MENTI-CLASS-X

INTRODUCTION TO TRIGONOMETRY
8
TRIGONOMETRY
BY B K SINGH OBJECTIVE SECTION
[BASIC/STANDARD]

I. MULTIPLE CHOICE QUESTIONS

1. The value of (sin 30° + cos 30°) – (sin 60° + cos 60°) is
(a) –1 (b) 0 (c) 1 (d) 2
tan 30°
2. The value of is
cot 60°
1 1
(a) (b) (c) 3 (d) 1
2 3
3. The value of (sin 45° + cos 45°) is
1 3
(a) (b) 2 (c) (d) 1
2 2
4
4. If cos A = , then the value of tan A is
3 5 3 4 5
(a) (b) (c) (d)
5 4 3 3
1
5. If sin A = , then the value of cot A is
2
1 3
(a) 3 (b) (c) (d) 1
3 2
6. The value of the expression cosec (75° + θ) – sec (15° – θ) – tan (55° + θ)
+ cot (35° – θ)] is
3
(a) –1 (b) 0 (c) 1 (d)
2
a
7. Given that sin θ = , then cos θ is equal to
b
b b b2 − a 2 a
(a) (b) (c) (d)
2
b − a 2 a b b − a2
2

8. If cos (α + β) = 0, then sin (α – β) can be reduced to

(a) cos β (b) cos 2β (c) sin α (d) sin 2 α
9. The value of tan 1° tan 2° tan 3° ... tan 89° is
1
(a) 0 (b) 1 (c) 2 (d)
2
10. If cos 9α = sin 9α and α < 90°, then the value of tan 5α is
1
(a) (b) 3 (c) 1 (d) 0
3
11. If ∆ ABC is right angled at C, then the value of cos (A + B) is
Real Numbers 43
 1 3
(a) 0 (b) 1 (c) (d)
2 2
12. If sin A + sin2 A = 1, then the value of the expression (cos2 A + cos4 A) is
1
(a) 1 (b) (c) 2 (d) 3
2
1 1
13. Given that sin α = and cos β = , then the value of (α + β) is
2 2
(a) 0° (b) 30° (c) 60° (d) 90°

 sin 2 22° + sin 2 68° 

14. The value of the expression  + sin 2
63° + cos 63°sin 27° is

2 2
 cos 22° + cos 68° 
(a) 3 (b) 2 (c) 1 (d) 0

 4 sin θ − cos θ 
15. If 4 tan θ = 3, then   is equal to
 4 sin θ + cos θ 
2 1 1 3
(a) (b) (c) (d)
3 3 2 4
16. If sin θ – cos θ = 0, then the value of (sin4 θ + cos4 θ) is
3 1 1
(a) 1 (b) (c) (d)
4 2 4
17. sin (45° + θ) – cos (45° – θ) is equal to
(a) 2 cos θ (b) 0 (c) 2 sin θ (d) 1
18. The value of cos226° – sin2 64° is
(a) 1 (b) –1 (c) 0 (d) 2
19. If sec θ + tan θ = m, then tan θ is equal to
m2 − 1 m2 + 1 m2 − 1 m2 + 1
(a) (b) (c) (d)
2m 2m m m
sin θ
20. is equal to
1 + cos θ
1 + cos θ 1 − cos θ 1 + cos 2 θ 1 − sin 2 θ
(a) (b) (c) (d)
sin θ sin θ sin θ sin θ
21. If triangle ABC is right angled at C, then the value of sec (A + B) is [Cbse (sp) 2019]
2
(a) 0 (b) 1 (c) (d) not defined
3
22. If sinθ + cosθ = 2 cosθ, (θ ≠ 90°), then the value of tanθ is [Cbse (sp) 2019]
(a) 2 −1 (b) 2 +1 (c) 2 (d) – 2

3
23. Given that sin α = and cos β = 0, then the value of β – α is [Cbse (sp) 2019]
2
(a) 0° (b) 90° (c) 60° (d) 30°
21 ,
24. Given cot θ = then cosec θ =
20
44 Mathematics - 10
 21 29 20 29
(a) (b) (c) (d)
20 21 29 20
P
25. In the figure, tanP – cotR =
(a) –1 (b) 1 cm
12 13 cm

