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611 questions with worked solutions.
Q1. Find the period of the function |sin³(x/2)| + |cos⁵(x/5)|.
Answer: 10π
The period of the function is determined by the individual periods of its components. The period of | sin³(x/2) is 4 and the period of | cos⁵(x/5) is 10, making the least common multiple 10.
Q2. For which values of a does the equation sin⁴ x + cos⁴ x = a admit at least one solution?
Answer: when 1/2 ≤ a ≤ 1
The expression sin⁴ x + cos⁴ x can be rewritten using the identity sin² x + cos² x = 1, leading to a maximum value of 1 and a minimum value of 1/2. Therefore, the equation admits solutions only for values of a within the range 1/2 to 1.
Answer: All of these
Using tan(theta/2)(1+sec theta) = tan theta repeatedly, the telescoping product gives f_n(theta) = tan(2^n theta). Then f_2(pi/16) = tan(pi/4) = 1, f_3(pi/32) = tan(pi/4) = 1, and f_4(pi/64) = tan(pi/4) = 1, so all of these are true.
Answer: 2cos x
Using cos terms with binomial coefficients, numerator = 2^? form reduces so that the ratio simplifies to 2cos x (verified numerically, e.g. at x=0.4 the ratio = 1.8421 = 2cos0.4). The expression equals 2cos x.
Q5. For angles α, β, γ all lying in the interval (0, π/2), the value of (sin(α+β+γ))/(sinα+sinβ+sinγ) is
Answer: less than 1
Using sin(a+b) < sin a + sin b for acute angles repeatedly, sin(a+b+g) is strictly less than sin a + sin b + sin g, so the ratio is always less than 1.
Q6. Evaluate the product (1 + cos(π/10))(1 + cos(3π/10))(1 + cos(7π/10))(1 + cos(9π/10)).
Answer: 1/16
Pair the terms: cos(7π/10)=-cos(3π/10) and cos(9π/10)=-cos(π/10). The product becomes (1-cos^2(π/10))(1-cos^2(3π/10))=sin^2(π/10)sin^2(3π/10). Using sin(π/10)sin(3π/10)=1/4 gives (1/4)^2=1/16.
Q7. If sin A - √(6)cos A = √(7)cos A, then the value of cos A + √(6)sin A is:
Answer: √(7)sin A
The equation can be rearranged to isolate terms involving sin A and cos A, leading to a relationship that allows us to express cos A + sqrt{6} sin A in terms of sin A. By substituting and simplifying, we find that it equals sqrt{7} sin A, confirming option B as the correct answer.
Q8. The general solution of the trigonometric equation (√3 - 1) sin θ + (√3 + 1) cos θ = 2 is
Answer: None
The equation does not simplify to a standard form that matches any of the provided options, indicating that none of the choices accurately represent the general solution.
Answer: π/2
The equations given can be manipulated to find the relationships between angles A and B. By solving these equations, we find that A and B satisfy specific trigonometric identities that lead to the conclusion that A + 2B equals π/2.
Q10. What are the maximum and minimum values of the expression sin x cos x?
Answer: 1/2, -1/2
sin x cos x = (1/2) sin 2x, and sin 2x ranges from -1 to 1, so the expression ranges from -1/2 to 1/2. Maximum is 1/2 and minimum is -1/2.
Q11. If tan(cot x)=cot(tan x), then which of the following is true?
Answer: sin 2x=(4)/((2n+1)π)
tan(cot x) = cot(tan x) = tan(π/2 - tan x), so cot x = π/2 - tan x + nπ, giving cot x + tan x = (2n+1)π/2. Since cot x + tan x = 2/sin2x, we get sin2x = 4/((2n+1)π).
Q12. If sin θ = 1/2 √(x/y) + √(y/x), then it must follow that:
Answer: x is equal to y
By AM-GM, sqrt(x/y)+sqrt(y/x) >= 2, so (1/2)(sqrt(x/y)+sqrt(y/x)) >= 1. Since sin theta <= 1, equality must hold, which requires sqrt(x/y)=sqrt(y/x), i.e. x = y.
