Download Elementary Set

Chapter 7 Trigonometry
Chapter 7
Trigonometry
WARM - UP E XERCISE
1. Simplify the following.
(a)
(b)  3
24
2. Rationalize the following.
1
(a)
2
(b)
5
36
4 5
15
3. Which of the following is a right-angled triangle? Explain your answer.
A
5
(a) C
(b)
A
8
4
B
7
12
B
4
C
4. Find the values of x in the following figures.
(a) B
(b)
B
x
x
5
A
A
4
9
D
12
25
C
C
5. Simplify the following.
(a) 2 sin 2   2 sin 2 (90  )
6. Simplify the following.
tan 
(a)
1  tan 2 
(b)
tan(90  )
cos 
(b) (sin   cos ) 2  (cos   sin ) 2
1
2
New Trend Mathematics S4B — Supplement
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. You may choose to complete any ONE set according to your need. ]
Exercise 7A
  El em en tar y S et
 
Level 1
1. In each of the following figures, find . (Correct your answers to 1 decimal place.)
(a)
(b)
5
7


8
4
(c)
3
1

In each of the following, find sin , cos  and tan . Leave your answers in surd form if necessary.
(2  5)
2.
3.
8
5
5
4
Ex.7A Elementary Set


y
4.
y
5.
P
20
(24, 7)

O
x

O
6. Find the value of sin 30  cos 60.
7. If cos  
2
where 0    90, find sin  and tan .
3
8. If sin  
4
where 0    90, find cos  and tan .
5
9. If tan   2 where 0    90, find sin  and cos .
Level 2
10. If cos  
12
where 0    90, find sin   tan .
13
16
x
3
Chapter 7 Trigonometry
1
2
where 0    90.
12. Find  such that cos(  10) 
1
where 0    90. (Correct your answer to 1 decimal place.)
3
13. In the figure, ABC is a right-angled triangle. AB  9 cm, AC  6 cm,
BC  CD and ABD  85.
(a) Find ABC.
(b) Find CBD.
(c) Find the length of CD.
(Correct your answers to 1 decimal place.)
D
Ex.7A Elementary Set
11. Find  such that sin  
C
6 cm
A
85
9 cm
B
 Intermediate Set

Level 1
14. In the figure, find . (Correct your answer to 1 decimal place.)
17

In each of the following, find sin , cos  and tan . Leave your answers in surd form if necessary.
(15  18)
15.
16. y
5
(3, 7)

10

x
O
y
17.
18.
4

12
P
25

O
15
x
5
Ex.7A Intermediate Set
15
4
New Trend Mathematics S4B — Supplement
19. In the figure, find the unknowns. (Correct your answers to 1 decimal place.)
40
12
x

6
20. If sin  
5
8 7
where 0    90, find cos  and tan .
21. Find the value of
tan 2 45 cos 60
. (Leave your answer in surd form.)
sin 45
Level 2
22. If cos  
3
10
where 0    90, find 2 tan 2   sin 2   cos2  .
23. If tan   2 where 0    90, find sin 2  2 cos 2 .
Ex.7A Intermediate Set
24. Find  such that cos(90  ) 
3
where 0    90.
2
25. Find  such that tan(  20) 
2
where 0    90. (Correct your answer to 1 decimal place.)
3
26. In the figure, AB  8 cm, AD  5 cm and
ABC  BAD  BDC  90.
(a) Find DBC.
(b) Find the length of DC.
(Correct your answers to 1 decimal place.)
C
D
5 cm
A
27. In the figure, ABCDE is a straight line and CF  AE.
(a) Find the length of AE.
(b) Find AFE.
(Correct your answers to 1 decimal place.)
F
8 cm
120 cm
60
A
B
40 cm
B
25
C
D
80 cm
E
5
Chapter 7 Trigonometry
 Advanced Set

Level 1
In each of the following, find sin , cos  and tan . Leave your answers in surd form if necessary.
(28  31)
y
28. A
29.

P (5, 11)
3
B
C
7

x
O
y
30.
31. BCD is a straight line.
A
P
O
14
12

x
8
B
45
D
6
C
A
32. In the figure, BCD is a straight line. Find the
unknowns. (Correct your answers to 1 decimal
place.)
Ex.7A Advanced Set

y
40 x
20
B
C
D
7
33. Find the value of
3 (cos 2 45  4 sin 2 45) tan 45
2 sin 60
. (Leave your answer in surd form.)
Level 2
34. If tan  
2 3
5
where 0    90, find 2 sin  cos . (Leave your answer in surd form.)
35. If tan  3 where 0    90, find
2 sin   cos 
.
sin   3 cos 
36. Find  such that sin 3  1 where 0    90.

