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

Options

A. 5
B. 10
C. 7
D. 13

Answer

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

### Step 1: Rewrite the trigonometric expression
We can express \( 3 \cos x + 4 \sin x \) in the form \( R \sin(x + \alpha) \) or \( R \cos(x + \beta) \). To do this, we first calculate \( R \):
\[ R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 2: Determine the angles
Next, we need to find \( \sin \alpha \) and \( \cos \alpha \) such that:
\[ \cos \alpha = \frac{3}{5}, \quad \sin \alpha = \frac{4}{5} \]
This means we can rewrite the expression as:
\[ 3 \cos x + 4 \sin x = 5 \left( \frac{3}{5} \cos x + \frac{4}{5} \sin x \right) = 5 \sin(x + \alpha) \]
where \( \alpha \) is such that \( \tan \alpha = \frac{4}{3} \).

### Step 3: Substitute back into the function
Now we can rewrite \( f(x) \):
\[ f(x) = 5 \sin(x + \alpha) + 8 \]

### Step 4: Find the maximum value
The sine function \( \sin(x + \alpha) \) has a maximum value of 1. Therefore:
\[ \text{Maximum of } f(x) = 5 \cdot 1 + 8 = 5 + 8 = 13 \]

### Conclusion
Thus, the maximum value of the expression \( 3 \cos x + 4 \sin x + 8 \) is:
\[ \boxed{13} \]