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SURAJ BHAN DAV PUBLIC SCHOOL
TRIGONOMETRIC FUNCTIONS
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

1. Find the degree measure corresponding to the following radian measures  Use  
 3 
(i) 

 4 
c
c
(ii) 3
 7 
(iii)  

 5 
22 

7 
c
(iv) 1c
2. Find the radian measure corresponding to the following degree measures
(i) 135
(ii) 4330
(iii) 45
(iv) 10045
3. Find the angle between minute hand of a clock and the hour hand when the time is 7:20 AM. Why time
management is important? (Value Based)
4. A railway train is traveling on a circular curve of 1500 metres radius at the rate of 66km/hr. Through what
angle has it turned in 10 seconds?
5. A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight
and describes 88 metres when it has traced out 72 at the centre, find the length of the rope. Why
should we protect our wild life? (Value Based)
6. Find the value of the following:7
 13 
(i) sin 300
(ii) tan(1410)
(iii) tan135
(iv) tan
(v) sin 

4
 6 
7. If A, B.C and D are the angles of a cyclic quadrilateral then prove that cos A  cos B  cos C  cos D  0
8. Find the value of the (i) sin 75 (ii) cos105 (iii) tan

12
9. Prove that tan 7 A  tan 5 A  tan 2 A  tan 7 A tan 5 A tan 2 A
10. Prove that cotAcot2 A  cot2 Acot3 A  cot3 AcotA  1
11. If cot A cot B  3 ,prove that 2cos  A  B   cos  A  B  (HOTS)
12. If cos(   )  cos       cos      
(HOTS)
13. Prove that
3
then prove that sin   sin   sin   cos   cos   cos   0
2
sin  B  C  sin  C  A  sin  A  B 


0
cos B cos C cos C cos A cos A cos B
14. If cos  α  β  sin  γ  δ   cos  α  β  sin  γ  δ  prove that cotαcotβcotγ  cotδ


then show that (i) 1  tan A1  tan B   2 (ii)  cot A  1 cot B  1  2 (iii)Find tan (HOTS)
4
8
16. If sin   sin   a and cos   cos   b then prove that
15. If A  B 
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SURAJ BHAN DAV PUBLIC SCHOOL
TRIGONOMETRIC FUNCTIONS
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2ab
b2  a 2
(ii) tan      2
(HOTS)
2
2
a  b2
b a
17. Prove that tan 57  tan17  2 tan 40
cos A  sin A
cos A  sin A




18. Prove that (i)
(ii)
 tan   A 
 tan   A 
cos A  sin A
cos A  sin A
4

4

cos 7  sin 7
 tan 38
19. Prove that
cos 7  sin 7
3
20. Prove that sin 20 sin 40 sin 60 sin 80 
16
21. Prove that 4 cos12 cos 48 cos 72  cos 36
(i) cos     
1
22. Prove that sinAsin  60ο  A  sin  60ο  A   sin3 A
4
ο
ο
23. Prove that tanθtan  60  θ  tan  60  θ   tan3θ
sin5 A  sin3 A
 tanA
cos5 A  cos3 A
sinA  sin3 A
 tan2 A
25. Prove that
cosA  cos3 A
24. Prove that
26. Prove that cot4 x  sin5 x  sin3x   cotx  sin5 x  sin3x 
27. Prove that sinx  sin3x  sin5 x  sin7 x  4cosxcos2 xsin4 x
2
2
 α β 
28. Prove that  cosα  cosβ    sinα  sinβ   4sin 2 

 2 
cos4 x  cos3x  cos2 x
 cot3x
29. Prove that
sin4 x  sin3 x  sin2 x
sinA  sin3 A  sin5 A  sin7 A
 tan4 A
30. Prove that
cosA  cos3 A  cos5 A  cos7 A
cos8 Acos5 A  cos12 Acos9 A
 tan4 A
31. Prove that
sin8 Acos5 A  cos12 Asin9 A
cos2 Acos3 A  cos2 Acos7 A  cosAcos10 A
 cot6 Acot5 A
32. Prove that
sin4 Asin3 A  sin2 Asin5 A  sin4 Asin7 A
sinA  sinC
33. If three angles A, B, C are in A.P. prove that cotB 
cosC  cosA
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SURAJ BHAN DAV PUBLIC SCHOOL
TRIGONOMETRIC FUNCTIONS
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 α β  β  γ   γ α 
34. Prove that sinα  sinβ  sinγ  sin  α  β  γ   4sin 
 sin 
 sin 

 2   2   2 
tan  A  B  λ  1
35. If sin2 A  λsin2 B , prove that:

tan  A  B  λ  1
cos  A  B  C   cos   A  B  C   cos  A  B  C   cos  A  B  C 
 cotC (HOTS)
sin  A  B  C   sin   A  B  C   sin  A  B  C   sin  A  B  C 
sin2θ
 tanθ
37. Prove that
1  cos2θ
1  sin2θ  cos2θ
 cotθ
38. Prove that
1  sin2θ  cosθ
cosθ
π θ
39. Prove that
 tan   
1  sinθ
 4 2
36. Prove that
40. Prove that 2  2  2  2cos8θ  2cosθ (HOTS)




