Question

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Answer 1

Step-by-step explanation:

Given,

3 sinθ + 5cosθ = 5

Squaring on both sides.

(3 sinθ + 5cosθ)² = 5²

(3sinθ)² + (5cosθ)² + 2× 3sinθ 5cosθ = 25

------- [a+b= a²+b²+2ab]

9sin²θ + 25cos²θ + 30sinθcosθ = 25

9 (1-cos²θ) + 25(1-sin²θ) + 30sinθcosθ = 25

------[sin²θ + cos²θ =1]

9 - 9cos²θ + 25 - 25sin²θ + 30sinθcosθ = 25

9 + 25 - (9cos²θ + 25sin²θ - 30sinθcosθ) = 25

34 - (9cos²θ + 25sin²θ - 30sinθcosθ) = 25

- (25sin²θ + 9cos²θ - 30sinθcosθ) = 25 - 34

(25sin²θ + 9cos²θ - 30sinθcosθ) = 9

(5sinθ - 3cosθ)² = 9

(5sinθ - 3cosθ) = √9

(5sinθ - 3cosθ) = ±3

Hence, proved.

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