The minimum value of `(1)/(3sintheta-4costheta+7),` is

Options

A. `1/12`
B. `5/12`
C. `7/12`
D. `1/6`

Answer

To find the minimum value of the expression \( \frac{1}{3 \sin \theta - 4 \cos \theta + 7} \), we can follow these steps:

### Step 1: Identify the function in the denominator
We need to analyze the expression in the denominator:
\[ 3 \sin \theta - 4 \cos \theta + 7. \]

### Step 2: Find the maximum value of \( 3 \sin \theta - 4 \cos \theta \)
To find the maximum value of \( 3 \sin \theta - 4 \cos \theta \), we can use the formula for the maximum and minimum values of the expression \( a \sin \theta + b \cos \theta \). The maximum value is given by:
\[ \sqrt{a^2 + b^2}. \]
Here, \( a = 3 \) and \( b = -4 \).

### Step 3: Calculate \( \sqrt{a^2 + b^2} \)
Now, we calculate:
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]

### Step 4: Determine the maximum value of the entire expression
The maximum value of \( 3 \sin \theta - 4 \cos \theta \) is \( 5 \). Therefore, the maximum value of the entire expression in the denominator is:
\[ 5 + 7 = 12. \]

### Step 5: Find the minimum value of the original expression
Since the minimum value of \( \frac{1}{x} \) occurs when \( x \) is maximized, we find:
\[ \text{Minimum value} = \frac{1}{12}. \]

### Final Answer
Thus, the minimum value of \( \frac{1}{3 \sin \theta - 4 \cos \theta + 7} \) is:
\[ \frac{1}{12}. \]

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