12
(c) 0 (d)
13 R Q

26. The value of cos267° – sin223° =

1
(a) 1 (b) –1 (c) 0 (d)
2
 Α + Β
27. A, B, and C are interior angles of ∆ABC, then cosec  =
 2 
C
(a) sin C (b) tan B (c) cos A (d) sec
2 2 2 2
28. If tan 2A = cot (A – 24°), then A =
(a) 32° (b) 35° (c) 37° (d) 38°
29. The value of sin233° + sin257° =
(a) 1 (b) –1 (c) 1 (d) 0
2
tan 36°
30. The value of =
cot 54°
(a) 0 (b) 1 (c) –2 (d) 3
31. If cosec2θ (1 + cosθ) (1 – cosθ) = k, then the value of k is
(a) 0 (b) –1 (c) 1 (d) 1
2
4
32. If tan A = , then sin A + cos A =
3
2 3 4 7
(a) (b) (c) (d)
5 5 11 5
33. If ∆ACB is right angled at C and AB = 29 units, BC = 21 units, and ∠ABC = θ, then
sin2θ + cos2θ =
29 21
(a) (b) (c) 0 (d) 1
21 29
34. In right triangle ABC, right angled at B, if tan A = 1, then 2sin A.cos A =
1
(a) 1 (b) –1 (c) 0 (d)
2
35. In ∆OPQ, right angled at P, OP = 7 cm and OQ – PQ = 1 cm, then sin Q + cos Q =
7 24 31 7
(a) (b) (c) (d)
25 25 25 24
36. If 15 cot A = 8, then sec A =
8 17 15 8
(a) (b) (c) (d)
17 8 8 15

Real Numbers 45
 P
37. In the given figure, ∆PQR is right-angled at Q, PQ = 3 cm,
PR = 6 cm. The value of ∠QPR – ∠PRQ =
(a) 15° (b) 30°

(c) 45° (d) 10° Q R

1 1
38. If sin(A – B) = ; cos(A + B) = , then sin(A + B) =
2 2
1 1 3
(a) (b) (c) (d) 1
2 2 2

39. In ∆ABC, right-angled at B, AB = 5 cm A

BC
and ∠ACB = 30°, then =
AC
1 3
(a) (b)
2 2 B
30°
C
(c) 5 3 (d) 10 cm
1
40. If tan(A + B) = 3 and tan(A – B) = , then sec(A – B) =
3
3 2 1
(a) (b) (c) 3 (d)
2 3 3
41. If sin 3A = cos(A – 26°), then the value of A is
(a) 29° (b) 32° (c) 36° (d) 42°
42. If tan 2A = cot(A – 18°), then the value of A is
(a) 32° (b) 35° (c) 36° (d) 38°
43. If sec 4A = cosec(A – 20°), then measurement of ∠A =
(a) 19° (b) 21° (c) 22° (d) 26°

Β+C
44. If A, B and C are interior angles of a triangle ABC, then sin  =
 2 
(a) cosec A (b) tan A (c) sec A (d) cos A
2 2 2 2
45. secθ(1 – sinθ) (secθ + tanθ) =
(a) 0 (b) 1 (c) 1 (d) none of these
2
sin 2 52° + sin 2 38°
46. =
cos 2 26° + cos 2 64°
(a) 0 (b) 1 (c) 1 (d) none of these
2
47. sin 72°.cos 18° + cos 72°. sin 18° =
(a) 1 (b) 1 – sin2θ (c) cos2θ (d) 0
48. 9 sec2θ –9 tan2θ =
(a) 1 (b) 9 (c) 8 (d) 0
46 Mathematics - 10
 49. (1 + tanθ + secθ) (1 + cotθ – cosecθ) =
(a) 0 (b) 1 (c) 2 (d) –1
2
1 + tan θ
50. =
1 + cot 2 θ
(a) sec2θ (b) –1 (c) cot2θ (d) tan2θ
51. (secθ + tanθ) (1 – sinθ) =
(a) secθ (b) sinθ (c) cosecθ (d) cosθ
2
52. 2cos2θ +
1 + cot 2 θ
(a) 1 (b) 2 (c) 0 (d) 1
2
3 − tan θ
53. Simplified form of is
3 cosec θ − sec θ
(a) cosθ (b) sinθ (c) cosecθ (d) tanθ
2 sin 2 63° + 1 + 2 sin 2 27°
54. =
3 cos 2 17° − 2 + 3 cos 2 73°
(a) 1 (b) 3 (c) 0 (d) 3
2 2
cos 2 40° + cos 2 50°
55. cos(40° + θ) – sin(50° – θ) + =
sin 2 40° + sin 2 50°
(a) 1 (b) 1 (c) 0 (d) none of these
2

