Question

3 sin theta + 5 cos theta = 4 find the value of 5 sin theta - 3 cos theta

Answer 1

Let ... 3 sin x + 5 cos x = 5.

∴ ( 3 sin x + 5 cos x )² = 5²

∴ 9 sin² x + 25 cos² x + 30 sin x cos x = 25

∴ 9(1-cos² x) + 25(1-sin² x) + 30 sin x cos x = 25

∴ 9 - 9cos² x + 25 - 25sin² x + 30 sin x cos x = 25

∴ 9 = 25 sin² x + 9 cos² x - 30 sin x cos x

∴ 9 = ( 5 sin x - 3 cos x )²

∴ 5 sin x - 3 cos x = ± √9

∴ 5 sin x - 3 cos x = ± 3 ............. Q.E.D.

Answer 2

Given:

(3 sinθ+5cosθ)²= 5²

Squaring on both sides.

(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

L.H.S = R.H.S

this is your answer..