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JEE Main Trigonometry questions

78 Mathematics questions on Trigonometry as asked in JEE Main, newest first. 60 come with a worked solution.

  1. Let f(θ) = 3 (sin^4 ((3π)/2 - θ ) + sin^4(3π + θ) ) - 2(1 - sin^2 2θ) and S = \θ ∈ [0, π] : f'(θ) = -frac√(3)2 \. If 4β = sum_θ ∈ S θ, then f(β) is equal to

    2023

  2. Let f: R → R be a function defined by f(x) = log_√(2)[√(2)(sin x - cos x) + m - 2], for some m, such that the range of f is [0, 2]. Then the value of m is

    2023

  3. Let S = \θ ∈ [0, 2π) : tan(π cosθ) + tan(π sinθ) = 0\. Then displaystylesum_θ ∈ S sin^2 (θ + (π)/4 ) is equal to __________.

    2023

  4. If m and n respectively are the numbers of positive and negative values of θ in the interval [-π, π] that satisfy the equation cos 2θ cos (θ)/2 = cos 3θ cos (9θ)/2, then mn is equal to _______.

    2023 · with solution

  5. The maximum value of 3cosθ + 5sin (θ - (π)/6 ) for any real value of θ is

    2019 · with solution

  6. The value of sin 10° sin 30° sin 50° sin 70° is

    2019 · with solution

  7. The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45° from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of…

    2019

  8. Let S be the set of all α ∈ R such that the equation, cos 2x + α sin x = 2α - 7 has a solution. Then, S is equal to

    2019 · with solution

  9. With the usual notation, in triangle ABC, if ∠ A + ∠ B = 120°, a = √(3) + 1 and b = √(3) - 1, then the ratio ∠ A : ∠ B is

    2019 · with solution

  10. The angles A, B and C of a triangle ABC are in AP and a:b=1:√(3). If c=4 cm, then the area (in sq cm) of this triangle is

    2019

  11. All the pairs (x, y) that satisfy the inequality 2^√(sin^2 x - 2sin x + 5) · dfrac14^sin^2 y ≤ 1 also satisfy the equation

    2019

  12. The equation y = sin x sin(x+2) - sin^2(x+1) represents a straight line lying in

    2019

  13. The value of cos^2 10° - cos 10° cos 50° + cos^2 50° is

    2019 · with solution

  14. Given, (b+c)/11 = (c+a)/12 = (a+b)/13 for a triangle ABC with usual notation. If (cos A)/(α) = (cos B)/(β) = (cos C)/(γ), then the ordered triad (α, β, γ) has a value

    2019 · with solution

  15. The number of solutions of the equation 1 + sin^4 x = cos^2 3x, x ∈ [-(5π)/2, (5π)/2 ] is

    2019 · with solution

  16. ABC is a triangular park with AB = AC = 100 m. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are cot^-1(3√(2)) and cosec^-1(2√(2)) respectively, then…

    2019 · with solution

  17. If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is

    2019 · with solution

  18. Let S = \θ ∈ [-2π, 2π] : 2cos^2θ + 3sinθ = 0\, then the sum of the elements of S is

    2019

  19. In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x^2 - c^2 = y, where c is the length of the third side of the triangle, then the circumradius of the…

    2019

  20. If cos(α + β) = 3/5, sin(α - β) = 5/13 and 0 < α, β < (π)/4, then tan(2α) is equal to

    2019 · with solution

  21. Two poles standing on a horizontal ground are of heights 5 m and 10 m, respectively. The line joining their tops makes an angle of 15° with the ground. Then, the distance (in m) between the poles, is

    2019

  22. If sin^4 α + 4cos^4 β + 2 = 4√(2)sinαcosβ; α, β ∈ [0, π], then cos(α + β) - cos(α - β) is equal to

    2019

  23. Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in m) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal…

    2019 · with solution

  24. Let f_k(x) = 1/k(sin^k x + cos^k x) for k = 1, 2, 3 … Then, for all x ∈ R, the value of f_4(x) - f_6(x) is equal to

    2019 · with solution

  25. If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30° and the angle of depression of reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters) from the…

    2019 · with solution

  26. The value of cos(π)/(2^2)·cos(π)/(2^3)·s·scosdfracπ2^10·sindfracπ2^10 is

    2019 · with solution

  27. The sum of all values of θ ∈ (0, (π)/2 ) satisfying sin^2 2θ + cos^4 2θ = 3/4 is

    2019 · with solution

  28. Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the mid-point D of AC subtends an angle 30° at B. The height (in m) of the lamp-post is

    2019 · with solution

  29. If 0 ≤ x ≤ (π)/2, then the number of values of x for which sin x - sin 2x + sin 3x = 0, is

    2019

  30. PQR is a triangular park with PQ = PR = 200 m. A TV tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45°, 30° and 30°, then the height of the tower…

    2018 · with solution

  31. If sum of all the solutions of the equation 8cos x · (cos ((π)/6 + x ) · cos ((π)/6 - x ) - 1/2 ) = 1 in [0, π] is kπ, then k is equal to

    2018 · with solution

  32. Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If ∠ BPC = β, then tanβ is equal to

    2017 · with solution

  33. If 5(tan^2 x - cos^2 x) = 2cos 2x + 9, then the value of cos 4x is

    2017 · with solution

  34. The value of sum_k=1^13 1/(sin ((π)/4 + ((k-1)π)/6 )sin ((π)/4 + (kπ)/6 )) is equal to

    2016 · with solution

  35. A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30°. After walking for 10 min from…

    2016 · with solution

  36. If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0, is

    2016 · with solution

  37. Let S = \x ∈ (-π, π): x ≠ 0, ±(π)/2 \. The sum of all distinct solutions of the equation √(3)sec x + csc x + 2(tan x - cot x) = 0 in the set S is equal to

