Find sin θ such that 3cos θ + 4sin θ = 4.

1.	Find sin θ such that 3cos θ + 4sin θ = 4.
Answer» 3cos θ + 4sin θ = 4 ∴ 3cos θ = 4(1 – sin θ) Squaring both the sides, we get . 9cos² θ = 16(1 – sin θ)² ∴ 9(1 – sin² θ) = 16(1 + sin² θ – 2sin θ) ∴ 9 – 9sin² θ = 16 + 16sin² θ – 32sin θ ∴ 25sin² θ – 32sin θ + 7 = 0 ∴ 25sin² θ – 25sin θ – 7sin θ + 7 = 0 25sin θ (sin θ – 1) – 7 (sin θ – 1) = 0 ∴ (sin θ – 1) (25sin θ – 7) = 0 ∴ sin θ – 1 = 0 or 25 sin θ – 7 = 0 ∴ sin θ = 1 or sin θ = 7/25 Since, -1 ≤ sin θ ≤ 1 ∴ sin θ = 1 or 7/25

Answer»

3cos θ + 4sin θ = 4

∴ 3cos θ = 4(1 – sin θ)

Squaring both the sides, we get .

9cos² θ = 16(1 – sin θ)²

∴ 9(1 – sin² θ) = 16(1 + sin² θ – 2sin θ)

∴ 9 – 9sin² θ = 16 + 16sin² θ – 32sin θ

∴ 25sin² θ – 32sin θ + 7 = 0

∴ 25sin² θ – 25sin θ – 7sin θ + 7 = 0

25sin θ (sin θ – 1) – 7 (sin θ – 1) = 0

∴ (sin θ – 1) (25sin θ – 7) = 0

∴ sin θ – 1 = 0 or 25 sin θ – 7 = 0

∴ sin θ = 1 or sin θ = 7/25

Since, -1 ≤ sin θ ≤ 1

∴ sin θ = 1 or 7/25

Discussion