Question

If ( 3 sin theta + 5 cos theta ) = 5 , prove that ( 5 sin there - 3 cos theta ) = +3 , -3.

Answer 1

Hey!!.

We have: (3 sin ¢ + 5 cos ¢)² + ( 5 sin¢ - 3 cos¢)² = 9 ( sin² ¢ + cos² ¢ ) + 25 ( Sin²¢ + Cos²¢ ) = ( 9 + 25 ) = 34.



Therefore,


( 3 sin¢ + 5 cos¢)² + ( 5 sin¢ - 3 cos¢)² = 34



5² + ( 5 sin¢ - 3 cos¢ )² = 34


( 5 sin ¢ - 3 cos¢ ) = +- 3.

Answer 2

Given Equation is 3 sin θ + 5 cos θ = 5.

On squaring both sides, we get

⇒ (3 sin θ + 5 cos θ)² = (5)²

⇒ 9 sin²θ + 25cos²θ + 30 sinθcosθ = 25

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

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

⇒ 9sin²θ - 25sin²θ + 30 sinθcosθ = 0

⇒ -16sin²θ = -30 sinθcosθ

⇒ 16sin²θ = 30 sinθcosθ.

LHS:

⇒ 5 sinθ - 3 cosθ.

On squaring, we get

⇒ (5 sinθ - 3 cosθ)²

⇒ 25sin²θ + 9cos²θ - 30sinθcosθ

⇒ 25sin²θ + 9cos²θ - 16sin²θ

⇒ 9sin²θ + 9cos²θ

⇒ 9(sin²θ + cos²θ)

⇒ 9.

Hence, (5 sinθ - 3 cosθ) = +3,-3.

Hope this helps!