Trigonometric Ratios of Allied Angles & Basic Fundamentals PDF

## Topic 1: Trigonometrical Ratios of Allied Angles and Basic Fundamentals ### Question 1 The sum of the series ∑n=1∞ sin(nπ/180) is: * a) sin(π/180) + sin(π/360) + sin(π/120) + sin(π/360) + sin(π/720) * b) sin(π/6) + sin(π/30) + sin(π/120) + sin(π/360) + sin(π/720) * c) sin(π/6) + sin(π/30) + sin(π/120) + sin(π/360) + sin(π/720) * d) sin(π/6) + sin(π/6) + sin(π/30) + sin(π/120) + sin(π/360) + sin(π/720) **Answer**: (d) ### Question 2 If cos x = tan y, cot y = tan z and cot z = tan x, then sin x is equal to: * a) (√5 + 1) / 4 * b) (√5 - 1) / 4 * c) (√5 + 1) / 2 * d) (√5 - 1) / 2 **Answer**: (a) ### Question 3 The value of 2cot2(π/6) + 4tan2(π/6) - 3cosec(π/6) is: * a) 0 * b) 1 * c) √3/2 * d) 1/2 **Answer**: (a) ### Question 4 Find the value of cos(29π/3) * a) 0 * b) 1 * c) √3/2 * d) 1/2 **Answer**: (d) ### Question 5 The expression (tan A)/(1-cot A) + (cot A)/(1- tan A) can be written as: * a) sin A cos A + 1 * b) sec A cosec A + 1 * c) tan A + cot A * d) sec A + cosec A **Answer**: (b) ### Question 6 Which one of the following is not correct? * a) sin x ≤ 1 * b) - 1 ≤ cos x ≤ 1 * c) sec x < 1 * d) cosec x ≥ 1 or cosec x ≤ - 1 **Answer**: (c) ### Question 7 In a ∆PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then ∠R is equal to: * a) 5π/6 * b) π/4 * c) π/3 * d) π/2 **Answer**: (a) ### Question 8 If sin A + sin B + sin C = 3, then cos A + cos B + cos C is equal to: * a) 3 * b) 2 * c) 1 * d) 0 **Answer**: (d) ### Question 9 If cos α + 2 cos β + 3 cos γ= 0, sin α + 2 sin β + 3 sin γ = 0 and α + β + γ = π, then sin 3α +8 sin 3β +27 sin 3γ is equal to: * a) -18 * b) 0 * c) 3 * d) 9 **Answer**: (b) ### Question 10 If sin θ = 2t/(1 + t2) and θ lies in second quadrant , then cos θ is equal to: * a) (1 - t2)/(1 + t2) * b) (t2 - 1)/(1 + t2) * c) -(1 - t2)/(1 + t2) * d)-(t2 - 1)/(1 + t2) **Answer**: (c) ### Question 11 If cos A cos B = 1/√2, π/4 ≤ A ≤ 0 and -π/2 ≤ B ≤ 0, then the value of 2 sin A + 4 sin B is: * a) 4 * b) -20 * c) -4 * d) 0 **Answer**: (a) ### Question 12 The value of cot 54° tan 20° / tan 36° cot 70° + (π/2) is: * a) 0 * b) 2 * c) 3 * d) 1 **Answer**: (d) ### Question 13 In a ∆ABC , if ∠A = π/2, then cos2 B + cos2 C equals: * a) 2 * b) 4 * c) 4/3 * d) 3/4 **Answer**: (b) ### Question 14 If ∆ABC is a right angled at C, then tan A + tan B is equal to: * a) -ac/a² * b) (a + b)/bc * c) a²/bc * d) bc/a² + b² **Answer**: (d) ### Question 15 Which one of the following is possible? * a) sin θ = a² / (a² + b²) * b) sec θ = 2 * c) tan θ = 45 * d) cos θ = 7/3 **Answer**: (b) ### Question 16 3(sin x - cos x) + 6(sin x + cos x)2 + 4(sin x + cos x) is equal to: * a) 12 * b) 13 * c) 14 * d) 11 **Answer**: (a) ### Question 17 If sin A - √6 cos A = √7 cos A, then cos A + √6 sin A is equal to: * a) √6 sin A * b) -√7 sin A * c) √6 cos A * d) √7 cos A * e) √42 cos A **Answer**: (c) ### Question 18 The value of (1 + cos(1 + cos(1 + cos(π/3))) is: * a) 3 * b) 16/6 * c) 3/4 * d) 3/2 **Answer**: (c) ### Question 19 If