Q13. If f(x) = cos(log x), then the value of f(x)f(y) - 1/2 [f(x/y) + f(xy)] is:
Answer: 0
The expression simplifies to zero due to the properties of the cosine function and logarithms, specifically using the identity for cosine of sums and the fact that the cosine function is even, leading to cancellation in the terms.
Answer: Statement-1 is true and Statement-2 is true, and Statement-2 correctly explains Statement-1
The function is defined as a rational function where the denominator can take values that approach zero, leading to unbounded behavior. Since the function can become infinitely large or small, it does not attain maximum or minimum values, making Statement-1 true and Statement-2 true, with the latter explaining the former.
Q15. Evaluate the product sin 12° · sin 24° · sin 48° · sin 84°.
Answer: cos 20° · cos 40° · cos 60° · cos 80°
sin12 sin24 sin48 sin84 = 1/16 numerically. The product cos20 cos40 cos60 cos80 also equals 1/16, while sin20 sin40 sin60 sin80 = 3/16. Hence the equal expression is cos20 cos40 cos60 cos80.
Answer: [-1,1]
Since sqrt(1+tan^2 x)=1/|cos x| and sqrt(1+cot^2 x)=1/|sin x|, f(x)=sin x|cos x| - cos x|sin x|. In quadrant II this equals -2 sin x cos x in (0,1], in quadrant IV it is in [-1,0), and 0 at the axes, so the value set is [-1, 1].
Q17. If (sin(x+y))/(sin(x-y)) = (a+b)/(a-b), then the value of (tan x)/(tan y) is
Answer: (a)/(b)
By componendo and dividendo, [sin(x+y)+sin(x-y)]/[sin(x+y)-sin(x-y)] = (a+b+a-b)/(a+b-a+b) = a/b. The left side is (2 sinx cosy)/(2 cosx siny) = tanx/tany, so tanx/tany = a/b.
Answer: Statement-1 is correct, Statement-2 is correct; Statement-2 does not correctly explain Statement-1
Writing a cosx + b sinx = R cos(x - phi), the two roots are phi +/- t, so their average is phi, independent of c; Statement-1 is true. Statement-2 (solution exists iff |c| <= sqrt(a^2+b^2)) is also true, but it concerns existence, not the sum of roots, so it does not explain Statement-1.
Q19. Evaluate the expression tan² θ · sec² θ · (cot² θ − cos² θ).
Answer: 1
The expression simplifies to tan² θ · sec² θ · (cot² θ − cos² θ), where cot² θ can be rewritten as 1/tan² θ. This leads to a cancellation that ultimately results in a value of 1 when evaluated.
Q20. If cosθ + cos 2θ + cos 3θ = 0, then the general solution for θ is:
Answer: θ = 2mπ ± (2π)/(3)
cosT+cos2T+cos3T = cos2T(2cosT+1) = 0. The branch 2cosT+1=0 gives cosT=-1/2, i.e. T = 2m*pi +/- 2pi/3. (Option D, T=m*pi+(-1)^m*pi/3, fails since at T=60 deg the sum equals -1, not 0.)
Answer: π/12
The expression sin(x + π/6) + cos(x + π/6) can be maximized by recognizing that it is equivalent to the sine of a shifted angle. The maximum occurs when the angle x + π/6 is equal to π/4, which corresponds to x = π/12, thus yielding the highest value in the given interval.
Answer: 2√(1+x)
The four least positive angles with sine x are alpha, pi-alpha, 2pi+alpha, 3pi-alpha (increasing). Substituting their half-angles into 4 sin(a/2)+3 sin(b/2)+2 sin(g/2)+sin(d/2) and simplifying gives 2 sqrt(1+x), confirmed numerically for several x.
Answer: (0, π/2)
For (3,2): 3+2-1 = 4 > 0, so need sin t + cos t > 1, i.e. sqrt(2) sin(t + pi/4) > 1. This holds for t + pi/4 in (pi/4, 3pi/4), giving t in (0, pi/2).
Answer: 1
f(x) = sin^2 x + sin^2(x+pi/3) + cos x cos(x+pi/3) simplifies to the constant 5/4 for all x. Hence gof(x) = g(f(x)) = g(5/4) = 1.
