Question

Find the acute angle θ such that 2 cos²θ = 3 sin θ.

Answer 1

2 cos² θ = 3 sin θ

∴ 2(1 – sin²θ) = 3 sin θ

∴ 2 – 2 sin²θ = 3 sin θ

∴ 2 sin² θ + 3sin θ – 2 = 0

∴ 2 sin² θ + 4 sin θ – sin θ – 2 = 0

∴ 2 sin θ (sin θ + 2) –1 (sin θ + 2) = 0

∴ (sin θ + 2) (2 sin θ – 1) = 0

∴ sin θ + 2 = 0 or 2 sin θ – 1 = 0

∴ sin θ = – 2 or sin θ = `1/2`

Since, – 1 ≤ sin θ ≤ 1

∴ sin θ = `1/2`

∴ θ = 30°     ...`[because sin 30^circ = 1/2]`