1
37. Find  such that tan(  15) 
where 0    90.
2
3
38. In the figure, ACF and CEF are two
right-angled triangles. AB  150 m, DE  200 m,
CF  180 m, FAB  30 and FED  20. Find
the value of x. (Correct your answer to the nearest
integer.)
F
180 m
A
30
B
150 m
20
D
C
xm
200 m
E
6
New Trend Mathematics S4B — Supplement
39. In the figure, AFE and BCDE are straight lines
where CD  15 cm and DE  25 cm. If AC  BE,
DF  AE, AB  20 cm and ABC  50,
(a) find the length of AD.
(b) find DAF.
(c) find the length of DF.
(Correct your answers to 1 decimal place.)
A
F
20 cm
50
B
C
D
E
15 cm
200 m
A
Ex.7A Advanced Set
40. In the figure, A and B are two points in a valley
lying 200 m apart and are opposite to each other.
Their altitudes above the bottom C of the valley
are both h m. If BAC  50 and ABC  40, find
(a) the distance between C and B.
(b) the distance between A and C.
(c) the value of h.
(Correct your answers to 1 decimal place.)
25 cm
B
40
50
hm
hm
C
41. In the figure, two ladders AB and AC lean against
the steps of a stair. The height and width of each
step are 15 cm and 25 cm respectively. If
DAB  30,
(a) find the length of AB.
(b) find DAC.
(Correct your answers to 1 decimal place if necessary.)
B
25 cm
15 cm
C
15 cm
15 cm
D
30
A
Exercise 7B
[ In this exercise, leave your answers in surd form if necessary. ]
Ex.7B Elementary Set
  El em en tar y S et
Level 1
1. In each of the following figures, find .
y
(a)
(b)
y
230
145
x

 

x
Chapter 7 Trigonometry
7
y
(c)

x
30
2. Find the reference angle corresponding to each of the following angles.
(a) 123
(b) 217
(c) 297
4. Find sin , cos  and tan  in the figure.
y

x
O
Ex.7B Elementary Set
3. If sin   0 and tan   0, determine the quadrant in which  lies.
(6, 4)
5. Express sin 160 in terms of acute angle.
6. Express cos(20 + 150) in terms of acute angle.
Find the value of each of the following trigonometric ratios. (7  9)
7. tan 135
8. sin 210
9. cos (225)
 Intermediate Set

Level 1
13. Find the reference angle corresponding to each of the following angles.
(a) 112
(b) 180
(c) 329
14. If sin  cos   0, determine the quadrant(s) in which  lies.
Ex.7B Intermediate Set
Level 2
Find the value of each of the following expressions. (10  12)
10. sin 45cos120
11. 2 sin 135  3 cos 225
12. 2 (cos 330  sin 150)
8
New Trend Mathematics S4B — Supplement
15. Find sin , cos  and tan  in the figure.
y
(7, 3)

x
O
Ex.7B Intermediate Set
16. Express tan 350 in terms of acute angle.
17. Express sin(250  30) in terms of acute angle.
Find the value of each of the following trigonometric ratios. (18  20)
18. sin 225
19. cos 330
20. tan (135)
Level 2
Find the value of each of the following expressions. (21  25)
21. tan 330 sin 300
22. 5 tan 45  6 tan 315
23.
1
1
sin 225  cos 315
3
4
1
24.  tan(225)  4 sin 135
2
25. cos 120 sin 225  sin 330 cos 225
 Advanced Set
Level 1
sin 2 
 0 , determine the quadrant(s) in which  lies.
26. If
tan 
Ex.7B Advanced Set
27. Find sin , cos  and tan  in the figure.
y

x
O
(5, 5)
28. Express cos (120) in terms of acute angle.

Chapter 7 Trigonometry
9
29. Express tan (130  210) in terms of acute angle.
Find the value of each of the following trigonometric ratios. (30  32)
30. tan (30)
32. cos (210)
31. tan 405
34. 3 cos (60)  4 tan (225)
33. sin (30) cos (120)
35. 
2
36. cos150 sin 240  3 sin 120
tan150   5 sin 240 
3
37. In the figure, AOC is a straight line. ADB is a straight line perpendicular to the x-axis and
AB  BC. Find sin .
Ex.7B Advanced Set
Level 2
Find the value of each of the following expressions. (33  36)
y
A (3, 4)
D

x
O
B
C (3, 4)
Exercise 7C
[ In this exercise, leave your answers in surd form if necessary. ]
  El em en tar y S et
Level 1
1. If sin   0, find the range of  where 0    360.
 
3. If tan   0, find the range of  where 0    360.
4. Find sin  and tan  if cos   
2
where 90    180.
3
5. Find cos  and tan  if sin   
1
where 270    360.
3
6. Find sin  and cos  if tan  
1
where 180    270.
3
7. Find sin  and cos  if tan  
3
where  lies in quadrant III.
4
Ex.7C Elementary Set
2. If cos   0, find the range of  where 0    360.
10
New Trend Mathematics S4B — Supplement
8. Find sin and tan if cos   
12
where  lies in quadrant II.
13
Ex.7C Elementary Set
3
9. Find cos  and tan  where  lies in quadrant IV if sin    .
5
Level 2
10. If tan  
5
and 180    270, find the value of 3 sin   2 cos .
12
11. If sin  
1
and 90    180, find the value of sin  tan .
3
12. If cos  
1
and 270    360, find the value of 4 tan   sin .
2
13. If tan   1 and cos   0, find sin .
 Intermediate Set
Level 1
1
14. Find sin  and tan  if cos   where 270    360.
2
Ex.7C Intermediate Set
15. Find cos  and tan  if sin   
1
where 180    270.
2
16. Find sin  and cos  if tan   
1
where 90    180.
3
17. Find sin  and cos  if tan   
5
where  lies in quadrant IV.
2
18. Find sin  and tan  if cos   
3
where  lies in quadrant III.
7
19. Find cos  and tan  if sin  
Level 2
20. If sin   

