Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Main · Mathematics · Trigonometric Equations

The complete interval of values of x in (- (π)/(2), (π)/(2))...

The complete interval of values of xx in (−π2,π2)\left(- \dfrac{π}{2}, \dfrac{π}{2}\right) satisfying the equations~cos⁡x⋅cos⁡2x>−14\cos x \cdot \cos 2x > - \dfrac{1}{4} and 3cos⁡x>1,3\cos x > 1, is

Options

  1. A(0,π3)\left(0, \dfrac{π}{3}\right)
  2. B(−π2,π4)\left(- \dfrac{π}{2}, \dfrac{π}{4}\right)
  3. C(−π3,π3)\left(- \dfrac{π}{3}, \dfrac{π}{3}\right)
  4. D(−π3,5π12)\left(- \dfrac{π}{3}, \dfrac{5\pi}{12}\right)

Correct answer

C. (−π3,π3)\left(- \dfrac{π}{3}, \dfrac{π}{3}\right)

Step-by-step solution

4cos⁡x⋅cos⁡2x+1>04\cos x \cdot \cos 2x + 1 > 0
4cos⁡x(2cos⁡2x−1)+1>04\cos x\left(2\cos^{2}x - 1\right) + 1 > 0
8cos⁡3x−4cos⁡x+1>08\cos^{3}x - 4\cos x + 1 > 0
8cos⁡2x(cos⁡x−12)+4cos⁡x(cos⁡x−12)−2(cos⁡x−12)>08\cos^{2}x\left(\cos x - \dfrac{1}{2}\right) + 4\cos x\left(\cos x - \dfrac{1}{2}\right) - 2\left(\cos x - \dfrac{1}{2}\right) > 0
Let cos⁡x=t\cos x = t
\left(t - \dfrac{1}{2}\right)\left(8t^{2} + 4t - 2\right) > 0<br/>(t−12)(4t2+2t−1)>0<br />\left(t - \dfrac{1}{2}\right)\left(4t^{2} + 2t - 1\right) > 0
t=−1±54t = \dfrac{- 1 \pm \sqrt{5}}{4}
∴cos⁡x>13∴1≥cos⁡x>12∴ \cos x > \dfrac{1}{3} ∴ 1 \ge \cos x > \dfrac{1}{2}
∴x∈(−π3,π3)∴ x \in \left(\dfrac{- \pi}{3}, \dfrac{π}{3}\right)
Practice Trigonometric Equations on Quantrex Academy →

More from Trigonometric Equations

Match each entry in List-I to the correct entry in List-II and choose the correct option.… JEE Advanced 2026Passage: Consider the curve C_1 given by y = e^ -x for x ∈ [0, 10π] , and the curve C_2… JEE Advanced 2026Let S = \ θ ∈ (-2π, 2π) : θ + 1 = 3 θ\ . Then ∑_ θ ∈ S θ is equal to: JEE Main 2026If S = \ θ ∈ [-π, π] : θ (5θ)/(2) = 7θ (7θ)/(2)\ , then n(S) is equal to _______. JEE Main 2026Let S = \ x ∈ [-π, π] : x ( x + x) = a, a ∈ ℤ\ . Then n(S) is equal to : JEE Main 2026The number of elements in the set \ x ∈[0,180^ ∘ ]: (x+100^ ∘ )= (x+50^ ∘ ) x (x-50^ ∘ )\… JEE Main 2026Number of solutions of 3 2 θ+8 θ+3 3 =0, θ ∈[-3 π, 2 π] is: JEE Main 2026The number of solutions of the equation 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3 (5 θ)/(2) in… JEE Main 2025If for θ ∈[-(π)/(3), 0] , the points (x, y)=(3 (θ+(π)/(3)), 2 (θ+(π)/(6))) lie on x y+α… JEE Main 2025If 10 ^4 θ+15 ^4 θ=6 , then the value of 27 cosec^6 θ+8 ^6 θ 16 ^8 θ is: JEE Main 2025The number of solutions of equation (4- 3 ) x -2 3 ^2 x=- 4 1+ 3 , x ∈[-2 π, (5 π)/(2)] is JEE Main 2025The number of solutions of the equation 2 x+3 x=π, x ∈[-2 π, 2 π]-\ ± (π)/(2), ± (3… JEE Main 2025If θ ∈[-(7 π)/(6), (4 π)/(3)] , then the number of solutions of 3 cosec^2 θ-2( 3 -1)… JEE Main 2025If θ ∈[-2 π, 2 π] , then the number of solutions of 2 2 ^2 θ+(2- 6 ) θ- 3 =0 , is equal… JEE Main 2025The sum of all values of θ ∈[0,2 π] satisfying 2 ^2 θ= 2 θ and 2 ^2 θ=3 θ is JEE Main 2025Let f: ℝ arrow ℝ be a function defined by f(x)=\ array cc x^2 ((π)/(x^2)), & if x ≠ 0 0… JEE Advanced 2024 Full Trigonometric Equations list Trigonometric Equations JEE Maths PYQsJEE Mathematics PYQs All 99 Percentile Qs Bank for JEE Main PYQs