1. Convert the following angles into radian measure. (Give your answers correct to 3 significant
figures.)
(a) 188.2°
(b) 12.6°
2. Convert the following angles into radian measure. (Express your answers in terms of π.)
(a) 15°
(b) 330°
3. Convert the following angles into degree measure.
7π
3π
(b)
6
8
4. Convert the following angles into degree measure. (Give your answers correct to 2 decimal
places.)
(a)
(a) 0.84c
(b) 3.14c
⌢
5. In the figure, O is the centre of the circle. OA = 6 cm and AB = 9 cm.
6 cm
O
A
θ
9 cm
B
(a) Find θ. (Express your answer in radian measure.)
(b) Find the area of minor sector OAB.
119
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⌢
6. In the figure, PQ = 8 cm and ∠POQ =
your answers in terms of π.)
π
. Find the radius and the area of sector OPQ. (Express
6
P
8 cm
π
6
O
⌢
Q
7. The figure shows a sector where AB = 30 cm. It is given that the area of the sector is 200 cm2.
Find θ. (Express your answer in radian measure.)
30 cm
A
B
θ
O
8. An iron wire of 30 cm long is bent into a sector as shown in the figure. ∠AOB = 72° and OA = r cm .
A
r cm
72°
O
B
(a) Find the value of r.
(b) Find the area of the sector.
(Give your answers correct to 3 significant figures.)
9. In the figure, O is the centre of the circle. R is a point on OP such that QR ⊥ OP. ∠POQ = 45°.
It is given that the area of the minor segment cut off by PQ is 10 cm2 . Find the radius. (Give
your answer correct to 3 significant figures.)
O
R
45 °
P
Q
120
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10. In the figure, O is the centre of the circle. The radius is 10 cm. Radius OC and chord AB are
perpendicular to each other and intersect at D. CD = 4 cm .
A
O
4 cm
C
D
10 cm
B
(a) Find ∠AOB. (Express your answer in radian measure and correct to 3 significant figures.)
(b) Find the area of the major segment cut off by AB.
11. In the figure, O is the centre of the circle with the radius of 7 cm. C is a point outside the
circle such that AC and BC touch the circle. AC = BC = 24 cm . Find the area of the shaded
region. (Give your answer correct to 3 significant figures.)
A
7 cm
O
24 cm
B
C
12. In the figure, ABCD is a rectangle where AD = 30 cm . E is a point on AD such that BCE is a
sector.
A
B
E
30 cm
D
C
π
, find the length of AB.
6
(b) Hence find the area of the shaded region. (Express your answer in terms of π.)
(a) If ∠EBC =
121
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13. In the figure, ABCD is a square with sides of 6 cm each. O is the centre of sector OEFG where
O and F are the mid-points of AD and BC respectively.
A
O
D
E
G
B
F
C
(a) Find ∠EOG. (Express your answer in terms of π and in radian measure.)
(b) Find the perimeter of the shaded region. (Give your answer correct to 3 significant
figures.)
14. In the figure, O is the centre of the circle with the radius of 8 cm. ∠COD = 60°. B is a point on
the circumference such that AB // OC. E is a point on AD such that BE ⊥ AD.
B
A
C
60 °
E
O
8 cm
D
(a) Find ∠AOB. (Express your answer in terms of π and in radian measure.)
(b) Find the length of BE. (Give your answer correct to 3 significant figures.)
(c) Find the area of the minor segment cut off by AB. (Give your answer correct to
3 significant figures.)
15. In the figure, O is the centre of the semi-circle with the radius of r cm. S is a point on PR such
that QS ⊥ PR. ∠QOS = θ .
Q
θ
R
r cm
O
S
P
(a) Express the area of ΔOQR in terms of r and θ.
(b) If the area of sector OPQ and the area of the segment cut off by QR are equal, show that
sin θ + 2θ = π .
122
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⌢
16. In the figure, OAB is an equilateral triangle with sides of 1 cm each. PQ is an arc with O as
the centre and divides ΔOAB into two parts with equal areas. C is a point on AB such that
AB ⊥ OC .
O
r cm
P
Q
A
C
B
(a) Find the value of r.
⌢
(b) Find the length of PQ .
(Give your answers correct to 3 significant figures.)
17. In the figure, O and H are the centres of the larger circle and the smaller circle respectively.
AC and BD are the diameters of the larger circle and ∠DOC = 90° . The two circles touch each
other at E. AC and BD touch the smaller circle at F and G respectively.
E
A
B
H
F
O
D
G
C
(a) If the radius of the smaller circle is 5 cm, find the radius of the larger circle.
(b) Find the area of the shaded region.
(Give your answers correct to 3 significant figures.)