1
where  lies in quadrant II.
3
5
sin   2 cos 
and 270    360, find the value of
.
2 sin   cos 
13
21. If tan   
1
sin   2 cos 
and 90    180, find the value of
.
2
2 sin   3 cos 
22. If cos   
2
3 sin 
and 180    270, find the value of
 4 tan  .
3
cos 
23. If sin   
1
5
and tan   0, find the value of
5 (sin   cos ) .
11
24. (a) If cos   k and 270    360, express sin  and tan  in terms of k.
(b) If sin   k and 90    180, express cos  and tan  in terms of k.
1
(c) If tan   and 180    270, express sin  and cos  in terms of k.
k
5
3
25. If cos A   , cos B   and both A and B are obtuse angles, find the values of the following.
6
5
(a) cos Acos B
(b) 2 sin A cos B  2 sin B cos A
tan A  tan B
(c)
1  tan A tan B
Ex.7C Intermediate Set
Chapter 7 Trigonometry
26. A(1, 0) and B(x, y) are two points on the unit circle with centre at the origin and AOB  . If
y  2x, find the possible values of sin , cos  and tan .
 Advanced Set
Level 1
2
27. Find sin  and tan  if cos   
where 90    180.
5
28. Find cos  and tan  if sin   
29. Find sin  and cos  if tan  
3
17

where  lies in quadrant IV.
3
where  lies in quadrant III.
2
30. If tan   
1
3
 2 tan 2  .
and 90    180, find the value of
2
2
cos 
31. If sin   
2
32. If cos  
33. If cos  
13
and 180    270, find the value of 2 cos 2   tan  .
2
3 tan 2   1
and 270    360, find the value of
.
2
sin 
3
1
and sin   0, find the value of
 tan  .
10
tan 
34. If tan   k where 90    180, express sin  and cos  in terms of k.
1
where  lies in quadrant IV.
k
(a) find tan  in terms of k.
(b) find sin  in terms of k.
2
(c) Hence find tan 2  
in terms of k.
sin 2 
35. It is given that cos  
Ex.7C Advanced Set
Level 2
12
New Trend Mathematics S4B — Supplement
b
where c  b  0 and  lies in quadrant III,
c
(a) express sin  and tan  in terms of b and c.
3
(b) find the value of 3 tan 2  
.
cos 2 
36. If cos   
37. A(1, 0) and B(x, y) are two points on the unit circle with centre at the origin and AOB  . If
y  4x, find the possible values of sin , cos  and tan .
Ex.7C Advanced Set
2 sin   cos  1
 and 90    180.
cos   3 sin  4
(a) Find the value of tan .
(b) Hence find . (Correct your answer to 1 decimal place.)
38. It is given that
1
39. If sin A   , tan B  3 where A and B lie in quadrants IV and III respectively, find the values of
3
the following.
(a) cos A sin B
(b) tan A tan B  cos A cos B
40. It is given that sin   p and tan   2p where p  0.
(a) In which quadrant does the angle  lie?
(b) Find cos .
(c) Find .
(d) Hence find the value of p.
Exercise 7D
  El em en tar y S et
Level 1
1. Express the following expressions in terms of sin .
Ex.7D Elementary Set
(a) cos  tan 
(b) cos2 
2. Express the following expressions in terms of cos .
(a)
sin 2 
tan 
(b)  sin 2   1
3. Express the following expressions in terms of tan .
sin 
cos 
(a) 4 tan  
(b)
cos 
sin 
 
Chapter 7 Trigonometry
13
4. sin(180  )
5. cos(180  )
6. tan(360  )
7. cos(90  2)
8. sin(270  2)
9. tan(90  2)
10.
sin( 90  )
sin( 90  )
11.
cos(90  )
cos(90  )
12. Find the value of cos 330  sin150 without using a calculator.
Prove the following identities. (13 – 14)
13. sin 2 (90  ) tan 2 (180   )  sin 2 
Level 2
15. Express
14. cos(90  ) tan(270  )  cos 
Ex.7D Elementary Set
Simplify the following expressions. (4  11)
cos(90  ) cos()
in terms of sin .
cos(180   )
Simplify the following expressions. (16 – 17)
16. a cos(180  )  a sin(180  )
17. cos2 (270   )  sin( 90  ) cos(180   )
21.
tan(180   )
sin( 270   )
24.
1
 tan 2 (90  )
cos (90  )
22.
sin( 90  3 )
cos(270  

)
3


20. cos(270  )
2
23. tan 2  
1
cos2 
2
Prove the following identities. (25 – 26)
sin(180   )
 2 cos 
25. cos() 
tan(360   )
26.
cos(180   ) tan(180   )
sin( )

sin( 90  )
cos(180   )
Level 2
27. Express 3sin(180  ) cos(90  )  4 sin( 270  ) cos(360  ) in terms of sin .
28. Find the value of
tan 240  cos150   3 sin 120 
without using a calculator.
cos 240 
Ex.7D Intermediate Set
 Intermediate Set
Level 1
Simplify the following expressions. (18  24)


18. sin( 270   )
19. tan(270  )
2
2
14
New Trend Mathematics S4B — Supplement
Simplify the following expressions. (29 – 30)
ab
ab
29.

