The maximum value of `3 cos x+4sinx+5,` is

Options

A. 5
B. 9
C. 7
D. none of these

Answer

To find the maximum value of the expression \(3 \cos x + 4 \sin x + 5\), we can follow these steps:

### Step 1: Identify the coefficients
We have the expression in the form \(a \cos x + b \sin x + c\), where:
- \(a = 3\)
- \(b = 4\)
- \(c = 5\)

### Step 2: Use the formula for maximum value
The maximum value of the expression \(a \cos x + b \sin x + c\) can be found using the formula:
\[ \text{Maximum value} = c + \sqrt{a^2 + b^2} \]

### Step 3: Calculate \(a^2 + b^2\)
Now, we will calculate \(a^2 + b^2\):
\[ a^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]
\[ a^2 + b^2 = 9 + 16 = 25 \]

### Step 4: Calculate \(\sqrt{a^2 + b^2}\)
Now, we find the square root:
\[ \sqrt{a^2 + b^2} = \sqrt{25} = 5 \]

### Step 5: Find the maximum value
Now we can substitute back into the formula for the maximum value:
\[ \text{Maximum value} = c + \sqrt{a^2 + b^2} = 5 + 5 = 10 \]

### Conclusion
Thus, the maximum value of \(3 \cos x + 4 \sin x + 5\) is \(10\).