Explanation:

To solve this trigonometric problem:

- Start with the given equation: \(3 \sin \theta + 5 \cos \theta = 5\).

- We need to find the value of \(5 \sin \theta - 3 \cos \theta\).

Let's use a geometric approach:

- Step 1: Assume a solution in the form \(R \sin (\theta + \alpha)\).

- Step 2: Set \(\sin \alpha = \frac{3}{R}\) and \(\cos \alpha = \frac{5}{R}\) from the identity.

Using the Pythagorean identity:

- \(R = \sqrt{3^2 + 5^2}\), which simplifies to \(R = \sqrt{34}\).

Now simplify:

- Finding: \(5 \sin \theta - 3 \cos \theta = R (\sin(\theta) \cos(\beta) - \cos(\theta) \sin(\beta))\).

\( \beta \) angle will adjust the equation to a value that satisfies the new cosine and sine components.

If we equate, we find correct options:

- Option 1: -3

This is calculated by substituting expected angles into the sine function, verifying trigonometric identities.

- Option 2: -2

- Option 3: 5

- Option 4: 8

Correct Answer: Option 1: -3