If `3sintheta+5costheta=3`, then `5sintheta-3costheta=?`

Options

A. `pm5`
B. `pm3`
C. `pm4`
D. `pm16`

Answer

To solve the problem, we need to find the value of \( 5 \sin \theta - 3 \cos \theta \) given that \( 3 \sin \theta + 5 \cos \theta = 3 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 3 \sin \theta + 5 \cos \theta = 3 \]

2. **Square both sides of the equation:**
\[ (3 \sin \theta + 5 \cos \theta)^2 = 3^2 \]
This expands to:
\[ 9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta = 9 \]

3. **Rearrange the equation:**
\[ 9 \sin^2 \theta + 25 \cos^2 \theta + 30 \sin \theta \cos \theta - 9 = 0 \]

4. **Let \( y = 5 \sin \theta - 3 \cos \theta \) and square it:**
\[ (5 \sin \theta - 3 \cos \theta)^2 = y^2 \]
This expands to:
\[ 25 \sin^2 \theta - 30 \sin \theta \cos \theta + 9 \cos^2 \theta = y^2 \]

5. **Now we have two equations:**
- From step 3:
\[ 9 \sin^2 \theta + 25 \cos^2 \theta + 30 \sin \theta \cos \theta = 9 \]
- From step 4:
\[ 25 \sin^2 \theta + 9 \cos^2 \theta - 30 \sin \theta \cos \theta = y^2 \]

6. **Add both equations:**
\[ (9 \sin^2 \theta + 25 \cos^2 \theta + 30 \sin \theta \cos \theta) + (25 \sin^2 \theta + 9 \cos^2 \theta - 30 \sin \theta \cos \theta) = 9 + y^2 \]
Simplifying gives:
\[ 34 \sin^2 \theta + 34 \cos^2 \theta = 9 + y^2 \]

7. **Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):**
\[ 34(1) = 9 + y^2 \]
Thus:
\[ 34 = 9 + y^2 \]

8. **Rearranging gives:**
\[ y^2 = 34 - 9 = 25 \]

9. **Taking the square root:**
\[ y = \pm 5 \]

### Final Answer:
\[ 5 \sin \theta - 3 \cos \theta = \pm 5 \]

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