PRACTICE SHEET

TRIGONOMETRY - I

2
c c
1. a, b, c are the sides of a triangle ABC which is right angled at C, then the minimum value of   
a b
is
(A) 16 (B) 4 (C) 6 (D) 8

2. If x + y = 3 – cos4 and x – y = 4 sin2 then

(A) x4 + y4 = 9 (B) x  y 16
(C) x3 + y3 = 2(x2 + y2) (D) x  y 2

n sin A cos A
3. If tanB = then tan(A + B) equals
1  n cos2 A
sin A ( n  1) cos A sin A sin A
(A) (B) (C) (D)
(1  n ) cos A sin A (n  1) cos A (n  1) cos A

FG  (a  x)IJ = 0 then, which of the following holds good?

4. Given a2 + 2a + cosec2 H2 K
x x
(A) a = 1 ; I (B) a = –1 ; I
2 2
(C) a  R ; x  (D) a , x are finite but not possible to find

5. The minimum value of the function

f (x) = (3sin x – 4 cos x – 10)(3 sin x + 4 cos x – 10), is
195  60 2
(A) 49 (B) (C) 84 (D) 48
2

6. Which value of  listed below leads to

2sin  > 1 and 3cos  < 1?
(A) 70° (B) 140° (C) 210° (D) 280°

7. In the inequality below the value of the angle is expressed in radian measure. Which one of the inequalities
below is true?
(A) sin 1 < sin 2 < sin 3 (B) sin 3 < sin 2 < sin 1
(C) sin 2 < sin 1 < sin 3 (D) sin 3 < sin 1 < sin 2

2 3 6 9 18 27 
8. The exact value of cos cos ec  cos cos ec  cos cos ec is equal to
28 28 28 28 28 28
(A) – 1/2 (B) 1/2 (C) 1 (D) 0

9. If the expression, 2 cos10° + sin 100° + sin 1000° + sin 10000° is simplified, then it simplifies to
(A) cos 10° (B) 3 cos 10° (C) 4 cos 10° (D) 5 cos 10°

1
10. Number of ordered pair(s) of (x, y) where x, y  [–2, 2] satisfying the equation
sin(x  y)  1 2 cos(2x  y)  1 = 6 is
(A) 1 (B) 2 (C) 3 (D) 4

11. If 3 tan A + cot A = 5cosec A, then the value of (sec A + 4sin2A) is equal to
(A) 3 (B) 4 (C) 5 (D) 6

2   2  
12. The maximum value of expression cos   x   sin   x  is equal to
4  4 
3 
(A) 1 (B) 2 (C) (D)
2 2

13. A regular decagon A0, A1, A2......A9 is given in the xy plane. Measure of the A0A3A7 in degrees is
(A) 108° (B) 96°
(C) 72° (D) 54°

14. Number of real solutions to the equation

sin (6x) = x, is
(A) 13 (B) 11 (C) 9 (D) 7

15. The value of x satisfying the equation, x = 2  2  2  x is

(A) 2 cos 10° (B) 2 cos 20° (C) 2 cos 40° (D) 2 cos 80°

16. An equilateral triangle has sides 1 cm long. An ant walks around the triangle, maintaining a distance of
1 cm from the triangle at all time. Distance travelled by the ant in one round is
3
(A) 3  3 3 (B) 3  6 3 (C) 3 + 2 (D) 3 +
2

17. If A, B, C and D denotes the interior angles of a quadrilateral then

 tan A   tan A
(A)
 cot A (B)  tan A   tan A   cot A 
(C)  cot A   tan A   tan A (D)  tan A   tan A   cot A

    3   7 
tan x   cos  x   sin 3   x
 2  2   2  when simplified reduces to
18.
    3 
cos x   tan  x 
 2  2 
(A) sin x cos x (B)  sin2 x (C)  sin x cos x (D) sin2x

cos 96  sin 96

19. The smallest natural 'n' such that tan(107n)° = , is
cos 96  sin 96
(A) n = 2 (B) n = 3 (C) n = 4 (D) n = 5

2
20. If in a ABC, sin3A + sin3B + sin3C = 3 sinA · sinB · sinC then
(A) ABC may be a scalene triangle (B) ABC is a right triangle
(C) ABC is an obtuse angled triangle (D) ABC is an equilateral triangle

