NTA Abhyas JEE Main2020MathematicsTrigonometric EquationsPractice
If A = θ : 2 cos 2 θ + sin θ ≤ 2 and B = θ : π 2 ≤ θ ≤ 3 π 2 , then A ∩ B is equal to
Options
- Aθ : π 2 ≤ θ ≤ 5 π 6
- Bθ : π ≤ θ ≤ 3 π 2
- Cθ : π 2 ≤ θ ≤ 5 π 6 or θ : π ≤ θ ≤ 3 π 2
- Dθ : 3 π 4 ≤ θ ≤ 7 π 4
Correct answer
C. θ : π 2 ≤ θ ≤ 5 π 6 or θ : π ≤ θ ≤ 3 π 2
Step-by-step solution
2 - 2 sin 2 ⁡ θ + sin ⁡ θ ≤ 2 2 sin 2 ⁡ θ - sin ⁡ θ ≥ 0 sin ⁡ θ 2 sin ⁡ θ - 1 ≥ 0 Case I: sin ⁡ θ ≥ 0 ,   sin ⁡ θ ≥ 1 2 ⇒ sin ⁡ θ ≥ 1 2 ⇒ θ ∈ π 2 , 5 π 6 Case II: sin ⁡ θ ≤ 0 ,   sin ⁡ θ ≤ 1 2 ⇒ sin ⁡ θ ≤ 0 ⇒ θ ∈ π , 3 π 2 ⇒ A ∩ B = θ : `