The minimum value of `3 cos x+4 sin x+8` is:

Options

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

Answer

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
The given expression can be rewritten as:
\[ f(x) = 3 \cos x + 4 \sin x + 8 \]
Here, \(a = 4\) and \(b = 3\).

### Step 2: Calculate the magnitude
To find the minimum value of \(3 \cos x + 4 \sin x\), we first calculate the magnitude of the coefficients:
\[ R = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Determine the maximum and minimum values
The maximum value of \(3 \cos x + 4 \sin x\) is \(R\) and the minimum value is \(-R\):
\[ \text{Maximum value} = 5 \]
\[ \text{Minimum value} = -5 \]

### Step 4: Add the constant
Now, we need to find the minimum value of the entire expression \(f(x)\):
\[ f(x) = 3 \cos x + 4 \sin x + 8 \]
The minimum value of \(3 \cos x + 4 \sin x\) is \(-5\), so:
\[ \text{Minimum value of } f(x) = -5 + 8 = 3 \]

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

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