2005
 2 n 
21. The value of the sum   2 sin
2
 1 equals
n 1001  
(A) 2007 (B) 2008 (C) 2010 (D) 20113

22. Number of degrees in the smallest positive angle x such that

8 sin x cos5x – 8 sin5x cos x = 1, is
(A) 5° (B) 7.5° (C) 10° (D) 15°

23 Given that the side length of a rhombus is the geometric mean of the lengths of its diagonals. The degree
measure of the acute angle of the rhombus is
(A) 15° (B) 30° (C) 45° (D) 60°

24. In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B satisfy the
equation, 3 sin x  4 sin3 x  K = 0, 0 < K < 1, then the measure of angle C is
(A) /3 (B) /2 (C) 2/3 (D) 5/6

25. The set of angles btween 0 and 2 satisfying the equation 4 cos2  2 2 cos  1 = 0 is
RS  , 5 , 19 , 23 UV  7 17 23 
(B)  ,
(A)
T12 12 12 12 W ,
12 12
,
12

12 

(C) S
R 5 , 13 , 19 UV RS  , 7 , 19 , 23 UV
T 12 12 12 W (D)
T 12 12 12 12 W
26. If cos ( + ) = 0 then sin ( + 2) =
(A) sin  (B)  sin  (C) cos  (D)  cos 

1  sin 2 1    3   
27. The expression  sin 2   cot  cot     when simplified

cos 2  2  . tan   3
4  4  2 2 2
reduces to
(A) 1 (B) 0 (C) sin2 /2 (D) sin2 

28. The number of ordered pairs (x, y) of real numbers satisfying 4x2 – 4x + 2 = sin2y and x2 + y2  3,
is equal to
(A) 0 (B) 2 (C) 4 (D) 8

29. Let y = cos x (cos x  cos 3 x). Then y is

(A)  0 only when x  0 (B)  0 for all real x
(C)  0 for all real x (D)  0 only when x  0

30. Range of the expression y = cos x (cos x  cos 3 x) is equal to

 5
(A) [0, 1] (B) [0, 2] (C) 0,  (D) [0, )
 2

3
 1 1
31. If tan x – tan y = a and cot y – cot x = b (a, b  0), then the value of  =
a b
(A) cot (x – y ) (B) tan (x – y) (C) tan ( x + y) (D) cot (x + y )

32. The simplified value of


2 sin2 2cos 2 1 
is
cossincos3sin3
(A) sin  (B) cos  (C) cosec  (D) sec 

33. In a triangle ABC, angle B < angle C and the values of B and C satisfy the equation
2 tan x – k (1 + tan2 x) = 0 where (0 < k < 1) . Then the measure of angle A is :
(A) /3 (B) 2/3 (C) /2 (D) 3/4

34. If a sin x + b cos x = 1 and a2 + b2 = 1 (a, b > 0), then consider the following statements:
I sin x = a II tan x = a/b III tan x = b
(A) only III is false (B) only I is true
(C) All of I, II, III must be true (D) None of I, II or III is correct.

35. The value of log2 [cos2 ( + ) + cos2 ()  cos 2 . cos 2]
(A) depends on  and  both (B) depends on  but not on 
(C) depends on  but not on  (D) independent of both  and  .

