If `3sintheta + 5 cos theta = 5`, then the value of `5sintheta - 3cos theta` is:

Options

A. 5
B. 3
C. 4
D. None of these

Answer

To solve the equation \(3 \sin \theta + 5 \cos \theta = 5\) and find the value of \(5 \sin \theta - 3 \cos \theta\), we can follow these steps:

### Step 1: Write down the given equation
We start with the equation:
\[ 3 \sin \theta + 5 \cos \theta = 5 \]

### Step 2: Define the expression we want to find
Let:
\[ x = 5 \sin \theta - 3 \cos \theta \]

### Step 3: Use the identity for the sum of squares
We can use the identity that relates the two expressions:
\[ A^2 + B^2 = x^2 + y^2 \]
where:
- \(A = 3 \sin \theta\)
- \(B = 5 \cos \theta\)
- \(x = 5 \sin \theta\)
- \(y = -3 \cos \theta\)

### Step 4: Calculate \(A^2 + B^2\)
Calculate \(A^2\) and \(B^2\):
\[ A^2 = (3 \sin \theta)^2 = 9 \sin^2 \theta \]
\[ B^2 = (5 \cos \theta)^2 = 25 \cos^2 \theta \]
Thus,
\[ A^2 + B^2 = 9 \sin^2 \theta + 25 \cos^2 \theta \]

### Step 5: Substitute the value of \(\sin^2 \theta + \cos^2 \theta\)
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ A^2 + B^2 = 9 \sin^2 \theta + 25 (1 - \sin^2 \theta) \]
\[ = 9 \sin^2 \theta + 25 - 25 \sin^2 \theta \]
\[ = (9 - 25) \sin^2 \theta + 25 \]
\[ = -16 \sin^2 \theta + 25 \]

### Step 6: Calculate \(x^2 + y^2\)
Now calculate \(x^2\) and \(y^2\):
\[ x^2 = (5 \sin \theta)^2 = 25 \sin^2 \theta \]
\[ y^2 = (-3 \cos \theta)^2 = 9 \cos^2 \theta \]
Thus,
\[ x^2 + y^2 = 25 \sin^2 \theta + 9 \cos^2 \theta \]

### Step 7: Set the two expressions equal
Now we set the two expressions equal:
\[ -16 \sin^2 \theta + 25 = 25 \sin^2 \theta + 9 \cos^2 \theta \]
Substituting \(\cos^2 \theta = 1 - \sin^2 \theta\):
\[ -16 \sin^2 \theta + 25 = 25 \sin^2 \theta + 9(1 - \sin^2 \theta) \]
\[ -16 \sin^2 \theta + 25 = 25 \sin^2 \theta + 9 - 9 \sin^2 \theta \]
\[ -16 \sin^2 \theta + 25 = (25 - 9) \sin^2 \theta + 9 \]
\[ -16 \sin^2 \theta + 25 = 16 \sin^2 \theta + 9 \]

### Step 8: Solve for \(\sin^2 \theta\)
Now, rearranging gives:
\[ -16 \sin^2 \theta - 16 \sin^2 \theta = 9 - 25 \]
\[ -32 \sin^2 \theta = -16 \]
\[ \sin^2 \theta = \frac{1}{2} \]

### Step 9: Find \(x\)
Now substituting back to find \(x\):
\[ x^2 = 9 \implies x = \sqrt{9} = 3 \text{ or } -3 \]

### Step 10: Conclusion
The value of \(5 \sin \theta - 3 \cos \theta\) is:
\[ \boxed{3} \text{ or } \boxed{-3} \]