cos 20° -sin 20° = p, then cos 40° is equal to: * a) p2√2 - p2 * b) p√2 - p2 * c) p + √2 - p2 * d) p -√2 - p2 **Answer**: (b) ### Question 20 If Sn=cosn θ + sinn θ, then the value of (3S4 - 2S2) is given by: * a) 4 * b) 0 * c) 1 * d) -1 **Answer**: (a) ### Question 21 If tan θ = 1. then ((cosec2 θ- sec2 θ) / (cosec2 θ + sec2 θ)) is equal to: * a) 1 * b) 3 * c) -5 * d) 7 **Answer**: (b) ### Question 22 If cos θ = -√3/3 and sin α = -2/5, where θ does not lie and α lies in the third quadrant, then (2 tan α + √3 tan θ)/(cot2 θ + cos α) is equal to: * a) 7/22 * b) 5/22 * c) 9/22 * d) 1 **Answer**: (a) ### Question 23 The value of tan 67° + cot 67° - (1/√2) is: * a) √2 * b) 3/2 * c) 2√2 * d) (2 - √2)/√2 **Answer**: (c) ### Question 24 If x = h + a sec θ and y = k + b cosec θ, then * a) (x + h)2/a2 + (y + k)2/b2 = 1 * b) (x - h)2/a2 + (y - k)2/b2 = 1 * c) (x - h)2/a2 + (y + k)2/b2 = 1 * d) (x - h)2/a2 + (y - k)2/b2 = 1 * e) None of the above **Answer**: (b) ### Question 25 If sec θ = m and tan θ = n, then (1/m + (m + n)/(m + n)) is equal to: * a) 2 * b) 2m * c) 2n * d) mn **Answer**: (a) ### Question 26 The value of cos2 (π/16) + cos2 (3π/16) + cos2 (5π/16) + cos2 (7π/16) is: * a) 2 * b) 1 * c) 0 * d) None of these **Answer**: (a) ## Topic 2: Trigonometrical Ratios of Sum and Difference of Two and Three Angles ### Question 1 If sin θ = sin α, then θ: * a) is any odd multiple of π/2 and θ = α * b) is any odd multiple of π/2 and θ = α * c) is any multiple of π/2 and θ = α * d) is any odd multiple of π/2 and θ is any multiple of π **Answer**: (d) ### Question 2 tan 81° - tan 63° - tan 27° + tan 9° equals: * a) 6 * b) 0 * c) 2 * d) 4 **Answer**: (d) ### Question 3 If sin A + cos A = √2, then the value of cos2 A is: * a) √2 * b) 1/2 * c) 4 * d) -1 **Answer**: (b) ### Question 4 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: * a) (p2 + q2)sin θ / ( p cos θ + q sin θ) * b) √(p² + q²) * c) p2 + q² / (p cos θ + q sin θ) * d) (p2 q2)sin θ / (p cos θ + q sin θ)2 **Answer**: (a) ### Question 5 If 0 < β < α < π/4, cos (α + β) = 4/5 and cos (α - β) = 3/5 then sin 2α is equal to: * a) 0 * b) 1 * c) 2 * d) None of these **Answer**: (b) ### Question 6 If cos A = m cos B and cot (A + B)/2 = (2/m) cot(B - A)/2, then A is equal to: * a) (m + 1)/(m - 1) * b) (m - 1)/(m + 1) * c) (m + 1)/(1 -m) * d) (m - 1)/(1 - m) * e) None of these **Answer**: (d) ### Question 7 If sin θ = 3 sin (θ + 2x), then the value of tan (θ + x) + 2 tan x is: * a) 3 * b) 2 * c) -1 * d) 0 * e) 1 **Answer**: (a) ### Question 8 If α, β, γ ∈ [0, π] and α, β, γ are in AP, then (sin α - sin γ)/ cos γ cos α is equal to: * a) sin β * b) cos β * c) cot β * d) cosec β * e) 2 cos β **Answer**: (c) ### Question 9 The value of tan 40° + tan 20° + √3 tan 20° tan 40° is: * a) √12 * b) -√155/√3 * c) √3 * d) -√155/2 **Answer**: (c) ### Question 10 The value of sin 55° cos 55° / sin 10° is: * a) 1/√2 * b) 1/2 * c) √2 * d) √3/2 **Answer**: (c) ### Question 11 If α, β ∈ [0, π/2), sin α = 4/5 and cos (α + β) = -12/13, then sin β is equal to: * a) 63/65 * b) 61/65 * c) 3/5 * d) 5/13 * e) 8/13 **Answer**: (a) ### Question 12 If cos (α + β) = -√2/5 and sin (α - β) = √3/5, where 0 ≤ α, β ≤ π/4, then tan 2α is equal to: * a) 25/16 * b) 56/33 * c) 19/12 * d) 20/7 **Answer**: (d) ### Question 13 Let A and B denote the statements A: cos α + cos β + cos γ = 0 B: sin α + sin β + sin γ = 0 If cos (β - γ) + cos(γ - α) + cos(α - β) = - 3/2, then: * a) A is correct and B is incorrect * b) A is incorrect and B is correct * c) Both A and B are correct * d) Both A and B are incorrect **Answer**: (a) ### Question 14 The value of sin 50° - sin 70° + sin 10° is: * a) 0 * b) 1 * c) 1/2 * d) 1/√2 **Answer**: (a) ### Question 15 If tan A and tan B are the roots of abx2 - c2 x + ab = 0, where a, b and c are the sides of a ∆ABC, then the value of sin2 A +sin2 B + sin2 C is: * a) 1 * b) 0 * c) 3 * d) 2 * e) 5 **Answer**: (d) ### Question 16 If A + B = 45°, then (cot A - 1)(cot B - 1) is equal to: * a) 3 * b) 1/2 * c) -1 * d) -2 * e) 2 **Answer**: (c) ### Question 17 If tan α = k cot β, then cos (α - β) / cos ( α + β) is equal to: * a) (1 - k)/(1 + k) * b) (1 + k) / (1 - k) * c) (k + 1) / (k - 1) * d) (k - 1)/(k + 1) **Answer**: (d) ### Question 18 If α, β, γ ∈ (0, π/2) then the value of [sin(α + β + γ)/(sin α + sin β + sin γ)] is: * a) < 1 * b) 2 * c) = -1 * d) None of these **Answer**: (d) ### Question 19 The value of cos2 (A - B) + cos2 B - 2 cos (A - B) cos A cos B is: * a) sin A * b) sin2A * c) cos2 A * d) cos A **Answer**: (b) ### Question 20 If cos x / cos (x - 2y) = λ, then tan (x - y) tan y is equal to: * a) (1 + λ) / λ * b) λ / (1 + λ) * c) (1 - λ) / (1 + λ) * d) (1 + λ) / (1 - λ) **Answer**: (c) ### Question 21 sin 47° + sin 61° - sin 11° - sin 25° is equal to: * a) sin 7° * b) cos 7° * c) sin 36° * d) cos 36° **Answer**: (d) ### Question 22 If sin (x + 3α) . = 3 sin ( α - x), then: * a) tan x = tan a * b) tan x . = tan2 a * c) tan2 x = tan2 a * d) tan x = 3 tan a **Answer**: (a) ### Question 23 The value of cos (π/15) cos (2π/15) cos (4π/15) cos (8π/15) is: * a) -1 * b) 1 * c) 1/16 * d) 0 **Answer**: (d) ### Question 24 If A = 35°, B = 15° and C = 40°, then tan A tan B + tan B tan C + tan C tan A is equal to: * a) 0 * b) 1 * c) 2 * d) 3 **Answer**: (d) ### Question 25 If α + β + γ = 20°, then cos α + cos (θ - α) + cos (θ - β) + cos (θ - γ) is equal to: * a) sin (α/2) cos (β/2) cos (γ/2) * b) 4 cos (α/2) cos (β/2) cos (γ/2) * c) 4 sin (α/2) sin (β/2) sin (γ/2) * d) 4 sin (α/2) cos (β/2) cos (γ/2) **Answer**: (b) ### Question 26 If cos x = 3 cos y, then 2 tan(y - x)/4 is equal to: * a) cot(y/2) * b) cot(x/2) * c) cot (y-x)/2 * d) cot(y + x)/2 **Answer**: (d) ### Question 27 If sin A + cos B = a and sin B + cos A = b, then sin (A + B) is equal to: * a) a2 + b2 / 2 * b) (a2 - b2 + 2) / 2 * c) a2 + b2 - 2 / 2 * d) None of these **Answer**: (a) ### Question 28 If sin A = 1/√10 and sin B = 1/√5, where A and B are positive acute angles, then A + B is equal to: * a) π * b) π/2 * c) π/3 * d) π/4 **Answer**: (c) ### Question 29 If cos (θ + α) = a and cos (θ - β) = b, then sin2 (α - β) + 2ab cos (α - β) is equal to: * a) a2 + b2 * b) a2 - b2 * c) b2 - a2 * d) - a2 - b2 **Answer**: (a) ### Question 30 If tan A = 2 tan B + cot B, then 2 tan (A - B) is equal to: * a) tan B * b) 2 tan B * c) cot B * d) 2 cot B **Answer**: (b) ### Question 31 If cos (A - B) = 3/5 and tan A tan B = 2, then which one of the following is correct? * a) sin (A + B) = 1/5 * b) sin (A + B) = -1/5 * c) cos (A - B) = 1/5 * d) cos (A + B) = -1/5 **Answer**: (a) ### Question 32 If sin A + sin B = √3(cos B - cos A), then sin 3A + sin 3B is equal to: * a) 0 * b) 2 * c) 1 * d) -1 **Answer**: (b) ### Question 33 If tan 80° - tan 10° / tan 70° is equal to: * a) 0 * b) 1 * c) 2 * d) 3 **Answer**: (d) ### Question 34 The value of cos2 (π/4 + θ). - sin2 (- π/4 - θ) is: * a) 0 * b) cos 2θ * c) sin 2θ * d) cos θ **Answer**: (b) ### Question 35 If sin (x + y) / (a + b) = sin (x - y) , then x / tan y is equal to: * a) (a2 - b2)/ b * b) a2 / b2 * c) (a2 + b2)/ b * d) (a2- b2)/ a2 +b2 * e) (a2 - b2)/(a2 + b2) **Answer**: (a) ### Question 36 If tan α = m / (m + 1) and tan β = 1 / (2m + 1), then α + β is equal to: * a) 3π/2 * b) π/2 * c) 0 * d) π/4 **Answer**: (d) ### Question 37 If x = tan 15°, y = cosec 75° and z = 4 sin 18°, then: * a) x < y < z * b) y < z < x * c) z < x < y * d) x < z < y **Answer**: (a) ### Question 38 sin 120° cos 150° - cos 240° sin 30° is equal to: * a) 1 * b) -1 * c) 2 * d) -(√3 + 1)/3 **Answer**: (c) ### Question 39 If p = cos 55°, q = cos 65° and r = cos 175°, then the value of pq + r/pq is: * a) 0 * b) -1 * c) 1 * d) None of these **Answer**: (a) ### Question 40 If tan (3π/4 + A)= a, then sin 3A/sin A is equal to: * a) 2a / (a + 1) * b) 2a / (a - 1) * c) a / (a + 1) * d) a / (a - 1) **Answer**: (a) ### Question 41 For any angle θ, the expression (2cos θ + 1)/ (2 cos 2θ + 1) (2 cos 4θ + 1) is equal to: * a) (2 cos θ + 1) (2 cos 2θ + 1) (2 cos 4θ + 1) * b) (cos θ - 1) (cos 2θ - 1) (cos 4θ - 1) * c) (2 cos θ - 1) (2 cos 2θ - 1) (2 cos 4θ - 1) * d) (2 cos θ + 1)(2 cos 2θ + 1)(cos 4θ + 1) **Answer**: (a) ## Topic 4: Maximum Value, Minimum Value and Conditional Trigonometrical Identities ### Question 1 Let n be a positive integer, such that sin (π/2n)+ cos (π/2n) = √2/2, then: * a) 2 ≤ n ≤ 8 * b) 3 ≤ n ≤ 8 * c) 0 ≤ n ≤ 8 * d) 4 ≤ n ≤ 8 **Answer**: (a) ### Question 2 The least value of 3 sin2 θ + 4 cos2 θ is: * a) 2 * b) 3 * c) 0 * d) 1 **Answer**: (d) ### Question 3 If A = sin2 x + cos4 x, then for all real x: * a) -13/16 ≤ A ≤ 1 * b) 1 ≤ A ≤ 2 * c) 3/4 ≤ A ≤ 13/16 * d) 1/4 ≤ A ≤ 1 **Answer**: (d) ### Question 4 The maximum value of f(x) = sin x (1 + cos x) is: * a) 3√3/4 * b) 3√3/2 * c) 3√3 * d) √3 **Answer**: (a) ### Question 5 The minimum value of 27cos2 x - 81 sin2 x is: * a) - 5 * b) -5/12 * c) -1/243 * d) 1/27 **Answer**: (a) ### Question 6 The smallest value of 5 cos θ + 12 is: * a) 5 * b) 12 * c) 7 * d) 17 **Answer**: (a) ### Question 7 The value of 1 + cos 56° + cos 58°cos 66° is: * a) 4 cos 28° cos 29° sin 33° * b) cos 28° cos 