Answer: ((n-1)d)/(1+a₁aₙ)
The correct option is derived from the properties of the tangent addition formula and the structure of the terms in the series. Each term in the sum represents an angle whose tangent can be expressed in terms of the arithmetic progression, leading to a cumulative result that simplifies to ((n-1)d)/(1+a₁aₙ), reflecting the relationship between the first and last terms of the sequence.
Q26. In triangle ABC, if angle A = tan⁻¹(2) and angle B = tan⁻¹(3), then angle C equals
Answer: π/4
tan^-1 2 + tan^-1 3 = 3pi/4 (since the tangent sum (2+3)/(1-6) = -1 and both angles lie in the first quadrant). Then C = pi - A - B = pi - 3pi/4 = pi/4.
Answer: 1
The maximum value of the function occurs when the numerator, sin 2x, reaches its peak of 1, while the denominator, sin(x + π/4), remains positive and does not exceed 1 in the given interval, ensuring that the overall value does not exceed 1.
Answer: 15x² + 8x − 16 = 0
From 5cosA+3=0, cosA=-3/5; since A is a triangle angle it is obtuse, sinA=4/5 and tanA=-4/3. Sum of roots = -8/15, product = -16/15, giving 15x^2+8x-16=0.
Answer: None of these
In triangle PAB the angles give PB=AB=200, and in triangle PBC the sine rule gives sin3a=2sin a, so a=30deg. Then h=200 sin(2a)=200 sin60=100*sqrt(3) m (about 173 m), which is not among the listed values, so 'None of these'.
Answer: 2
Using A+B+C=pi, sin^2A+sin^2B+sin^2C = 2 + 2cosA cosB cosC, so sin^2A+sin^2B+sin^2C - 2cosA cosB cosC = 2.
Q31. In triangle ABC, if 2b² = a² + c², then the value of sin 3B divided by sin B is
Answer: ((c² − a²)/(2ca))²
sin3B/sinB=3-4sin^2 B=4cos^2 B-1. With 2b^2=a^2+c^2, cosB=(a^2+c^2-b^2)/(2ac)=(a^2+c^2)/(4ac). Then 4cos^2 B-1 simplifies to ((c^2-a^2)/(2ca))^2.
Answer: 109.28 m
Using trigonometric relationships, the height of the tower and the angles of elevation allow us to calculate the distances from each observer to the base of the tower. By applying the tangent function for both angles and solving the resulting equations, we find the total distance between the two observers to be approximately 109.28 m.
Answer: 20 m and 60 m
The correct option is right because the lower segment of the pole is one-third of the total height, and using the tangent of the angle from the observation point, we can derive two possible total heights that satisfy the given conditions, which are 20 m and 60 m.
Q34. For an equilateral triangle, what is the ratio of the inradius, circumradius, and one exradius?
Answer: 1: 2: 3
In an equilateral triangle, the inradius is half the circumradius, and the exradius is one and a half times the inradius, leading to the ratio of 1: 2: 3 for the inradius, circumradius, and one exradius respectively.
Answer: 1.5 min
The angle of depression indicates the observer's line of sight to the car, and as the angle increases from 30° to 60°, the car is getting closer to the tree. By using trigonometric relationships, we can determine the distances covered by the car during the initial time and the subsequent time, leading to the conclusion that it will take an additional 1.5 minutes to reach the base of the tree.
Answer: 52 m
Shadow at 30deg is h*cot30 = h*sqrt(3); at 60deg it is h*cot60 = h/sqrt(3). The decrease is h*sqrt(3) - h/sqrt(3) = h*(2/sqrt(3)) = 60, so h = 30*sqrt(3) ~= 52 m.
Q37. In a triangle ABC, if cos A = (sin B)/(2sin C), then which of the following is true?
Answer: c = a
The equation ( rac{ ext{sin} B}{2 ext{sin} C} = ext{cos} A) implies a specific relationship between the angles and sides of triangle ABC, leading to the conclusion that sides a and c must be equal, thus confirming that c = a.
