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

If (1)/(6) θ, θ and θ are in geometric progression, then the...

If 16sin⁡θ,cos⁡θ\frac{1}{6} \sin \theta, \cos \theta and tan⁡θ\tan \theta are in geometric progression, then the solution set of θ\theta is

Options

  1. A2nπ±(π6)2 n \pi \pm \left(\frac{\pi}{6}\right)
  2. B2nπ±(π3)2 n \pi \pm \left(\frac{\pi}{3}\right)
  3. Cn π+\pi+1(-1)^n \left(\frac{\pi}{3}\right)
  4. Dnπ±(π3)n \pi \pm \left(\frac{\pi}{3}\right)

Correct answer

B. 2nπ±(π3)2 n \pi \pm \left(\frac{\pi}{3}\right)

Step-by-step solution

Given that, terms 16sin⁡θ,cos⁡θ\frac{1}{6} \sin \theta, \cos \theta and tan⁡θ\tan \theta are in GP. ∴cos⁡2θ=16sin⁡θ×tan⁡θ⇒cos⁡2θ=sin⁡2θ6cos⁡θ⇒6cos⁡3θ−sin⁡2θ=0⇒6cos⁡3θ−1+cos⁡2θ=0\begin{aligned} & \therefore \quad \cos ^2 \theta=\frac{1}{6} \sin \theta \times \tan \theta \\ & \Rightarrow \quad \cos ^2 \theta=\frac{\sin ^2 \theta}{6 \cos \theta} \\ & \Rightarrow \quad 6 \cos ^3 \theta-\sin ^2 \theta=0 \\ & \Rightarrow \quad 6 \cos ^3 \theta-1+\cos ^2 \theta=0 \end{aligned}⇒6cos⁡3θ+1+cos⁡2θ=0⇒6cos⁡3θ+cos⁡2θ−1=0⇒6cos⁡3θ+4cos⁡2θ−3cos⁡2θ+2cos⁡θ−2cos⁡θ−1=0⇒6cos⁡3θ−3cos⁡2θ+4cos⁡2θ−2cos⁡θ+2cos⁡θ−1=0⇒3cos⁡2θ(2cos⁡θ−1)+2cos⁡θ(2cos⁡θ−1)+1(2cos⁡θ−1)=0⇒(2cos⁡θ−1)(3cos⁡2θ+2cos⁡θ+1)=0\begin{aligned} & \Rightarrow \quad 6 \cos ^3 \theta+1+\cos ^2 \theta=0 \\ & \Rightarrow \quad 6 \cos ^3 \theta+\cos ^2 \theta-1=0 \\ & \Rightarrow \quad 6 \cos ^3 \theta+4 \cos ^2 \theta-3 \cos ^2 \theta+2 \cos \theta-2 \cos \theta-1=0 \\ & \Rightarrow \quad 6 \cos ^3 \theta-3 \cos ^2 \theta+4 \cos ^2 \theta-2 \cos \theta+2 \cos \theta-1=0 \\ & \Rightarrow \quad 3 \cos ^2 \theta(2 \cos \theta-1)+2 \cos \theta(2 \cos \theta-1)+1(2 \cos \theta-1) \\ & \quad=0 \\ & \Rightarrow \quad(2 \cos \theta-1)\left(3 \cos ^2 \theta+2 \cos \theta+1\right)=0 \end{aligned}ForFor3 \cos ^2 \theta+2 \cos \theta+1=0valueofvalue of\cos \thetawillbeimaginarysoonlywill be imaginary so only2 \cos \theta-1=0willbeconsideredtofindwill be considered to find\cos \theta..⇒2cos⁡θ−1=0⇒cos⁡θ=12⇒cos⁡θ=cos⁡π3∴θ=2nπ±π3\begin{aligned} & \Rightarrow \quad 2 \cos \theta-1=0 \Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \cos \theta=\cos \frac{\pi}{3} \\ & \therefore \quad \theta=2 \mathrm{n} \pi \pm \frac{\pi}{3} \end{aligned}$
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