18. The figure shows four circles with the radii of r cm each. Each circle touches its adjacent
circles, and a square can be formed by joining the centres of the four circles. Express the area
of the shaded region in terms of π and r.
r cm
123
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19. In the figure, the radii of the three circles are 4 cm and their centres are A, B and C.
A
B
C
⌢
(a) Find the area of ΔABC.
(b) Find the area of the minor segment bounded by AB and AB .
(c) Find the area of the shaded region.
(Give your answers correct to 3 significant figures.)
20. In the figure, a conical paper cup is formed by sector OAB. It is given that the height of the
cup is 12 cm and the radius of its opening is 5 cm.
5 cm
A
12 cm
B
O
(a) Find ∠AOB. (Express your answer in terms of π and in radian measure.)
(b) Find the area of sector OAB. (Express your answer in terms of π.)
21. In the figure, a machine is made of two rollers bounded by a belt. The centres of the two
rollers are A and B, and AB = 50 cm. The radii of the two rollers are 15 cm and 10 cm. Let the
belt touches the larger roller at C and D, and the smaller one at E and F.
C
E
A
α
50 cm
15 cm
D
B β
10 cm
F
(a) Find α and β. (Express your answers in radian measure.)
(b) Find the length of the belt.
(c) Find the area bounded by the belt.
(Give your answers correct to 3 significant figures.)
124
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tan θ
⋅ csc θ .
sec θ
1
23. Simplify (cot θ +
) cos θ .
cot θ
24. Simplify sin θ(sec3 θ − sec θ).
22. Simplify
25. Simplify tan 2 θ(csc2 θ + sec2 θ).
25
26. If sec θ = −
and θ lies in quadrant II, find the values of the other five trigonometric ratios of
24
θ without using a calculator. (Leave your answers in surd form if necessary.)
27. If cot θ = 5 and 180° ≤ θ ≤ 270° , find the values of sin θ and sec θ without using a calculator.
(Leave your answers in surd form if necessary.)
9
3π
and
≤ θ ≤ 2π, find the value of sec θ + tan θ without using a calculator. (Leave
7
2
your answer in surd form if necessary.)
28. If csc θ = −
1
and θ lies in quadrant III, find the value of sin θ − cot θ without using a calculator.
3
(Leave your answer in surd form if necessary.)
29. If cos θ = −
30. If csc θ = 4 and tan θ < 0 , find the value of (cos θ − cot θ) (sin θ + sec θ) without using a calculator.
(Leave your answer in surd form if necessary.)
3 sin θ − csc θ
1
and cos θ < 0, find the value of
without using a calculator. (Leave
sec θ + 2 tan θ
5
your answer in surd form if necessary.)
25
3π
3 cot θ + 2 csc θ
32. If csc2 θ =
and π ≤ θ ≤
, find the value of
without using a calculator.
9
2
cos θ
(Leave your answer in surd form if necessary.)
31. If sin θ = −
π
≤ θ ≤ π , find the value of (sec θ + tan θ) sin θ without using a calculator.
2
(Leave your answer in surd form if necessary.)
33. If sec2 θ = 2 and
34. If cot 2 α + cot 2 β = 1, prove that csc2 α + csc2 β = 3.
6
.
2 csc2 θ + 1
5
36. Find the minimum value of y = 3 −
.
2 sec2 θ − 1
35. Find the maximum value of y =
125
© 2009 Chung Tai Educational Press. All rights reserved.
csc θ
.
1 + tan 2 θ
sin 2 θ tan θ − cos2 θ cot θ = tan θ − cot θ.
cos θ + cot θ
.
cos θ cot θ =
tan θ + sec θ
sin θ
tan θ
−
= 2 − sec θ sin 2 θ.
csc θ − cot θ csc θ + cot θ
cot θ
1 + csc θ
.
=−
1 − csc θ
cot θ
csc A + csc B cot A − cot B
.
=
cot A + cot B csc A − csc B
37. Prove that (1 − sin θ)(1 + csc θ) =
38. Prove that
39. Prove that
40. Prove that
41. Prove that
42. Prove that
43. If cos 2 α + sec2 β = 3, prove that cos 2 α(2 − sin 2 β) = 3 cos 2 β − sin 2 α.
44. If sec2 α − csc2 β = 2 , find the value of tan 2 α tan 2 β − 2 tan 2 β.
45. If tan 2 α + sec2 β = 3, find the value of cos 2 α − 4 cos 2 α cos 2 β + cos 2 β.
3
13
46. If sin α = and sec β = − , where α and β lie in the same quadrant, find the value of
5
12
cos α + csc β
without using a calculator. (Leave your answer in surd form if necessary.)
tan α − tan β
3π
3π
and
≤ β ≤ 2π , find the value of
2
2
(tan α + cosβ)(sin α − tan β) without using a calculator. (Leave your answer in surd form if
necessary.)