56. If 3 tanθ = 3sinθ, then sin2θ – cos2θ =

1
(a) 1 (b) 1 (c) 1 (d)
2 3 3
sin θ 1 + cos θ
57. + =
1 + cos θ sin θ
(a) 2sinθ (b) 2cosθ (c) 2tanθ (d) 2cosecθ
a
58. If sinθ = , then tanθ =
a + b2
2

(a) b (b) a (c) – a (d) –b

a b b a
cos θ cos θ
59. + =
1 − sin θ 1 + sin θ
(a) 2sinθ (b) 2cosθ (c) 2cosθ (d) 2secθ
sin θ − 2 sin 3 θ
60. =
2 cos3 θ − cos θ
1
(a) cotθ (b) tanθ (c) (d) secθ
sin θ
61. If tanθ + cotθ = 2, then tan1000θ + cot100θ =

Real Numbers 47
 1
(a) 100 (b) (c) –2 (d) 2
100
62. If sinθ + cosθ = 2 sin(90° – θ), then cot θ =
1
(a) 2 (b) 2–1 (c) 2 +1 (d)
2
63. If a cosθ – b sinθ = c, then a sinθ + b cosθ =
(a) ± a 2 + b2 + c2 (b) ± a 2 − b 2 + c 2 (c) ± a 2 − b2 − c2 (d) ± a 2 + b2 − c2
sin 50° cosec 40°
64. + – 4cos50°.cosec40° =
cos 40° sec 50°
(a) 1 (b) 2 (c) –2 (d) 0
65. If cosθ + sinθ = 2 cosθ, then cosθ – sinθ =
1
(a) 2 tanθ (b) 2 cotθ (c) sinθ (d) 2 sinθ
2
p2 − 1
66. If secθ + tanθ = p, then =
p2 + 1
(a) tanθ (b) cosθ (c) sinθ (d) cosecθ
67. sin6θ + cos6θ + 3sin2θ.cos2θ =
(a) 0 (b) 1 (c) 2 (d) –2
68. If sinθ + cosθ = 3 , then tanθ + cotθ =
(a) 1 (b) –1 (c) 2 (d) –2
3
69. If tanA = , then sinA.cosA =
4
11 13 12 17
(a) (b) (c) (d)
25 25 25 25
70. If 3 tanθ = 1, then sin2θ – cos2θ =
–1 1 3 5
(a) (b) (c) (d)
2 2 2 2
71. (1 + tan2θ) (1 – sinθ) (1 + sin2θ) =
(a) 0 (b) –1 (c) 1 (d) 2
72. If 2sin2θ – cos2θ = 2, then the value of θ is =
(a) 60° (b) 30° (c) 0° (d) 90°
cos 2 (45° + θ) + cos 2 (45° − θ)
73. =
tan(60° + θ).tan(30° − θ)
(a) 0 (b) 1 (c) 2 (d) –1
2
p −1
74. If cosecθ + cotθ = p, then =
p2 + 1
(a) sinθ (b) tanθ (c) cosecθ (d) cosθ
75. If sinθ + 2cosθ = 1, then 2sinθ – cosθ =
(a) 1 (b) 0 (c) –2 (d) 2

48 Mathematics - 10
 p2 + 1
76. If tanθ + secθ = p, then =
2p
(a) sinθ (b) cosθ (c) secθ (d) tanθ
sin 3 θ + cos3 θ
77. + sinθ.cosθ =
sin θ + cos θ
1
(a) 0 (b) 1 (c) –1 (d)
2
sec θ − 1 sec θ + 1
78. + =
sec θ + 1 sec θ − 1
(a) 2 sinθ (b) 2 cosecθ (c) 2 tanθ (d) 2 secθ
(1 − cot θ)2
79. + 2 sinθ.cosθ =
cosec2 θ
(a) 1 (b) 0 (c) 2 (d) –2
sin 43° cot 30°
80. 2 − − 2 sin 45° =
cos 47° tan 60°
(a) 2 (b) –1 (c) 1 (d) 0
2 2
sin 47°   cos 43° 
81.  + – 2 cos2 45° =
 cos 43°   sin 47° 
(a) –2 (b) 2 (c) 0 (d) 1
cos 80°
82. + 2 sin 59°.cosec 31° – 3tan 49° tan 41° =
sin 10°
3
(a) 0 (b) 1 (c) (d) –1
2
83. cot4 θ – cosec4θ + cot2θ + cosec2θ =
(a) 0 (b) 1 (c) –1 (d) 2
1 1
84. If cot θ + = 2, then cot2θ + =
cot θ cot 2 θ
(a) 1 (b) 2 (c) –1 (d) –2
85. If sin θ + cos θ = 1, then the value of sin θ.cos θ =
(a) 1 (b) 0 (c) –1 (d) 2
86. If sin θ + cosec θ = 2, then the value of sin2θ + cosec2θ =
(a) 1 (b) 0 (c) 3 (d) 2