2
2
sin (180   ) cos (180   )
30. sin(360  ) cos(90  )  sin(270  ) cos(180  )
Ex.7D Intermediate Set
Prove the following identities. (31 – 33)
sin( 90  )
 sin 2   0
31. cos(90  ) 
tan(270   )
32. 2 tan(90  )  tan(360   ) 
33.
1  3 cos 2 
sin  cos 
sin( 90  ) cos(360   )

 sin   1
cos(180   ) tan(270   )
34. If sin 2 A 
64
and A lies between 180 and 270, find the values of the following.
289
(a) cos A
(b) sin(270  A)
(c)
sin 2 (90  A) cos(270  A)
tan 2 (270  A)
 Advanced Set

Level 1
Simplify the following expressions. (35  37)

35. tan(  180 )
36. cos(2  180)
3
Ex.7D Advanced Set
37. cos 2(450  )
Prove the following identities. (38  39)
sin( )
 tan 
38.
sin( 270   )
39.
sin( 90  ) tan()
 1
sin(180   )
Level 2
Simplify the following expressions. (40  42)
40.
cos(180   ) tan(270   )
sin( 90  )
41. 2 tan(180   ) 
sin(180   )
cos(360   )
Chapter 7 Trigonometry
42.
15
(a  b) tan(270   ) (a  b) tan(180   )

tan(90  )
tan 
43. Find the value of
44. Express
cos 40  tan140   sin 220
without using a calculator.
sin 40  tan 220   cos140
cos()[1  sin 2 (180   )]
1  cos 2 (180   )  sin 2 ()
in terms of cos .
Prove the following identities. (45  48)
1
45. tan(180  )  tan(270  ) 
sin  cos 
cos()
0
sin(   180 )
47.
1
1
2 tan 


1  sin  1  sin  cos 
48.
sin(180   )
 3 tan(180   )  tan()  tan(360   )
cos(360   )
49. Express the following expressions in terms of sin .
1
1
(a)
(b) (cos2   sin 2 ) cos 2 

1  cos  1  cos 
50. Express the following expressions in terms of cos .
(a) sin  tan 
(b) sin 2   tan2 
1
and 180  x  225.
4
(a) Find the values of the following expressions.
(i) sin x  cos x
(ii) sin x  cos x
(b) Hence find sin x and x.
51. Let sin x cos x 
52. Suppose sin  cos   2,
(a) find the value of (sin   cos ) 2.
(b) Can an angle  lying between 0 and 360 which satisfies sin  cos   2 be found? Explain
briefly.
Ex.7D Advanced Set
46. tan(270   ) 
16
New Trend Mathematics S4B — Supplement
Exercise 7E
  El em en tar y S et
Level 1
1. The figure shows the graph of y  sin x for 0  x  360.
 
y
1
y  sin x
x
O
90
180
270
360
1
(a) Find the value of sin 295 from the graph.
(b) If sin x  0.8, find x from the graph.
(c) Find the range of x from the graph such that sin x  0.7.
Ex.7E Elementary Set
2. The figure shows the graph of y  cos x for 0  x  360.
y
1
y  cos x
O
x
90
180
270
360
1
(a) Find the value of cos 250 from the graph.
(b) If cos x  0.6, find x from the graph.
(c) Find the range of x from the graph such that cos x  0.5.
Find the maximum and minimum values of the following expressions for 0     360. (3 – 8)
3. 2 sin 
4. 5 cos 
5. 4  2 sin 
6. 2  4 cos 
7. 3  2 cos 
8. 3 sin 2
Level 2
Sketch the graphs of the following trigonometric functions for 0   x  360. (9  11)
1
1
9. (a) y  2 sin x
(b) y  sin x 
(c) y  2 sin x 
2
2
Chapter 7 Trigonometry
10. (a) y  2 cos x
(b) y  2 cos(x  45)
(c) y  2 cos 3x
11. (a) y  3 tan x
(b) y  tan(x  45)
(c) y  tan(x  90)  2
17
12. The figure shows the graph of y  A cos x  B for 0  x  360.
y
y  A cos x  B
2
O
90
180
270
360
x
2
4
6
(a) From the graph, find the period and amplitude.
(b) Find the values of A and B.
(c) Find the range of values of y such that 0 < x < 90.
Ex.7E Elementary Set
Find the maximum and minimum values of the following expressions for 0  x  360. (13 – 14)
13.  4 cos2 x
14. (3 sin x  1) 2
15. The figure shows the graphs of y  sin x and y  cos x for 0  x  360.
y
1
y  sin x
O
x
30
60
90
120
150
180
210
y  cos x
1
(a) Find the values of the following from the graph.
(i) sin 156
(ii) cos 162
(b) Find x from the graph such that cos x  sin x.
(c) Find the range of x from the graph such that cos x  sin x.
240
270
300
330
360
18
New Trend Mathematics S4B — Supplement
 Intermediate Set
Level 1
16. The figure shows the graph of y  sin 2x for 0  x  180.

