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 \( y = 3 \cos x + 4 \sin x + 8 \), we can follow these steps:

### Step 1: Identify the trigonometric components
We can separate the constant from the trigonometric part:
\[ y = 3 \cos x + 4 \sin x + 8 \]
Here, we will focus on the expression \( 3 \cos x + 4 \sin x \).

### Step 2: Find the minimum value of \( 3 \cos x + 4 \sin x \)
We know that the expression \( a \cos x + b \sin x \) can be rewritten in the form \( R \cos(x - \phi) \), where:
\[ R = \sqrt{a^2 + b^2} \]
In our case, \( a = 3 \) and \( b = 4 \).

### Step 3: Calculate \( R \)
Calculate \( R \):
\[ R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Determine the range of \( 3 \cos x + 4 \sin x \)
The expression \( 3 \cos x + 4 \sin x \) will have a maximum value of \( R \) and a minimum value of \( -R \):
\[ \text{Minimum value} = -R = -5 \]

### Step 5: Incorporate the constant
Now, we add the constant \( 8 \) back to the minimum value:
\[ \text{Minimum value of } y = -5 + 8 = 3 \]

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