cosec2 θ − cot 2 θ 7
87. If 4 sin θ = 3 and 2 + 2cotθ = + cos θ, then the value of x is
sec θ − 1 x
1 5 4 3
(a) (b) (c) (d)
3 3 3 4
15 (2 + 2 sin θ)(1 − sin θ)
88. If cot θ = , then =
8 (1 + cos θ)(2 − 2 cos θ)
25 64 225 225
(a) (b) (c) (d) –
64 225 64 64
89. 5 tan2 θ – 5 sec2 θ =
(a) –5 (b) 5 (c) 0 (d) 1

Real Numbers 49
 3
90. If cosec θ = , then the value of 2(cosec2 θ + cot2 θ) =
2
7
(a) 5 (b) 6 (c) 7 (d)
5
2 , cosec2 θ − sec2 θ
91. If tan θ = then =
5 cosec2 θ + sec2 θ
1 5 1 5
(a) (b) (c) (d)
3 7 9 9
92. If cosec θ – sin θ = a, sec θ – cos θ = b, then a2 b2(a2 + b2 + 3) =
(a) 3 (b) –3 (c) 1 (d) –1
93. sec4 θ (1 – sin2 θ) – 2tan4 θ =
(a) 0 (b) 1 (c) –1 (d) none of these
94. cot 18°.cot 39°.cot 51°.cot 60°.cot 72° =
2 1
(a) 3 (b) (c) (d) 1
3 3
2 2
 x 3  y 3
95. If x = a cos3θ, y=b sin3θ,
then   +   =
a b
1 2 3
(a) (b) (c) (d) 1
3 3 2
sin 43° cot 30°
96. 2 − – 2 sin 45° =
cos 47° tan 60°
(a) 2 (b) –2 (c) 2 (d) 0
1 2 tan θ
97. If tan 30° = , then, using tan2θ = , the value of tan 60° =
3 1 − tan 2 θ
1
(a) (b) 3 (c) 0 (d) not defined
3
98. [cos(90° – θ) + sin(90° – θ)]2 + [sin(90° – θ) – cos(90° – θ)]2 =
(a) 0 (b) 1 (c) 2 (d) –1
cos(90° − θ).cos θ
99. + cos2(90° – θ) =
tan θ
(a) 1 (b) –1 (c) 0 (d) none of these
2 2
 sin 47°   cos 43° 
100.  + – 4cos2 45° =
 cos 43°   sin 47° 
(a) 1 (b) 0 (c) 2 (d) –1

II. FILL IN THE BLANKS

3 1
1. If sin A = , then tan A + = ______________
5 cos A

7,
2. If cos θ = then the value of tan θ + cot θ = ___________
25

50 Mathematics - 10
 13 ,
3. If cosec θ = then the value of cot θ + tan θ = ___________
12
3,
4. If sin θ = then the value of 2cot2 θ –1 = ___________
2
1 ,
5. If tan θ = then 7 sin2 θ + 3 cos2 θ = _____________
3
6. In a right triangle PQR, right angled at Q, if tan P = 1, then, 2 sin P. cos P = _______
7. If cos A = cos 60°.cos 30° + sin 60°.sin 30°, then the value of A = _________.
1 ,
8. If A and B are acute angle such that tan (A + B) = 3 and tan (A – B) = then
A + B = __________ 3

