Minimum value of `(1)/(3 sin theta - 4 cos theta + 7)` is

Options

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

Answer

To find the minimum value of the expression \( \frac{1}{3 \sin \theta - 4 \cos \theta + 7} \), we will first determine the maximum value of the expression in the denominator, \( 3 \sin \theta - 4 \cos \theta + 7 \).

### Step 1: Find the maximum value of \( 3 \sin \theta - 4 \cos \theta \)

To find the maximum value of the expression \( 3 \sin \theta - 4 \cos \theta \), we can rewrite it in the form \( R \sin(\theta + \phi) \), where \( R = \sqrt{a^2 + b^2} \) and \( a \) and \( b \) are the coefficients of \( \sin \theta \) and \( \cos \theta \) respectively.

Here, \( a = 3 \) and \( b = -4 \).

\[ R = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 2: Determine the angle \( \phi \)

Next, we calculate the angle \( \phi \) using the relationships:

\[ \sin \phi = \frac{-4}{5} \quad \text{and} \quad \cos \phi = \frac{3}{5} \]

### Step 3: Rewrite the expression

Now we can express \( 3 \sin \theta - 4 \cos \theta \) as:

\[ 3 \sin \theta - 4 \cos \theta = 5 \sin(\theta + \phi) \]

### Step 4: Find the maximum value

The maximum value of \( \sin \) function is 1. Therefore, the maximum value of \( 5 \sin(\theta + \phi) \) is:

\[ 5 \cdot 1 = 5 \]

### Step 5: Add 7 to find the maximum of the entire expression

Now we can find the maximum value of \( 3 \sin \theta - 4 \cos \theta + 7 \):

\[ \text{Maximum of } (3 \sin \theta - 4 \cos \theta + 7) = 5 + 7 = 12 \]

### Step 6: Find the minimum value of the original expression

Now that we have the maximum value of the denominator, we can find the minimum value of the original expression:

\[ \text{Minimum value of } \frac{1}{3 \sin \theta - 4 \cos \theta + 7} = \frac{1}{12} \]

Thus, the minimum value of \( \frac{1}{3 \sin \theta - 4 \cos \theta + 7} \) is:

\[ \boxed{\frac{1}{12}} \]