47. If csc α = −2 and cot β = − 2 , where π ≤ α ≤
k +3
3π
and
≤ θ ≤ 2π , where k > 0 , express csc θ + cot θ in terms of k.
k +1
2
π
sin θ
1
49. If
= , where < θ < π , find the value of sin θ .
2
5 − 2 csc θ 2
2
50. If sec θ = 6 tan θ − 9, where θ lies in quadrant IV, find the value of sec θ .
48. If sec θ =
51. If 9 cot2 θ − 3 csc θ − 11 = 0, where π < θ <
3π
, find the value of cot θ . (Leave your answer in
2
surd form.)
52. If
3π
tan θ
2
< θ < 2π , find the values of csc θ . (Leave your answers in surd
= , where
2
2
1 − 3 sec θ 11
form if necessary.)
53. If 2 tan θ + 1 = sec θ , where 0 < θ < π , find the values of sin θ , cos θ and tan θ .
54. If 3sec2 θ + 5 tan θ − 5 = 0, where 0 ≤ θ < 2π , find θ. (Give your answers correct to 3 significant
figures.)
126
© 2009 Chung Tai Educational Press. All rights reserved.
55. If 3 csc θ − 2 sin θ = 2 2 , where 0 ≤ θ < 2π , find θ. (Express your answers in terms of π.)
3π
, find θ. (Express your answer in terms of π.)
2
57. If csc θ + 7 cot θ = 4, where 0 < θ < π , find θ. (Give your answer correct to 3 significant figures.)
56. If 5 tan 2 θ + 3sec2 θ = 27, where π < θ <
58. It is given that sin θ + cos θ =
π
7
, where < θ < π .
2
13
(a) Find the value of sin θ − cos θ .
(b) Find the values of sin θ and cos θ .
(c) Find the value of cot θ .
59. (a) Given that tan α and cot α are the roots of the quadratic equation x2 − 3mx + m = 0, find
the value of m.
(b) Hence find a quadratic equation in x with roots tan 2 α and cot2 α.
60. Given that cos θ and csc θ are the roots of the equation 3x2 + kx − 4 = 0, where π < θ < 2π , find
the value of k.
61. It is given that sin θ and cos θ are the roots of the equation 5x 2 + x + k = 0.
(a) Find the value of k.
(b) Find the value of [sin θ(sec θ + 1)][cos θ(csc θ + 1)] .
62. If θ is an acute angle and 2 x 2 + (4 sin θ) x + 3 cos θ = 0 is an equation in x with two equal real
roots, find θ. (Express your answer in terms of π and in radian measure.)
cot 3 θ + 1 csc θ − sec θ + sec3 θ
.
=
cot 3 θ − 1 csc θ + sec θ − sec3 θ
sec θ + csc θ sec θ + 2 sin θ
64. Prove that
.
=
1 + cot θ
1 + cot 2 θ
1
65. If sec θ =
, prove that sin8 θ + sin 6 θ + sin 2 θ − 1 = 0.
sec θ − 1
63. Prove that
66. (a) Prove that (2 2 + 1) tan 2 θ − ( 2 + 2) tan θ sec θ + 1 = (2 tan θ − sec θ)( 2 tan θ − sec θ).
(b) Hence, solve the equation (2 2 + 1) tan 2 θ − ( 2 + 2) tan θ sec θ + 1 = 0, where 0 ≤ θ ≤ 2π .
127
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67. The figure shows three similar right-angled triangles, where ∠B1 A1 C1 = ∠B2 A 2 C2 = ∠B3 A 3 C3 = φ
and A1 C1 = A2 B2 = B3 C3 = 1.
A3
A2
A1
1
φ 1
B1
C1
φ
φ
B2
C2
B3
1
C3
B1C1 B2C2
correct?
=
B2C2 B3C3
BC
BC
(b) If 1 1 = 2 2 , find sin φ .
B2C2 B3C3
(a) If φ = 30°, is
68. Find the values of the following trigonometric ratios without using a calculator.
(a) cos 315°
(b) csc 240°
(c) cot(−150°)
69. Find the values of the following trigonometric ratios without using a calculator.
11π
3π
5π
(a) tan(− )
(b) sec
(c) csc( − )
6
4
4
tan(180° − θ)
70. Simplify
.
csc(90° − θ)
71. Simplify
sin(−θ) tan(θ − 270°)
.
sec(θ + 180°)
3π
π
+ θ) cos( + θ) .