2
2
ο
2
ο
41. Prove that cos A  cos A  120  cos A  120 
3
2
42. Prove that cos4 x  1  8sin 2 xcos 2 x
sinx sin3x sin9 x 1


  tan27 x  tanx  (HOTS)
cos3x cos9 x cos27 x 2
sin5 x  2sin3 x  sinx
 tanx
44. Prove that
cos5 x  cosx
sin2n A
2
3
n 1
45. Prove that cosAcos2 Acos2 Acos2 A cos 2 A  n
(HOTS)
2 sinA
4tanθ 1  tan 2θ 
46. Prove that tan4θ 
1  6tan 2θ  tan 4θ
47. Prove that tanα  2tan2α  4tan4α  8cot8α  cotα (HOTS)
43. Prove that
48. Prove that tan

8
 2 1
49. Find the value of sin

8
3 π
x
x
x
50. If cosx   ,  x  π , then find sin cos andtan
5 2
2,
2
2
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SURAJ BHAN DAV PUBLIC SCHOOL
TRIGONOMETRIC FUNCTIONS
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 π A
 π A 1
51. Prove that sin 2     sin 2    
sinA
2
8 2
8 2
π
 2  3  4  6 (HOTS)
52. Prove that cot
24
1
53. Prove that tan144 ο  2  2  3  6 (HOTS)
2
π
3π
5π
7π
 cos 2
 cos 2
2
54. Prove that cos 2  cos 2
8
8
8
8
55. Prove that cos4 A  1  8cos 2 A  8cos 4 A
56. Solve: sin 2 x  cos x  0
57. Solve sin 2 x  sin x  0
58. Solve: cos 5 x  cos 3x  cos x  0
59. Solve: cos 3x  cos x  cos 2 x  0
60. Solve: 2cos 2 x  3sin x  0
61. Solve: sec2 2 x  1  tan 2 x
62. Solve: 3 cos x  sin x  3
63. Solve: cos x  sin x  1
64. Solve: cos x  3 sin x  1
65. In any ABC prove that a sin  B  C   b sin  C  A  c sin  A  B   0
66. In any ABC prove that b cos B  c cos C  a cos  B  C 
b2  c 2
c2  a2
a 2  b2


 0 (HOTS)
cos B  cos C cos C  cos A cos A  cos B
A
 B C   b c 
68. In any ABC prove that sin 

 cos
2
 2   a 
67. In any ABC prove that
2
69. a  cos C  cos B   2  b  c  cos
A
2
A
B
1  tan tan
c
2
2

70. In any ABC prove that
a  b 1  tan A tan B
2
2
A
B
tan  tan
c
2
2

71. In any ABC prove that
A
B
a  b tan  tan
2
2
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TRIGONOMETRIC FUNCTIONS
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
72. In any ABC prove that a 2 sin  B  C   b 2  c 2 sin A
73. In any ABC prove that
b2  c 2 sin  B  C 

a2
sin  B  C 
74. In any ABC prove that
a 2  b2 1  cos  A  B  cos C
(HOTS)

a 2  c 2 1  cos  A  C  cos B
A
B
A
  c  a  cot   a  b  cot  0
2
2
2
cos 2 A cos 2 B 1 1

 2 2
76. In any ABC prove that
a2
b2
a b
77. If a cos A  b cos B then prove that triangle is either isosceles or right angled.
75. In any ABC prove that  b  c  cot
cos A cos B cos C a 2  b 2  c 2
78. In any ABC prove that



a
b
c
2abc
2
2
79. In any ABC prove that b  c cos A  a cos C   c  a
c  b cos A cos B

b  c cos A cos C
C
B

81. In any ABC prove that 2  b cos 2  c cos 2   a  b  c
2
2

A
B
C
2

82. In any ABC prove that 4  bc cos 2  ca cos 2  ab cos 2    a  b  c  (HOTS)
2
2
2

80. In any ABC prove that
 b2  c2 
 c2  a2 
 a 2  b2 
sin
2
A

sin
2
B





 sin 2C  0
2
2
2
 a 
 b

 c

83. In any ABC prove that 
a  b
2
cos 2
C
C
2
  a  b  sin 2  c 2
2
2
84. In any ABC if B  60 , prove that  a  b  c  a  b  c   3ac (HOTS)

86. In any ABC prove that  b





85. In any ABC prove that b 2  c 2  a 2 tan A  c 2  a 2  b 2 tan B  a 2  b 2  c 2 tan C (HOTS)
2
 c 2  cot A   c 2  a 2  cot B   a 2  b 2  cot C  0


87. Draw graph of (i) y  sin  2 x  (ii) y  2 cos 3x (iii) tan  x 



 (iv) cos   x 
4
2

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