y
1
y  sin 2x
x
O
45
90
135
180
1
(a) Find the value of sin 2(120) from the graph.
(b) If sin 2x  0.4, find x from the graph.
(c) Find the range of x from the graph such that sin 2x  0.8.
Ex.7E Intermediate Set
17. The figure shows the graph of y  cos
y
1
x
for 0  x  720.
2
x
y  cos 2
x
O
180
360
540
720
1
(a) Find the value of cos
(b) If cos
100 
from the graph.
2
x
 0.7 , find x from the graph.
2
x
(c) Find the range of x from the graph such that  0.5  cos  0 .
2
Find the maximum and minimum values of the following expressions for 0    360. (18 – 20)
1
5
18. 4 sin   5
19. cos 3
20. 2  sin
2
2
Level 2
Sketch the graphs of the following trigonometric functions for 0   x  360. (21  23)
21. (a) y  2 sin x
(b) y  2 sin x  1
(c) y  2 sin x  2
22. (a) y  3 cos x
(b) y  3 cos(x  45)
(c) y  3 cos 3x
Chapter 7 Trigonometry
23. (a) y  2 tan x
(b) y  tan(x  45)
19
(c) y  2 tan(x  90)  1
24. The figure shows the graph of y  A cos x  B for 0  x  360.
y
y  A cos x  B
3
2
1
x
O
90
180
270
360
1
(a) From the graph, find the period and amplitude.
(b) Find the values of A and B.
(c) Find the range of values of y such that x lies in quadrant II.
25. (a) 3sin 2 x  1
(b)  4 cos 2 x  6
26. (a) (cos x  2) 2
1
(b) ( sin x  3) 2
3
27. (a) (sin x  1) 2  2
(b) (2 cos x  1) 2  2
28. (a)
1
4 cos x  5
(b)
Ex.7E Intermediate Set
Find the maximum and minimum values of the following expressions for 0   x  360. (25 – 28)
1
4  sin 2 x
29. The figure shows the graphs of y  sin 2x and y  cos x for 0  x  360.
y
1
y  sin 2x
O
y  cos x
x
30
60
90
120
150
180
210
240
1
(a) Find the values of the following from the graph.
(i) sin 246
(ii) cos 216
(b) Find x from the graph such that sin 2x  cos x.
(c) Find the range of x from the graph such that cos x  sin 2x.
270
300
330
360
20
New Trend Mathematics S4B — Supplement
 Advanced Set
Level 1
30. The figure shows the graph of y  cos 2x for 0  x  180.

y
1
y  cos 2x
x
O
45
90
135
180
1
(a) Find the value of cos 110 from the graph.
(b) If cos 2x  0.8, find x from the graph.
(c) Find the range of x from the graph such that cos 2x  0.5.
31. The figure shows the graph of y  sin
Ex.7E Advanced Set
x
for 0  x  720.
2
y
1
O
x
y  sin 2
x
180
360
540
720
1
(a) Find the value of sin 40 from the graph.
x
(b) If sin  0.4 , find x from the graph.
2
(c) Find the range of x from the graph such that sin
x
 0.8 .
2
Level 2
Sketch the graphs of the following trigonometric functions for 0   x  360. (32  34)
1
1
1
1
1
32. (a) y  sin x
(b) y  sin x 
(c) y  sin x 
2
2
2
2
2
33. (a) y  cos x
(b) y  cos(x  45)
(c) y  cos 3x
34. (a) y  2 tan x  4
(b) y  3 tan(x  45)
(c) y  tan(x  90)  3
Chapter 7 Trigonometry
21
35. The figure shows the graph of y  A cos x  B for 0  x  360.
y
y  A cos x  B
8
6
4
2
O
90
180
270
x
360
(a) From the graph, find the period and amplitude.
(b) Find the values of A and B.
(c) Find the range of values of y such that x lies in quadrant IV.
36. (a) 3 4 sin 2 x
(b) 3  4 cos3 x
37. (a) (5 cos x  1) 2
(b) (2 sin x  1) 2
38. (a)
3
(cos 2 x  5)
(b)
2
39. (a) ( 2 cos x  3) 2
(b)
Ex.7E Advanced Set
Find the maximum and minimum values of the following expressions for 0   x  360. (36 – 39)
1
(sin  3) 2
x
2
2
4  2 sin 2 x
40. By letting y  sin x and expressing the following expressions in the form of a(y  b) 2  c where a, b
and c are integers, find the maximum and minimum values of the following expressions for
0  x  360.
(a) 2 sin 2x  4 sin x  1
(b) 13  12 sin x  4 cos 2x
41. The figure shows the graphs of y  sin x and y  cos
x
for 0  x  360.
2
y
1
y  sin x
O
1
x
30
60
90
120
150
180
210
240
270
x
y  cos 2
300
330
360
22
New Trend Mathematics S4B — Supplement
Ex.7E Advanced Set
(a) Find the values of the following from the graph.
(i) sin 138
(ii) cos 117
x
(b) Find x from the graph such that sin x  cos .
2
x
(c) Find the range of x from the graph such that cos  sin x .
2
3
1
42. For 0  x  360, the maximum and minimum values of A  B sin x are and respectively
2
2
where A and B are positive numbers.
(a) Find the values of A and B.
(b) Explain briefly whether the maximum and minimum values of A  B sin x for 0  x  360
have the same values as those of A  B sin x.
Exercise 7F
[ In this exercise, correct your answers to 1 decimal place if necessary. ]
  El em en tar y S et
Level 1
Solve the following equations for 0    360. (1 – 15)
1. tan   tan 15
2. sin   sin 24
4. sin   sin 123
7. sin  
2
2
8. cos  
Ex.7F Elementary Set
10. sin 2  0
13. sin(   10) 
5. cos   cos 100
1
2
1
2
 
3. cos   cos 79
6. tan   tan 111
9. tan   3
11. sin   0.6
12. cos 2  0.1
14. 2 sin   1
15. sin   cos   0
Level 2
Solve the following equations for 0    360. (16 – 22)
16. 3 sin   tan 
17. cos 2  cos   0
18. 2 sin 2  sin   3  0
19. 2 sin 2  cos 2  0
20. cos   tan  sin 
21. sin 2  2 sin  cos   cos 2  0
22. sin 2  
1
4
cos   2 sin  3
 in the form of tan   k.
sin   cos  4
cos   2 sin  3
(b) Hence solve
 for 0    360.
sin   cos  4
23. (a) Rewrite the equation
Chapter 7 Trigonometry
 Intermediate Set
Level 1
Solve the following equations for 0    360. (24 – 35)
24. cos 50  cos 
25. sin   sin 165
27. cos   
30. sin
2
2
 1