36. In a triangle ABC, 3 sin A + 4 cos B = 6 and 3 cos A + 4 sin B = 1 then C can be
(A) 30° (B) 60° (C) 90° (D) 150°

37. If x sec + y tan = x sec + y tan = a, then sec  · sec  =

a 2  y2 a 2  y2 x 2  y2 x 2  y2
(A) 2 (B) 2 (C) 2 (D) 2
x  y2 x  y2 a  y2 a  y2

cos 3x 1  sin 3x
38. If = for some angle x, 0  x  , then the value of for same x, is
cos x 3 2 sin x
7 5 2
(A) (B) (C) 1 (D)
3 3 3

 C C A B
39. If A + B + C =  and sin  A   = k sin , then tan tan =
 2 2 2 2

k1 k1 k k1

(A) (B) (C) (D)
k1 k1 k1 k

40. The graphs of y = sin x, y = cos x, y = tan x and y = cosec x are drawn on the same axes from 0 to /
2. A vertical line is drawn through the point where the graphs of y = cos x and y = tan x cross, intersecting
the other two graphs at points A and B. The length of the line segment AB is:
5 1 5 1
(A) 1 (B) (C) 2 (D)
2 2

4
41. The expression wherever defined, sec4x – 4 tan3x + 4 tan x is always :
(A) positive (B) negative (C) non-positive (D) non–negative

1  1
42. If cos  =  a   then cos 3 in terms of ‘a’ =
2  a
1  3 1 1  3 1  1
a  3 
3
(A) a  3  (B) (C) 4  a  3  (D) none
4  a  2  a  a

5 1  sin x  1  sin x
43. If  x  3 , then the value of the expression is
2 1  sin x  1  sin x
x x x x
(A) – cot (B) cot (C) tan (D) – tan
2 2 2 2

44. If x, y  R and satisfy the equation xy(x2 – y2) = x2 + y2 where x  0 then the minimum possible value
of x2 + y2 is
(A) 1 (B) 2 (C) 4 (D) 8

 2  4
45. If x sin  = y sin     = z sin     then :
 3   3
(A) x + y + z = 0 (B) xy + yz + zx = 0 (C) xyz + x + y + z = 1 (D) none

sin 2 (212) cos 302  cos3 (148)

46. The value of is :
sin(82) cos(8)  sin 368 sin(172)  sin 58 sin 148
(A) cos 32º  sin 32º (B) sin 32º  cos 32º
(C) cos 32º + sin 32º (D) cos 32º sin 32º + 1

 2 4 8 16
47. The value of cos cos cos cos cos is
10 10 10 10 10

1 1 cos 10  10  2 5
(A) (B) (C) (D) –
32 16 16 64

48. If a cos3  + 3a cos  sin2  = m and a sin3  + 3a cos2  sin  = n . Then

(m + n)2/3 + (m  n)2/3 is equal to :
(A) 2 a2 (B) 2 a1/3 (C) 2 a2/3 (D) 2 a3

49. As shown in the figure AD is the altitude on BC and AD

produced meets the circumcircle of ABC at P where
DP = x. Similarly EQ = y and FR = z. If a, b, c respectively
a b c
denotes the sides BC, CA and AB then  
2 x 2 y 2z
has the value equal to
(A) tanA + tanB + tanC (B) cotA + cotB + cotC
(C) cosA + cosB + cosC (D) cosecA + cosecB + cosecC

5
50. The maximum value of (7 cos + 24 sin) × ( 7 sin – 24 cos) for every   R .
625 625
(A) 25 (B) 625 (C) (D)
2 4

51. If  is eliminated from the equations x = a cos( – ) and y = b cos ( – ) then

x2 y2 2xy
2
 2
 cos(  ) is equal to
a b ab
(A) cos2 (  – ) (B) sin2 ( – ) (C) sec2 (  – ) (D) cosec2 ( – )

tan 2 x  4 tan x  9
52. If M and m are maximum and minimum value of the function f (x) = , then (M + m)
1  tan 2 x
equals
(A) 20 (B) 14 (C) 10 (D) 8

