Find the acute angle θ such 2cos2 θ = 3sin θ.

1.	Find the acute angle θ such 2cos2 θ = 3sin θ.
Answer» 2cos² 0 = 3sin θ ∴ 2(1 – sin² θ) = 3sin θ ∴ 2 – 2sin² θ = 3sin θ ∴ 2sin² θ + 3sin 9-2 = θ ∴ 2sin² θ + 4sin θ – sin θ – 2 = θ ∴ 2sin θ(sin θ + 2) -1 (sin θ + 2) = θ ∴ (sin θ + 2) (2sin θ – 1) = 0 ∴ sin θ + 2 = 0 or 2sin θ – 1 = 0 ∴ sin θ = -2 or sin θ = 1/2 Since, -1 ≤ sin θ ≤ 1 ∴ Sin θ = 1/2 ∴ θ = 30° …[ ∵ sin 30 = 1/2]

Answer»

2cos² 0 = 3sin θ

∴ 2(1 – sin² θ) = 3sin θ

∴ 2 – 2sin² θ = 3sin θ

∴ 2sin² θ + 3sin 9-2 = θ

∴ 2sin² θ + 4sin θ – sin θ – 2 = θ

∴ 2sin θ(sin θ + 2) -1 (sin θ + 2) = θ

∴ (sin θ + 2) (2sin θ – 1) = 0

∴ sin θ + 2 = 0 or 2sin θ – 1 = 0

∴ sin θ = -2 or sin θ = 1/2

Since, -1 ≤ sin θ ≤ 1

∴ Sin θ = 1/2

∴ θ = 30° …[ ∵ sin 30 = 1/2]

Discussion