The minimum value of 3cox + 4 sinx + 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
We have the expression in the form \(a \cos x + b \sin x + c\), where:
- \(a = 3\)
- \(b = 4\)
- \(c = 8\)

### Step 2: Find the minimum value of \(a \cos x + b \sin x\)
The minimum value of the expression \(a \cos x + b \sin x\) can be found using the formula:
\[ \text{Minimum value} = -\sqrt{a^2 + b^2} \]
Calculating \(a^2 + b^2\):
\[ a^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]
\[ a^2 + b^2 = 9 + 16 = 25 \]
Now, taking the square root:
\[ \sqrt{a^2 + b^2} = \sqrt{25} = 5 \]
Thus, the minimum value of \(3 \cos x + 4 \sin x\) is:
\[ -\sqrt{25} = -5 \]

### Step 3: Add the constant term
Now, we need to add the constant term \(c = 8\) to the minimum value we found:
\[ \text{Minimum value of } (3 \cos x + 4 \sin x + 8) = -5 + 8 \]
Calculating this gives:
\[ -5 + 8 = 3 \]

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

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