Answer: c
OH parallel to BC requires equal heights above BC: R cosA = 2R cosB cosC, which simplifies to tanB tanC = 3 (NOT that tanA,tanB,tanC are in AP), so Statement 1 is false. Statement 2 is true: if tanA,tanB,tanC are in AP then tanA+tanB+tanC = 3tanB, and since this sum equals tanA tanB tanC, we get tanA tanC = 3. Hence Statement 1 false, Statement 2 true = option (c).
Q39. In triangle ABC, if angle C is an obtuse angle, which of the following is true?
Answer: tan A × tan B < 1
In a triangle with an obtuse angle, the sum of the angles A and B must be less than 90 degrees. This leads to the product of their tangents being less than 1, as the tangent function is positive and increasing in the first quadrant.
Answer: b((a²+b²)/(a²-b²))
The correct option is derived from the relationship between the angles subtended by the tower and the pole at point O. By applying the tangent function to the angles and using the properties of similar triangles, we can express the height of the pole in terms of the height of the tower and the distances involved, leading to the formula b((a²+b²)/(a²-b²)}.
Answer: 1/√6
The ratio of the sines of the angles ∠BAD and ∠CAD can be determined using the Law of Sines and the internal division of segment BC by point D. Given the angles of triangle ABC, the sine values can be calculated, leading to the conclusion that sin∠BAD/sin∠CAD equals 1/√6.
Answer: 2a² = 3h²
The correct option relates the height of the tower to the side length of the triangle through trigonometric relationships in an equilateral triangle, where the angles formed from the top of the tower to the triangle's sides lead to the derived equation, confirming that the height is proportional to the square of the side length.
Q43. In a right-angled triangle ABC with ∠C = 90°, the value of (a² - b²)/(a² + b²) is:
Answer: sin(A - B)
The expression (a² - b²)/(a² + b²) simplifies to sin(A - B) due to the relationships between the sides of the triangle and the angles, where a and b are the lengths of the sides opposite angles A and B respectively.
Answer: 2.05 sin 38° / sin 42°
The correct option uses the relationship between the lengths of the shadow and the pole, factoring in the angles of elevation and tilt. By applying the sine rule, we can relate the height of the pole to the length of the shadow and the angles involved, leading to the formula provided in option A.
Answer: K ≤ (1)/(3√(3))
The inequality holds because the product of the cotangents of angles that sum to π is maximized under the constraint of the angles being acute, leading to the conclusion that their product cannot exceed rac{1}{3 ext{√}3}.
Answer: 20(3+√3) m
The balloon descends at a constant rate, and the change in angle of elevation indicates a change in height and horizontal distance. By calculating the height after 10 minutes and using trigonometric relationships, we find the horizontal distance from the observer when the balloon touches the ground, which corresponds to the correct option.
Answer: Statement-1 is true, Statement-2 is true; Statement-2 correctly explains Statement-1
For a right triangle the hypotenuse a=2R, so a^2=4R^2 and b^2+c^2=a^2=4R^2, giving (a^2+b^2+c^2)/R^2 = 8R^2/R^2 = 8. Statement-1 is true only because a^2=b^2+c^2 (Statement-2) holds, so both are true and S2 correctly explains S1.
Answer: a sin(α + β) metre / sin(β − α)
The correct option is derived from the relationship between the angles of elevation and depression, which involves trigonometric functions. The height of the cloud can be expressed using the sine of the angles, leading to the formula that incorporates both angles in the correct manner.
Answer: √3/(√3+2)
The angles of the triangle are in the ratio 4:1:1, which means the angles measure 4x, x, and x. The longest side, opposite the largest angle, is proportional to the sine of that angle, leading to the ratio of the longest side to the perimeter being √3/(√3+2) when calculated using the sine rule.
Answer: 1/2
The equation e^(cos² x + cos⁴ x + cos⁶ x +...) simplifies to a geometric series, which converges to a specific value. The roots of the quadratic equation t² - 9t + 8 = 0 are t = 1 and t = 8, and substituting these values leads to the correct trigonometric ratio, yielding the final answer of 1/2.