4 2
33. tan(  60)  5
23

26. tan 195  tan 
3
2
28. tan   4
29. sin   
31. cos(  40)  0.2
32. tan(  35)  
34.  3 cos   2
35.
1
3
1
cos   2 sin   0
3
36. 2 sin   3 tan   0
37. 2 sin 2  3 sin 
38. 3 tan 2  1
39. 2 tan 2  3 tan   1  0
40. 2 cos 2  5 sin   1  0
41. 2 cos   3 tan 
42. 2 cos 2  3 cos  sin   2 sin 2  0
Ex.7F Intermediate Set
Level 2
Solve the following equations for 0    360. (36 – 42)
4 sin 
1
 in the form of tan   k.
sin   3 cos  2
4 sin 
1
(b) Hence solve
 for 0    360.
sin   3 cos  2
43. (a) Rewrite the equation
44. (a) Rewrite the equation 5 tan   2 cos   0 in the form of a sin 2  b sin   c  0, where a, b and
c are integers.
(b) Hence solve 5 tan   2 cos   0 for 0    360.
cos  1
 ,
sin 2  3
(a) rewrite the equation in the form of a cos 2  b cos   c  0 where a, b and c are integers.
cos  1
(b) Hence solve
 for 0    360.
sin 2  3
 Advanced Set

Level 1
Solve the following equations for 0    360. (46 – 51)
1

46. tan  
47. sin  sin 46
2
3
Ex.3A Advanced Set
45. Suppose
24
New Trend Mathematics S4B — Supplement
48. cos
3 2 3

4
5
49. tan
50. 3sin 2   3

2
3
51. cos(  50)  
3
2
Level 2
Solve the following equations for 0    360. (52 – 59)
52. (tan   1)sin   0
54.
3 sin 2   cos 2
56. sin 2


 sin  0
2
2
58. 2 cos 2  3 sin   3
53. 2 cos2   2
55.
2 sin  tan   2 sin   0
57. 4sin 2  2sin  2
59. 3 cos 2  4 sin  cos   4 sin 2  0
Ex.7F Advanced Set
60. (a) Rewrite the equation 4 tan   3 cos   0 in the form of a sin 2  b sin   c  0, where a, b and
c are integers.
(b) Hence solve 4 tan   3 cos   0 for 0    360.
1  5 cos 
, where 0    180.
sin 
(a) Rewrite the above equation in the form of a cos 2  b cos  c  0, where a, b and c are
integers.
1  5 cos 
(b) Hence solve tan  
.
sin 
61. It is given that tan  
sin 
3
,
cos  8
(a) rewrite the equation in the form of a sin 2  b sin  c  0.
sin 
3
 for 0    360.
(b) Hence solve
2
cos  8
62. Suppose
2

63. Solve the equation 4 sin 2  3 sin  cos   2 for 0    360.
64. Solve the equation 2 sin 2  cos (tan   4 cos )  0 for 0    360.
65. (a) Expand (x  1) 4.
(b) Hence solve cos 4  4 cos 3  6 cos 2  4 cos   1  0 for 0    360.
Chapter 7 Trigonometry
Exercise 7G
  El em en tar y S et
Level 1
1. The figure shows the graph of y  2 sin x for 0  x  360.
25
 
y
y  2 sin x
2
1
x
O
30
60
90
120
150
180
210
240
270
300
330
360
300
330
360
1
2
Solve the equation 2 sin x  0 graphically.
2. The figure shows the graph of y  cos 2x  1 for 0  x  360.
y  cos 2x  1
2
1
x
O
30
60
90
120
150
180
210
240
270
Solve the equation cos 2x  1  0 graphically.
3. The figure shows the graph of y  cos(x  60) for 0  x  360.
y
1
y  cos (x  60)
O
x
30
60
90
120
150
180
210
1
Solve the equation cos(x  60)  0 graphically.
240
270
300
330
360
Ex.7G Elementary Set
y
26
New Trend Mathematics S4B — Supplement
4. The figure shows the graph of y  4 cos 2x for 0  x  180.
y
4
y  4 cos 2x
2
x
O
30
60
90
120
150
180
2
4
Solve the equation 4 cos 2x  0 graphically.
Ex.7G Elementary Set
5. The figure shows the graph of y  1  2 sin(x  30) for 0  x  360.
y
3
y  1  2 sin (x  30)
2
1
O
x
30
60
90
120
150
180
210
1
Solve the equation 1  2 sin(x  30)  0 graphically.
240
270
300
330
360
Chapter 7 Trigonometry
27
Level 2
6. The figure shows the graph of y  2 tan x  sinx for 0  x  360.
y
y  2 tan x  sin x
20
10
x
O
30
60
90
120
150
180
210
240
270
300
330
360
10
Solve the following equations graphically.
(a) 2 tan x  sin x  20
(b) 2 tan x  sin x  15  0
(c) 4 tan x  2 sin x  36  0
7. The figure shows the graph of y  a sin x  2 cos x for 0  x  180.
y
2
y  a sin x  2 cos x
1
O
x
30
60
90
120
150
180
1
2
(a) Find the value of a.
(b) Solve the equation a sin x  2 cos x  1  0 graphically.
Ex.7G Elementary Set
20
28
New Trend Mathematics S4B — Supplement
8. The figure shows the graph of y  a sin x  b cos x for 0  x  180.
y
2
y  a sin x  b cos x
Ex.7G Elementary Set
1
x
O
30
60
90
120
150
180
1
2
(a) Find the values of a and b.
(b) Solve the equation 4 sin x  3 cos x  2 graphically.
 Intermediate Set
Level 1
1
9. The figure shows the graph of y  sin x for 0  x  360.
2

