Let `theta in [0,4pi]` satisfy the equation `(sintheta+2)(sintheta+3)(sintheta+4)=6.` If the sum of all the values of `theta` is of the form `kpi` , then the value of `k` is 6 (b) 5 (c) 4 (d) 2

Options

A. 6
B. 5
C. 4
D. 2

Explanation

To solve the equation \((\sin \theta + 2)(\sin \theta + 3)(\sin \theta + 4) = 6\) for \(\theta\) in the interval \([0, 4\pi]\), we can follow these steps:

### Step 1: Analyze the Equation
We start with the equation:
\[ (\sin \theta + 2)(\sin \theta + 3)(\sin \theta + 4) = 6 \]

### Step 2: Determine the Range of \(\sin \theta\)
We know that the sine function, \(\sin \theta\), takes values in the range \([-1, 1]\). Therefore:
- \(\sin \theta + 2\) will range from \(1\) to \(3\),
- \(\sin \theta + 3\) will range from \(2\) to \(4\),
- \(\sin \theta + 4\) will range from \(3\) to \(5\).

### Step 3: Evaluate the Left-Hand Side
To find the minimum value of the left-hand side, we can evaluate it at the endpoints of the sine function:
- When \(\sin \theta = -1\):
\[ (-1 + 2)(-1 + 3)(-1 + 4) = (1)(2)(3) = 6 \]
- When \(\sin \theta = 0\):
\[ (0 + 2)(0 + 3)(0 + 4) = (2)(3)(4) = 24 \]
- When \(\sin \theta = 1\):
\[ (1 + 2)(1 + 3)(1 + 4) = (3)(4)(5) = 60 \]

From this, we see that the left-hand side can equal \(6\) when \(\sin \theta = -1\).

### Step 4: Solve for \(\theta\)
The equation \(\sin \theta = -1\) occurs at:
\[ \theta = \frac{3\pi}{2} + 2k\pi \quad \text{for integers } k \]

### Step 5: Find Values of \(\theta\) in \([0, 4\pi]\)
Now we find the values of \(\theta\) within the interval \([0, 4\pi]\):
- For \(k = 0\):
\[ \theta = \frac{3\pi}{2} \]
- For \(k = 1\):
\[ \theta = \frac{3\pi}{2} + 2\pi = \frac{3\pi}{2} + \frac{4\pi}{2} = \frac{7\pi}{2} \]

Both values \(\frac{3\pi}{2}\) and \(\frac{7\pi}{2}\) lie within the interval \([0, 4\pi]\).

### Step 6: Calculate the Sum of Values of \(\theta\)
Now we calculate the sum of all values of \(\theta\):
\[ \text{Sum} = \frac{3\pi}{2} + \frac{7\pi}{2} = \frac{10\pi}{2} = 5\pi \]

### Step 7: Identify \(k\)
The sum of the values of \(\theta\) is of the form \(k\pi\), where \(k = 5\).

### Final Answer
The value of \(k\) is:
\[ \boxed{5} \]