Maximum value of ` 3 cos theta + 4 sin theta` is

Options

A. 3
B. 4
C. 5
D. 7

Answer

To find the maximum value of the expression \( 3 \cos \theta + 4 \sin \theta \), we can use the formula for the maximum value of a linear combination of sine and cosine functions.

### Step-by-Step Solution:

1. **Identify the coefficients**:
- Here, \( a = 4 \) (coefficient of \( \sin \theta \)) and \( b = 3 \) (coefficient of \( \cos \theta \)).

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

3. **Calculate \( a^2 + b^2 \)**:
- Calculate \( a^2 \) and \( b^2 \):
\[ a^2 = 4^2 = 16 \]
\[ b^2 = 3^2 = 9 \]
- Now, add these values:
\[ a^2 + b^2 = 16 + 9 = 25 \]

4. **Take the square root**:
- Now, find the square root of \( 25 \):
\[ \sqrt{25} = 5 \]

5. **Conclusion**:
- Therefore, the maximum value of \( 3 \cos \theta + 4 \sin \theta \) is \( 5 \).

### Final Answer:
The maximum value of \( 3 \cos \theta + 4 \sin \theta \) is \( 5 \).

---