Maximum value of `(2 sin theta + 3 cos theta)` is

Options

A. 2
B. `sqrt 13`
C. `sqrt15`
D. 1

Answer

To find the maximum value of the expression \(2 \sin \theta + 3 \cos \theta\), we can use the formula for the maximum value of an expression of the form \(a \sin \theta + b \cos \theta\).

### Step-by-Step Solution:

1. **Identify the coefficients**:
In the expression \(2 \sin \theta + 3 \cos \theta\), we identify \(a = 2\) and \(b = 3\).

2. **Use the formula for maximum value**:
The maximum value of the expression \(a \sin \theta + b \cos \theta\) is given by the formula:
\[ \text{Maximum Value} = \sqrt{a^2 + b^2} \]

3. **Calculate \(a^2\) and \(b^2\)**:
- Calculate \(a^2 = 2^2 = 4\)
- Calculate \(b^2 = 3^2 = 9\)

4. **Add \(a^2\) and \(b^2\)**:
\[ a^2 + b^2 = 4 + 9 = 13 \]

5. **Take the square root**:
\[ \sqrt{a^2 + b^2} = \sqrt{13} \]

6. **Conclusion**:
Therefore, the maximum value of \(2 \sin \theta + 3 \cos \theta\) is \(\sqrt{13}\).

### Final Answer:
The maximum value of \(2 \sin \theta + 3 \cos \theta\) is \(\sqrt{13}\).

---