y
0.5
Ex.7G Intermediate Set
O
y
1
sin x
2
120
150
x
30
60
90
180
0.5
Solve the equation
1
sin x  0 graphically.
2
210
240
270
300
330
360
Chapter 7 Trigonometry
29
10. The figure shows the graph of y  2 cos 2x  1 for 0  x  360.
y
3
y  2 cos 2x  1
2
1
x
O
30
60
90
120
150
180
210
240
270
300
330
360
1
Solve the equation 2 cos 2x  1  0 graphically.
11. The figure shows the graph of y  sin(x  30) for 0  x  360.
y
1
x
O
30
60
90
120
150
180
210
240
270
300
330
360
330
360
1
Solve the equation sin(x  30)  0 graphically.
Level 2
12. The figure shows the graph of y  2  3 cos(x  20) for 0  x  360.
y
5
y  2  3 cos (x  20)
4
3
2
1
O
1
x
30
60
90
120
150
180
210
240
270
300
Ex.7G Intermediate Set
y  sin (x  30)
30
New Trend Mathematics S4B — Supplement
Solve the following equations graphically.
2
1
(a) cos(x  20) 
(b) cos(x  20)  
3
3
(c) 3 cos(x  20)  1
13. The figure shows the graph of y  3 tan x  sin x for 0  x  360.
y
y  3 tan x  sin x
40
30
20
10
x
O
30
60
90
120
150
180
210
240
270
300
330
360
10
20
Ex.7G Intermediate Set
30
40
Solve the following equations graphically.
(a) 3 tan x  sin x  30
(b) 3 tan x  sin x  10  0
14. The figure shows the graph of y  a sin bx for 0  x  180.
y
y  a sin bx
4
2
O
2
4
x
30
60
90
120
150
180
(c) 6 tan x  12  2 sin x
Chapter 7 Trigonometry
31
(a) Find the values of a and b.
(b) Solve the equation a sin bx  2 graphically.
(c) Solve the equation 2 sin bx  1  0 graphically.
15. The figure shows the graph of y  a sin x  b cos x for 0  x  360.
y
y  a sin x  b cos x
2
1
O
x
30
60
90
120
150
180
210
240
270
300
330
360
1
2
(b) Solve the equation a sin x  b cos x  2  0 graphically.
14
(c) Solve the equation 2(a sin x  b cos x)   0 graphically.
5
16. The figure shows the graph of y  A tan x  B for 0  x  180.
y
6
4
2
y  A tan x  B
O
x
30
60
90
120
150
180
2
4
6
(a) Find the values of A and B.
(b) Solve the following equations graphically.
(i) A tan x  B  2
(ii) tan x  2
Ex.7G Intermediate Set
(a) Find the values of a and b.
32
New Trend Mathematics S4B — Supplement
 Advanced Set
Level 1

17. The figure shows the graph of y  4 sin 2x for 0  x  360.
y
4
y  4 sin2 x
2
x
O
30
60
90
120
150
180
210
240
270
300
330
360
300
330
360
Solve the equation 4 sin 2x  0 graphically.
Level 2
18. The figure shows the graph of y  2  3 sin 2x for 0  x  360.
y
Ex.7G Advanced Set
5
y  2  3 sin 2x
4
3
2
1
O
x
30
60
90
120
150
180
1
Solve the following equations graphically.
(a) 3 sin 2x  2
(b) 3 sin 2x  1
(c) sin 2x  0.6
210
240
270
Chapter 7 Trigonometry
33
19. The figure shows the graph of y  2 cos(x  45) for 0  x  360.
y
2
y  2cos (x  45)
1
x
O
30
60
90
120
150
180
210
240
270
300
330
360
1
2
Solve the following equations graphically.
(a) cos(x  45)  0
(b) cos(x  45)  0.8
(c) 4cos(x  45)  3.2  0
20. The figure shows the graph of y  3  2 sin(x  60) for 0  x  360.
5
Ex.7G Advanced Set
y
y  3  2 sin (x  60)
4
3
2
1
O
x
30
60
90
120
150
180
Solve the following equations graphically.
(a) sin(x  60)  0
(b) sin(x  60)  0.5
(c) 3 sin(x  60)  5
210
240
270
300
330
360
34
New Trend Mathematics S4B — Supplement
21. The figure shows the graph of y  2 tan x  10 cos x for 0  x  360.
y
y  2 tan x  10 cos x
40
30
20
10
x
O
30
60
90
120
150
180
210
240
270
300
330
360
300
330
360
10
20
30
40
Ex.7G Advanced Set
Solve the following equations graphically.
(a) 2 tan x  10 cos x  6
(b) 2 tan x  10 cos x  18  0
(c) tan x  10  5 cos x
22. The figure shows the graph of y  a cos bx  2 for 0  x  360.
y
5
y  a cos bx  2
4
3
2
1
O
x
30
60
90
120
150
180
210
1
(a) Find the values of a and b.
(b) Solve the equation a cos bx  2 graphically.
(c) Solve the equation 5a cos bx  2  0 graphically.
240
270
Chapter 7 Trigonometry
35
23. The figure shows the graph of y  a sin x  b cos x for 0  x  360.
y
y  a sin x  b cos x
4
3
2
1
x
O
30
60
90
120
150
180
210
240
270
300
330
360
1
2
3
4
Ex.7G Advanced Set
(a) Find the values of a and b.
(b) Solve the equation a sin x  b cos x  2  0 graphically.
a sin x  b cos x
(c) Solve the equation
 1  0 graphically.
3
24. The figure shows the graph of y  3 sin x  a cos x for 0  x  360.
y
4
y  3 sin x  a cos x
3
2
1
O
x
30
60
90
120
150
180
210
240
1
2
3
4
(a) Find the values of a.
(b) Solve the equation 3 sin x  a cos x  3  0 graphically.
(c) Find the range of x for which 3 sin x  a cos x  5  0.
270
300
330
360
36
New Trend Mathematics S4B — Supplement
25. The figure shows the graph of y  a sin(x  b) for 0  x  180.
y
4
y  a sin (x  b)
3
Ex.7G Advanced Set
2
1
x
O
30
60
90
120
150
180
1
2
(a) Find the values of a and b.
b
(b) Solve a sin( x  b) 
graphically.
40
C HAPTER T EST
(Time allowed: 1 hour)
Section A
1. In the figure, find sin , cos  and tan . (Leave your answers in surd form if necessary.) (3 marks)
3
4

