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Question
Find sinθ such that 3cosθ + 4sinθ = 4
Solution
3cosθ + 4 sinθ = 4
∴ 3cosθ = 4 – 4sinθ
∴ 3cosθ = 4(1 – sinθ)
Squaring both the sides, we get,
9cos2θ = 16(1 – sinθ)2
∴ 9(1 – sin2θ) = 16(1 + sin2θ – 2sinθ)
∴ 9 – 9sin2θ = 16 + 16sin2θ – 32sinθ
∴ 25sin2θ – 32sinθ + 7 = 0
∴ 25sin2θ – 25sinθ – 7sinθ + 7 = 0
∴ 25sinθ (sinθ – 1) – 7(sinθ – 1) = 0
∴ (sinθ – 1)(25sinθ – 7) = 0
∴ sinθ – 1 = 0 or 25sinθ – 7 = 0
∴ sinθ = 1 or sinθ = `7/25`
Since, – 1 ≤ sinθ ≤ 1
∴ sinθ = 1 or `7/25`
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