9. If 3 tan2 A – 1 = 0 (0° < A < 90°), then A = _________.

10. If 3 cot 2A = 1, [0° < A < 90°], then A = __________.

5
11. If cosec θ = , then 5 sin θ – 3 tan θ = ___________.
4
12. If cos x = cos 60°.cos 30° + sin 60°.sin 30°, then x = ___________.
13. If 3 sin θ + 5 cos θ = 5, then 5 sin θ – 3 cos θ = _________
2  2 1 
14. If cosec θ = 2x and cot θ = , then 2  x − 2  = ___________
x  x 
2  1 
15. If 2x = sec θ and = tan θ, then  x 2 − 2  = ___________
x  x 
x y x y x2 y 2
16. If sin θ – cos θ = 1 and cos θ + sin θ = 1, then 2 + 2 = _________.
a b a b a b
17. If 5 sin θ + 7 cos θ = 7, then 7 sin θ – 5 cos θ = ___________.
18. If cos θ + cos2 θ = 1, then the value of sin2 θ + sin4 θ = _____________
19. If 7 sin2 θ + 3 cos2 θ = 4, then tan θ = _____________
20. (sin4 θ – cos4 θ + 1) cosec2 θ = ___________
cos 70° cos 59°
21. + – 8 sin2 30° = ___________
sin 20° sin 31°
22. cot 1°.cot 2°.cot 3° ..... cot 89° = __________
23. If xsin (90° – θ). cot(90° – θ) = cos(90° – θ), then x = _____________
 sec2 59° − cot 2 31°  2 2 2 x
24. If 4   – 3 sin 90° + 3 tan 56°.tan 34° = 3, then x = ____________
 3 
25. (sec 13° – cot 77°) (sec 13° + cot 77°) = _________

III. VERY SHORT ANSWER QUESTIONS

1. Find the value of sin A sec A cot A.

Real Numbers 51
 2
2. Find the value of (1 + tan θ) .
sec2 θ
3
3. If sin A = , then find the value of cos A + sec A.
5
4. If θ ≤ 0° ≤ 90°, then what is the minimum value of sin θ.
1 1
5. As we know, can always be written as x–1, x ≠ 0. Can we also write as sin–1A?
x sin Α
6. Find the value of 2sin 30° cos 30°.
7. Find the value of cos 45° sin 30° + cos 30° sin 45°.
5
8. Can sin θ be equal to
for some angle θ?
3
5
9. Can tan θ be equal to for some angle θ?
3
sin 44°
10. Evaluate + tan 45°.
cos 46°
11. Express tan 72° + sin 69° in terms of t-ratios of angles between 0° and 45°.
12. Find the value of 4 sec2 θ – 4 tan2 θ
13. Evaluate sin2 60° + 2 tan 45° – cos2 30°. [Cbse 2019]
3
14. If sin A = , find sec A. [Cbse 2019]
4
15. Find A if tan 2A = cot(A – 24°). [Cbse 2019]
16. Find the value of (sin2 33° + sin2 57°) [Cbse 2019]
17. Find the value of (cos2 67° – sin2 23°) [Cbse 2018]

ANSWERS
I. Multiple Choice Questions :
1. (b) 2. (d) 3. (b) 4. (b) 5. (a) 6. (b) 7. (c)
8. (b) 9. (b) 10. (c) 11. (a) 12. (a) 13. (d) 14. (b)
15. (c) 16. (c) 17. (b) 18. (c) 19. (a) 20. (b) 21. (d)
22. (a) 23. (d) 24. (b) 25. (c) 26. (c) 27. (d) 28. (d)
29. (a) 30. (b) 31. (d) 32. (d) 33. (d) 34. (a) 35. (c)
36. (b) 37. (b) 38. (c) 39. (c) 40. (b) 41. (a) 42. (c)
43. (c) 44 (d) 45. (c) 46. (c) 47. (a) 48. (b) 49. (c)
50. (d) 51. (d) 52. (b) 53. (b) 54. (d) 55. (b) 56. (c)
57. (d) 58. (b) 59. (d) 60. (b) 61. (d) 62. (c) 63. (d)
64. (c) 65 (d) 66. (c) 67. (b) 68. (a) 69. (c) 70. (a)
71. (c) 72. (d) 73. (b) 74. (d) 75. (d) 76. (c) 77. (b)
78. (b) 79. (a) 80. (d) 81. (d) 82. (a) 83. (a) 84. (b)
85. (b) 86. (d) 87. (c) 88. (c) 89. (a) 90. (c) 91. (c)
92. (c) 93. (b) 94. (c) 95. (d) 96. (d) 97. (b) 98. (c)
99. (a) 100. (b)

52 Mathematics - 10
 II. Fill in the Blanks :
625 –1
1. 2 2. 3. 169 4. 5. 4
168 60 3
6. 1 7. 30° 8. 60° 9. 30° 10. 30°
11. 0 12. 30° 13. ± 3 14. 1 15. 1
2 4
1
16. 2 17. ± 5 18. 1 19. 20. 2
3
21. 0 22. 1 23. 1 24. 11 25. 1
III. Very Short Answer Questions :
41
1. 1 2. 1 3. 4. 0 5. no
20
3 2 +1
6. 7. 8. no 9. yes 10. 2
2 2 2
4
11. cot 18° + cos 21° 12. 4 13. 2 14.
7
15. 38° 16. 1 17. 0

Real Numbers 53