​If 3 sin θ + 5 cos θ = 5, then what is the value of 5 sin θ - 3 cos θ ?​

Correct option is A

Given:

3 sin θ + 5 cos θ = 5

Concept Used:

We can use the trigonometric identities and algebraic manipulation to solve the given equation.

Solution:

Let 3 sin θ + 5 cos θ = R

Given R = 5

We need to find the value of 5 sin θ - 3 cos θ

Square both sides of the given equation:

=> (3 sin θ + 5 cos θ)²= 5²

=> 9 sin²θ + 30 sin θ cos θ + 25 cos²θ = 25

Using the identity sin²θ + cos²θ = 1:

=> 9 (1 - cos²θ) + 30 sin θ cos θ + 25 cos²θ = 25

=> 9 - 9 cos²θ + 30 sin θ cos θ + 25 cos²θ = 25

=> 9 + 16 cos²θ + 30 sin θ cos θ = 25

=> 16 cos²θ + 30 sin θ cos θ = 16

Now, consider the expression we need to find:

Let 5 sin θ - 3 cos θ = K

Square both sides of this equation:

=> (5 sin θ - 3 cos θ)²= K²

=> 25 sin²θ - 30 sin θ cos θ + 9 cos²θ = K²

Using the identity sin²θ + cos²θ = 1:

=> 25 (1 - cos²θ) - 30 sin θ cos θ + 9 cos²θ = K²

=> 25 - 25 cos²θ - 30 sin θ cos θ + 9 cos²θ = K²

=> 25 - 16 cos²θ - 30 sin θ cos θ = K²

Since we know from earlier:

16 cos²θ + 30 sin θ cos θ = 16

Substitute this into the equation:

=> 25 - 16 = K²

=> 9 = K²

=> K = ±3

Since the options provided are negative, the correct answer is:

=> K = -3

∴ The value of 5 sin θ - 3 cos θ is -3.