    2016

  38. For x ∈ (0, π), the equation sin x + 2sin 2x - sin 3x = 3 has

    2014 · with solution

  39. In a triangle, the sum of two sides is x and the product of the same two sides is y. If x^2 - c^2 = y, where c is the third side of the triangle, then the ratio of the inradius to the circumradius of the triangle is

    2014 · with solution

  40. ABCD is a trapezium such that AB and CD are parallel and BC ⊥ CD, if ∠ ADB = θ, BC = p and CD = q, then AB is equal to

    2013

  41. The expression (tan A)/(1 - cot A) + (cot A)/(1 - tan A) can be written as

    2013 · with solution

  42. If P = \θ : sinθ - cosθ = √(2)cosθ\ and Q = \θ : sinθ + cosθ = √(2)sinθ\ be two sets. Then,

    2011 · with solution

  43. If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression a/csin 2C + c/asin 2A is

    2010 · with solution

  44. The number of solutions of the pair of equations 2sin^2θ - cos 2θ = 0 and 2cos^2θ - 3sinθ = 0 in the interval [0, 2π] is

    2007 · with solution

  45. The set of values of θ satisfying the inequation 2sin^2θ - 5sinθ + 2 > 0, where 0 < θ < 2π, is

    2006 · with solution

  46. In radius of a circle which is inscribed in a isosceles triangle one of whose angle is 2π/3, is √(3), then area of triangle (in sq units) is

    2006 · with solution

  47. In a triangle ABC, among the following which one is true?

    2005 · with solution

  48. The number of ordered pairs (α, β), where α, β ∈ (-π, π) satisfying cos(α - β) = 1 and cos(α + β) = 1/e is

    2005

  49. The sides of a triangle are in the ratio 1 : √(3) : 2, then the angles of the triangle are in the ratio

    2004 · with solution

  50. Given both θ and φ are acute angles and sinθ = 1/2, cosφ = 1/3, then the value of θ + φ belongs to

    2004

  51. If the angles of a triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is

    2003 · with solution

  52. The number of integral values of k for which the equation 7cos x + 5sin x = 2k + 1 has a solution, is

    2002 · with solution

  53. Which of the following pieces of data does not uniquely determine an acute angled triangle ABC (R being the radius of the circumcircle)?

    2002 · with solution

  54. If α + β = (π)/2 and β + γ = α, then tanα equals hspace2em

    2001 · with solution

  55. In a triangle ABC, let ∠ C = π/2. If r is the inradius and R is the circumradius of the triangle, then 2(r + R) is equal to

    2000 · with solution

  56. Let f(θ) = sinθ(sinθ + sin 3θ). Then, f(θ)

    2000 · with solution

  57. In a triangle PQR, ∠ R = (π)/2, if tan (P/2 ) and tan (Q/2 ) are the roots of the equation ax^2 + bx + c = 0 (a ≠ 0), then

    1999

  58. The number of values of x in the interval [0, 5π] satisfying the equation 3sin^2 x - 7sin x + 2 = 0 is

    1998 · with solution

  59. If in a triangle PQR, sin P, sin Q, sin R are in AP, then

    1998 · with solution

  60. Which of the following numbers is rational?

    1998 · with solution

  61. Let n be an odd integer. If sin nθ = sum_r=0^n b_r sin^r θ, for every value of θ, then

    1998 · with solution

  62. The graph of the function cos x cos(x+2) - cos^2(x+1) is

    1997 · with solution

  63. In a triangle ABC, ∠ B = (π)/3 and ∠ C = (π)/4. Let D divides BC internally in the ratio 1:3, then (sin ∠ BAD)/(sin ∠ CAD) is equal to

    1995 · with solution

  64. 3(sin x - cos x)^4 + 6(sin x + cos x)^2 + 4(sin^6 x + cos^6 x) equals

    1995 · with solution

  65. The general value of θ satisfying the equation 2sin^2θ - 3sinθ - 2 = 0, is

    1995 · with solution

  66. Number of solutions of the equation tan x + sec x = 2cos x lying in the interval [0, 2π] is

    1993 · with solution

  67. The number of solutions of the equation sin(e^x) = 5^x + 5^-x is

    1991 · with solution

  68. In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B satisfy the equation 3sin x - 4sin^3 x - k = 0, 0 < k < 1, then the measure of ∠ C is

    1990 · with solution

  69. The general solution of sin x - 3sin 2x + sin 3x = cos x - 3cos 2x + cos 3x is

    1989 · with solution

  70. The value of the expression √(3)cosec 20°-sec 20° is equal to

    1988 · with solution

  71. The smallest positive root of the equation tan x - x = 0 lies in

    1987 · with solution

  72. The number of all possible triplets (a_1, a_2, a_3) such that a_1 + a_2 cos(2x) + a_3 sin^2(x) = 0, forall x is

    1987 · with solution

  73. The expression 3 [sin^4 ((3π)/2-α )+sin^4(3π+α) ]-2 [sin^6 ((π)/2+α )+sin^6(5π-α) ] is equal to

    1986 · with solution

  74. (1 + cos(π)/8 ) (1 + cos(3π)/8 ) (1 + cos(5π)/8 ) (1 + cos(7π)/8 ) is equal to

    1984

  75. The general solution of the trigonometric equation sin x + cos x = 1 is given by

    1981 · with solution

  76. Given A = sin^2θ + cos^4θ, then for all real values of θ

    1980 · with solution

  77. If α + β + γ = 2π, then

    1979 · with solution

  78. If tanθ = -4/3, then sinθ is

    1978 · with solution

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