The minimum of ` 3cosx +4sin x+8` is

Options

A. 5
B. 9
C. 7
D. 3

Explanation

To find the minimum value of the expression \( 3\cos x + 4\sin x + 8 \), we can follow these steps:

### Step 1: Identify the coefficients
We have the expression:
\[ 3\cos x + 4\sin x + 8 \]
Here, we can identify:
- \( A = 3 \)
- \( B = 4 \)
- \( C = 8 \)

### Step 2: Calculate \( \sqrt{A^2 + B^2} \)
Next, we need to calculate \( \sqrt{A^2 + B^2} \):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Find the minimum value
The minimum value of the expression \( A\cos x + B\sin x + C \) can be found using the formula:
\[ \text{Minimum value} = C - \sqrt{A^2 + B^2} \]
Substituting the values we have:
\[ \text{Minimum value} = 8 - 5 = 3 \]

### Conclusion
Thus, the minimum value of \( 3\cos x + 4\sin x + 8 \) is:
\[ \boxed{3} \]

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