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

Question

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, θ : π 2 ≤ θ ≤ 5 π 6, θ : π ≤ θ ≤ 3 π 2, θ : π 2 ≤ θ ≤ 5 π 6 or θ : π ≤ θ ≤ 3 π 2, θ : 3 π 4 ≤ θ ≤ 7 π 4

MARKS App · Accepted Answer

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 : π 2 ≤ θ ≤ 5 π 6 or θ : π ≤ θ ≤ 3 π 2

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