The least vlaue of `(1)/(5cosx+12sinx+15),` is

Options

A. `-18`
B. `1//28`
C. `28`
D. `1//18`

Explanation

To find the least value of the expression \( \frac{1}{5\cos x + 12\sin x + 15} \), we can follow these steps:

### Step 1: Identify the function
We need to find the minimum value of the function \( 5\cos x + 12\sin x \).

### Step 2: Use the formula for maximum and minimum values
The maximum and minimum values of the expression \( a\sin x + b\cos x \) can be found using the formula:
- Maximum value = \( \sqrt{a^2 + b^2} \)
- Minimum value = \( -\sqrt{a^2 + b^2} \)

Here, \( a = 12 \) and \( b = 5 \).

### Step 3: Calculate \( \sqrt{a^2 + b^2} \)
Calculate:
\[ \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 4: Determine the minimum and maximum values
From the calculations:
- Minimum value of \( 5\cos x + 12\sin x = -\sqrt{169} = -13 \)
- Maximum value of \( 5\cos x + 12\sin x = \sqrt{169} = 13 \)

### Step 5: Add 15 to the function
Now, we need to find the minimum and maximum values of \( 5\cos x + 12\sin x + 15 \):
- Minimum value: \( -13 + 15 = 2 \)
- Maximum value: \( 13 + 15 = 28 \)

### Step 6: Find the least value of the reciprocal
We want to find the least value of \( \frac{1}{5\cos x + 12\sin x + 15} \). Since the minimum value of \( 5\cos x + 12\sin x + 15 \) is \( 2 \), the least value of the reciprocal is:
\[ \frac{1}{2} \]

### Step 7: Check the values
The maximum value of \( 5\cos x + 12\sin x + 15 \) is \( 28 \), so the maximum value of the reciprocal is:
\[ \frac{1}{28} \]

### Conclusion
Thus, the least value of \( \frac{1}{5\cos x + 12\sin x + 15} \) is:
\[ \frac{1}{28} \]