29° sin 33° * c) 4 cos 28° sin 29° cos 33° * d) 4 cos 28° sin 29° sin 33° **Answer**: (a) ### Question 8 The maximum value of 3 cos x + 4 sin x + 5 is: * a) 5 * b) 6 * c) 7 * d) None of these **Answer**: (c) ### Question 9 If P = sin2 θ + cos2 θ, then: * a) 1/2 ≤ P ≤ 3/2 * b) P2/2 ≤ P ≤ √13/6 * c) 2 ≤ P ≤ 3 * d) -√13/6 ≤ P ≤ √13/6 **Answer**: (a) ### Question 10 In ∆ABC, if sin2 A + sin2 B + sin2 C = 2, then the triangle is: * a) right angled but need not be isosceles * b) right angled and isosceles * c) isosceles but need not be right angled * d) equilateral **Answer**: (b) ### Question 11 The maximum value of cos2(-x)-cos2 (+x) is: * a) √3/2 * b) 1 * c) √3 * d) (√3 + 1)/√3 **Answer**: (a) ### Question 12 If A = {x: x2/6 ≤ x < π/2} and f(x) = cos x - x(1 + x), then f(A) is equal to: * a) π / 6 - 1 * b) π/3 - 1 * c) π / 6 + 1 / 2 * d) π/3 - √3/2 - 1 * e) π / 3 - √3/6 -1/6 **Answer**: (e) ### Question 13 The maximum value of sin(x + 1) + cos (x + π/6) in the interval (0, π/2) is attained at: * a) x = π/12 * b) x = π/6 * c) x = π/3 * d) x = π/2 **Answer**: (a) ### Question 14 The maximum value of 5 cos θ + 3 cos (θ + π/3) + 3 is: * a) 5 * b) 11 * c) 10 * d) -1 **Answer**: (b) ### Question 15 If 5 cos x + 12 cos y = 13, then the maximum value of 5 sin x + 12 sin y is: * a) 12 * b) √120 * c) √20 * d) 13 **Answer**: (b) ### Question 16 If f: R -> S defined by f(x) = sin x - √3 cos x + 1 is onto, then the interval of S is: * a) [0, 3] * b) [-1, 1] * c) [0, 1] * d) [-1, 3] **Answer**: (d) ### Question 17 If A + B + C = 180°, then ∑ tan(B/2)- tan (A/2) is: * a) 0 * b) 1 * c) 2 * d) 3 **Answer**: (a) ### Question 18 If α + β - γ = π, then sin2 α + sin2 β - sin2 γ is equal to: * a) 2 sin α sin β cos γ * b) 2 cos α cos β cos γ * c) 2 sin α sin β sin γ * d) None of the above **Answer**: (a) ### Question 19 The maximum value of 4 sin2 x - 12 sin x +7 is: * a) 25 * b) 4 * c) does not exist * d) None of these **Answer**: (a) ### Question 20 If y = cos2 x + sec2 x, then: * a) y ≤ 2 * b) y ≤ 1 * c) y ≥ 2 * d) 1 ≤ y ≤ 2 **Answer**: (c) ### Question 21 If A + B + C = π, then sin 2A + sin 2B + sin 2C is equal to: * a) 4 sin A sin B sin C * b) 4 cos A cos B cos C * c) 2 cos A cos B cos C * d) 2 sin A sin B sin C **Answer**: (d) ### Question 22 The minimum value of f(x) = sin x + cos x, 0 ≤ x ≤ π/2 is: * a) √2 / 2 * b) 1 / √2 * c) 1 * d) 1/2 **Answer**: (b) ### Question 23 The maximum value of the function 3 cos x - 4 sin x is: * a) 2 * b) 3 * c) 4 * d) 5 **Answer**: (d) ### Question 24 For all values of θ, the value of 3 - cos θ + cos (θ/2) lies in the interval: * a) [2, 3] * b) [-2, 1] * c) [2, 4] * d) [1, 5] **Answer**: (d) ### Question 25 The maximum value of sin θ + cos θ in [0, π/2] is: * a) √2 * b) 2 * c) 0 * d) -√2 **Answer**: (a) ### Question 26 If A + B + C = π, then cos2A + cos2B + cos2C -2 cos A cos B cos C is equal to: * a) 1 * b) 2 * c) 0 * d) 3 **Answer**: (a) ### Question 27 The maximum value of 4 sin2x + 3 cos2x is * a) 4

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