
(2 marks)
(d) Find the area of ABC.
(2 marks)
8. The figure shows a rectangular block ABCDEFGH with
dimensions 8 cm  6 cm  3 cm.
(a) Find the lengths of DH, DB and BH.
(3 marks)
(b) Find the area of DBH.
(3 marks)
(c) Find the angle between DBH and base EFGH.
(4 marks)
B
C
25 cm
N
N
18 km
B
70
C
12 km
A
3 cm
D
A
B
C
8 cm
F
E
G
6 cm
H
Multiple Choice Questions (3 marks each)
9. In the figure, find the value of sin A.
9
C
A.
24 23
115
B.
5 23
24
C.
5 23
24
D.
 24 23
115
A
4
10
B
□
Chapter 12 Application of Trigonometry
10. In the figure, B 
A. 4.30 cm 2
A
B. 8.03 cm 2
72
C. 8.55 cm 2
12
B
A. 50.0 (corr. to 1 d.p.).
14. In the figure, the height of the vertical pole
PO is
B. 58.5 (corr. to 1 d.p.).
C. 72.4 (corr. to 1 d.p.).
D. 121.6 (corr. to 1 d.p.).
□
D. 12.5 cm 2
C
15
P
□
11. What is the area of ABC if a  7, b  9 and
c  13?
20
30
A
O
E
50 m
B
S
A. 15.7 (corr. to 1 d.p.)
A. 15.4 m (corr. to 1 d.p.).
B. 30.0 (corr. to 1 d.p.)
B. 19.9 m (corr. to 1 d.p.).
C. 43.3 (corr. to 1 d.p.)
C. 29.2 m (corr. to 1 d.p.).
D. 60.1 (corr. to 1 d.p.)
□
D. 93.7 m (corr. to 1 d.p.).
N
A
A
55
35
60
b
65
B
O
60
a
□
15. In the figure, the true bearing of A from B is
12. In the figure, find a : b : c.
c
199
C
B
A. 55 : 65 : 60
1 1 1
B.
: :
55 65 60
A. 060.
C. sin 55 : sin 65 : sin 60
B. 050.
D. cos 55 : cos 65 : cos 60
□
13. In the figure, A and B lie on the
circumference of a circle with centre O and
radius 5 cm. OBA  20. Find the area of
OAB, correct to 3 significant figures.
C. 035.
D. 025.
□
16. In the figure, D is a point on AC.
AB  AC  6, BD  4 and BDC  80.  
B
6
A
A

4
80
20
B
O
6 D
C
200
New Trend Mathematics S4B — Supplement
A. 57.7 (corr. to 1 d.p.).
ABCD.
B. 61.2 (corr. to 1 d.p.).
A. BEC
C. 69.5 (corr. to 1 d.p.).
B. BED
□
D. 72.3 (corr. to 1 d.p.).
Questions 17  18 refer to the following
rectangular block ABCDEFGH with AB  6 cm,
BC  4 cm and CH  3 cm.
E
F
G
A
6 cm
□
18. Find FBH, correct to 1 decimal place.
A. 15.6
B. 31.2
H
D
C. EBC
D. EBD
C. 54.7
D. 74.4
3 cm
C
4 cm
□
B
17. Name the angle between line BE and plane
H INTS
(for questions with
in the textbook)
Revision Exercise 12
31. (c) (i) Key Information
 When x  2,   15.3 (corr. to nearest 0.1)
18
3 2 2
) 6 2
 x ( x 
x
x
3
 tan  
x  18
x
Analysis
As tan  
3
18
and 0 <  < 90, we know that if x 
attains its minimum value, then
18
x
x x
tan  is maximum and  is the maximum viewing angle.
Method
Find the minimum value of x 
3 2 2
18
18
by using the identity x   ( x 
) 6 2.
x
x
x
32. (b) (iii) Key Information
 ABCR is a regular tetrahedron with sides of 10 cm each.
10 3
cm
 AD  5 3 cm , AG 
3
 AG : GD = 2 : 1
 The number of fold of rotational symmetry for tetrahedron ABCR along RG is 3.
 Tetrahedron ABCR has 4 axes of rotation.
Chapter 12 Application of Trigonometry
201
Analysis
Since ARG is a right-angled triangle, we can use trigonometric ratios to find ARO and
AOR can then be found.
Method
Find the length of RG by means of DGR.