1 1
53. + =
cos 290 3 sin 250

2 3 4 3
(A) (B) (C) 3 (D) none
3 3

54. Let f (x) = sin 4 x  4 cos2 x – cos4 x  4 sin 2 x . An equivalent form of f (x) is
(A) 1 (B) – 1 + 2 sinx (C) cos 2x (D) – cos 2x

sin 8θ ·cos θ  sin 6θ ·cos 3θ

55. The value of when  = 7.5° is
cos 2θ ·cos θ  sin 3θ ·sin 4θ
(A) 2  1 (B) 2  3 (C) 2 1 (D) 2  3

56. The value of cot x + cot (60º + x) + cot (120º + x) is equal to :

(A) cot 3x (B) tan 3x (C) 3 tan 3x (D) 3cot 3x

x2  x 1 
57. If tan  = 2 and tan  = 2 (x  0, 1), where 0 < ,  < , then tan ( + ) has
x  x 1 2 x  2x  1 2
the value equal to
3
(A) 1 (B) – 1 (C) 2 (D)
4

58. If tan A and tan B are the roots of the quadratic equation x2  ax + b = 0, then the value of sin2 (A + B) is
a2 a2 a2 a2
(A) 2 (B) 2 (C) (D) 2
a  (1  b)2 a  b2 ( b  a) 2 b (1  a ) 2

x2
59. For every x  R the value of the expression y = + x cos x + cos 2x is never less than
8
(A) – 1 (B) 0 (C) 1 (D) 2

6
60. If  be an acute angle satisfying the equation 8 cos 2 + 8 sec 2 = 65, then the value of cos  is equal
to
1 2 2 3 3
(A) (B) (C) (D)
8 3 3 4

x y
61. If x sin  = y cos  then + is equal to
sec2 cosec2
(A) x (B) y (C) x2 (D) y2

62. If cos25° + sin25° = p, then cos50° is

(A) 2  p2 (B) – p 2  p 2 (C) p 2  p 2 (D) – p 2  p 2

63. In a triangle ABC if tan A < 0 then

(A) tan B . tan C > 1 (B) tan B . tan C < 1
(C) tan B . tan C = 1 (D) none

64. , ,  and  are the smallest positive angles in ascending order of magnitude which have their sines
equal to the positive quantity k . The value of
   
4 sin + 3 sin + 2 sin + sin is equal to :
2 2 2 2
(A) 2 1  k (B) 2 1  k (C) 2 k (D) 2 k

  x 1
65. Suppose that  is a positive acute angle such that sin    , then the value of tan  is
 2 2x

x 1 x2 1
(A) x (B) (C) 2
x 1 (D)
x 1 x

A
66. If A = 3400 then 2 sin is identical to
2
(A) 1  sin A  1  sin A (B)  1  sin A  1  sin A
(C) 1  sin A  1  sin A (D)  1  sin A  1  sin A

 
67. The value of cosec – 3 sec 18 is a
18
(A) surd (B) rational which is not integral
(C) negative natural number (D) natural number

68. The expression S = sec 11° sec 19° – 2 cot 71° reduces to
1
(A) 2 cot 11° (B) tan 19° (C) 2 tan 11° (D) tan 19°
2

7
69. The set of values of ‘a’ for which the equation, cos 2x + a sin x = 2a  7 possess a solution is
(A) (, 2) (B) [2, 6] (C) (6, ) (D) ()

70. If tan x + tan y = 25 and cot x + cot y = 30, then the value of tan(x + y) is
(A) 150 (B) 200 (C) 250 (D) 100

71. In a right angled triangle the hypotenuse is 2 2 times the perpendicular drawn from the opposite vertex.
Then the other acute angles of the triangle are
   3    3
(A) and (B) and (C) and (D) and
3 6 8 8 4 4 5 10

A B
 cot 2 2 . cot 2 2
72. In  ABC, the minimum value of is
A
 cot 2 2
(A) 1 (B) 2 (C) 3 (D) non existent

73. Minimum vertical distance between the graphs of y = 2 + sin x and y = cos x is
(A) 2 (B) 1 (C) 2 (D) 2 – 2

74. If sin () = a and sin () = b (0 <, ,</2)

then cos2 ()  4 ab cos() =
(A) 1  a2  b2 (B) 1  2a2  2b2 (C) 2 + a2 + b2 (D) 2  a2  b2

75. The exact value of cos273º + cos247º + (cos73º . cos47º) is

(A) 1/4 (B) 1/2 (C) 3/4 (D) 1