2. In the figure, find sin , cos  and tan . (Leave your answers in surd form if necessary.) (3 marks)
y

O
(4, 2)
x
Chapter 7 Trigonometry
3. Solve the equation
37
2 sin   3 cos  for 0  x  360. (Correct your answers to 1 decimal place.)
(3 marks)
3
where 90    180, find cos  and tan . (Leave your answers in surd form if
2
necessary. )
(4 marks)
4. If sin  
5. If tan   1, find the value of
6. Find the value of
cos 2 (180   )
.
tan()
(4 marks)
1
1
tan 240  sin 120 cos 300 . (Leave your answer in surd form.)
3
2
(4 marks)
7. Find the maximum and minimum values of the following expressions for 0   x  360.
(a) 1  sin x
(2 marks)
1
(b)
(2 marks)
3  2 cos x
Section B
8. (a) Rewrite 2 cos   3 tan  in the form of a sin 2  b sin   c  0, where a, b and c are integers.
(2 marks)
(b) Hence solve 2 cos   3 tan  for 0    360.
(2 marks)
(c) Let f()  2 cos 2  3 tan  cos .
(i) Rewrite f () in the form of a(sin   h) 2 + k, where a, h and k are constants.
(ii) Find the maximum and minimum values of f () for 0    360.
(6 marks)
3
9. The figure shows the graph of y  sin x  cos x for 0  x  180.
2
y
2
y  sin x 
3
cos x
2
1
O
x
30
60
90
120
150
180
1
2
3
(a) Find the maximum and minimum values of y  sin x  cos x from the graph for 0  x  180.
2
(2 marks)
38
New Trend Mathematics S4B — Supplement
(b) Solve the following equations graphically.
3
(i) sin x  cos x  1  0
2
3
(ii) sin x  cos x  1.4  0
2
(iii) 2 sin x  3 cos x  3  0
(3 marks)
3
(c) Find the range of values of k such that the equation sin x  cos x  k  0 does not have
2
solutions for 0  x  180.
(5 marks)
Multiple Choice Questions (3 marks each)
10. In the figure, sin  
10
.
cos 50
10 sin 25
B.
.
cos 50
C. 10 cos 25 sin 50 .
10
10
D.
.

tan 25 tan 50
A.
2

2
8
A.
1
8
.
B.
7.
C.
14
.
4
D.
13.
1
.
4
11. 2 cos 2  sin 2  1 
A. sin 2.
1
1


1  cos  1  cos 
A. 1.
2
B.
.
sin 
2
C.
.
sin 2 
1
D.
.
sin 2 
B. cos 2.
C. 2.
12. In the figure, ABC is a straight line. If
DAB  90, DBA  50, DCA  25 and
AD  10, BC 
D
10
50
A
3
where  lies in quadrant III,
5
what is the value of tan   cos ?
1
A. 
20
31
B. 
20
31
C.
20
1
D.
20
14. If sin   
D. 3 cos 2.
25
B
C
Chapter 7 Trigonometry
15. Which of the following statements is/are
true for 180  x  270?
A.
B.
C.
D.
I. sin x  cos x  0
II. sin x  tan x  0
III. cos x  tan x  0
A. I and II only
B. II only
y
D. II and III only
0    360 is
19
161
19 or 161
19, 161 or 270
19. What is the function represented by the
following graph for 0  x  180?
C. III only
16. The maximum value of
39
6
1
2 sin 2   1
for
4
A. 1.
B. 0.
2
C. 1.
4
D.
.
3
O
17. If sin   cos 70, then  
A. 20 or 160.
B. 70 or 110.
C. 110 or 200.
D. 120 or 240.
2
18. Solve the equation (3 sin   1)(sin   1)  0
for 0  x  360, correct your answers to the
nearest degree if necessary.
H INTS
(for questions with
x
30
A. y  sin x
B. y  cos x
C. y  tan x
1
D. y 
tan x
in the textbook)
Revision Exercise 7
35. (c) Key information
 f ()  (2 sin   3) 2  6
 Maximum value of f () for 0    360  31
 Minimum value of f () for 0    360  7
60
90
120
150
180
40
New Trend Mathematics S4B — Supplement
Analysis
The range of  may affect the maximum or minimum value of f (). Therefore, we have to
analyze whether the angles  corresponding to the maximum and minimum values for
0    360 lie in the range 0    180.
Method
Check the range of sin  for 0    180 so as to find the range of f () for 0    180.