2
2
3π
π
73. Simplify sec(θ − ) cot(θ + ).
2
2
π
74. Simplify sin(2θ − ) csc( π − 2θ).
2
72. Simplify sec(
75. Simplify sec2 (θ − 2π) − cot 2 (θ −
3π
).
2
π θ
76. Simplify 1 − sec2 ( + ) .
2 2
13
π
3π
77. If sec θ = − , where ≤ θ ≤ π , find the values of sin(π + θ) and cot( − θ) without using a
2
2
12
y
calculator.
P
13
N
128
© 2009 Chung Tai Educational Press. All rights reserved.
12
θ
O
x
78. If tan θ = 3 and θ lies in quadrant III, find the values of sec(π − θ) and sec(
using a calculator. (Leave your answers in surd form if necessary.)
π
− θ) without
2
y
θ
N 1
O
x
3
P
79. If csc(
π
4
+ θ) = − , find the value of cos(π + θ) without using a calculator.
2
3
3 7
, find the value of cot(3π + θ) without using a calculator. (Leave your
7
answer in surd form.)
80. If tan(π − θ) =
π
2
81. If cos(θ − ) = −
, find the value of sec 2 ( π − θ) without using a calculator.
2
2
82. Find the value of sin(−
17π
9π
11π
13π
43π
23π
) csc
− cos
cot
+ tan
cos( −
) without using a
6
2
3
4
6
6
calculator.
5π
π
4π
csc(θ + ) + tan
sec(θ − 2π) .
6
2
3
2π
23π
17π
π
5π
84. Simplify 2 csc( − ) cot
cot(5π − θ) − csc
sec( − ) tan( + θ) .
3
4
6
6
2
cos A csc C − cos( B + C ) sec( A + C )
85. Simplify
, where A + B + C = π .
cot( B + C ) sec B + cot A csc( A + B)
π
1
86. Prove that [sin(θ − 2π) sec(θ + )]2 − [
]2 = cos 2 θ .
3
π
2
sec( −θ) tan( 2 − θ)
83. Simplify 3 cos
87. Prove that
sec( −θ) + cos( π − θ)
= tan 3 θ .
csc( π − θ) − sin(π − θ)
88. Prove that
1 + cos(θ − 2π)
[1 + cos( π + θ)]2
{
1
+
} = 2 csc θ .
cos(θ + 32π )
sin 2 (−θ)
89. Prove that cos 2 (π − θ)[
90. If cot θ = −
sec2 θ − cos 2 θ
π
− 1] = cos 2 ( + θ)(1 + cot 2 θ) .
2 3π
2
cos ( 2 + θ)
π
3π
1
and sec θ > 0 , find the value of [sec( −θ) + sec( + θ)] sin( + θ) without using a
2
2
3
calculator.
129
© 2009 Chung Tai Educational Press. All rights reserved.
csc(θ − π)
3π
5
and sin θ > 0 , find the value of
without using a
+ θ) = −
2
2
sec(θ + 32π ) + cot(2π − θ)
calculator. (Leave your answer in surd form.)
3π
π
92. (a) If cos θ and csc( − θ) are the roots of the equation x2 − 3kx − 2k = 0 , where < θ < π ,
2
2
find the value of k.
91. If csc(
(b) Find the value of cos θ + sec θ .
1
3π
93. It is given that sin θ − cos θ = − , where π < θ <
.
4
2
(a) Find the value of sin θ + cos θ .
3π
(b) Hence, find the value of sin( − θ) .
2
(Leave your answers in surd form.)
3π
3
94. It is given that sin θ + cos θ =
, where
< θ < 2π .
2
2
(a) Find the value of sin θ − cos θ .
(b) Find the value of tan(π + θ) .
(Leave your answers in surd form.)
95. If sin 2 (π + θ) =
96. Prove that
97. If [
1
π
, where θ is an obtuse angle, find the value of cot 2 (π + θ) − sec( + θ) .
4
2
1 − csc( θ − 2π) + tan(θ +
1 + csc( π − θ) − tan(θ −
sin( 32π + θ)
cos( 4π − θ)
×
3π
)
2
π
)
2
=
1 + sec( π2 + θ)
cot(π + θ)
.
sec( π + θ) 2
] is a root of the equation 2 y 2 − 5 y + 2 = 0, where 0 < θ < 2π ,
cot( π2 + θ)
find θ.
98. It is given that y =
sin 4 x − sin 4 ( 32π − x) + 4
sin 2 x − sin 2 ( π2 − x) + 2
(a) Prove that y = 1 +
2
2
2 sin x + 1
.
.
(b) Hence find the maximum and minimum values of y.
130
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