If `log_(1/sqrt2) sinx> 0 ,x in [0,4pi]`, then the number values of x which are integral multiples of `pi/4` ,is

Options

A. 6
B. 12
C. 3
D. none of these

Explanation

To solve the problem, we need to analyze the inequality given by the logarithmic expression and find the integral multiples of \(\frac{\pi}{4}\) that satisfy the condition. Here’s the step-by-step solution:

### Step 1: Rewrite the logarithmic inequality
We start with the inequality:
\[ \log_{1/\sqrt{2}}(\sin x) > 0 \]
This can be rewritten using the property of logarithms:
\[ \sin x > 1 \]
However, since the base \(1/\sqrt{2}\) is less than 1, the inequality reverses when we exponentiate:
\[ \sin x < 1 \]

### Step 2: Determine the range of \(\sin x\)
We know that \(\sin x\) is always less than or equal to 1 for all \(x\). Therefore, we need to find when \(\sin x\) is strictly less than 1. The sine function equals 1 at:
\[ x = \frac{\pi}{2} + 2k\pi \quad \text{for integers } k \]
Thus, we avoid these points.

### Step 3: Identify the intervals for \(x\)
We are given that \(x\) is in the interval \([0, 4\pi]\). We need to find the values of \(x\) in this interval where \(\sin x\) is strictly less than 1. The points where \(\sin x = 1\) in the interval \([0, 4\pi]\) are:
\[ x = \frac{\pi}{2}, \frac{5\pi}{2} \]

### Step 4: Determine the integral multiples of \(\frac{\pi}{4}\)
Now we need to find the integral multiples of \(\frac{\pi}{4}\) in the interval \([0, 4\pi]\):
\[ x = 0, \frac{\pi}{4}, \frac{2\pi}{4} = \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{6\pi}{4} = \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi, \frac{9\pi}{4}, \frac{10\pi}{4} = \frac{5\pi}{2}, \frac{11\pi}{4}, 3\pi, \frac{13\pi}{4}, \frac{14\pi}{4} = \frac{7\pi}{2}, \frac{15\pi}{4}, 4\pi \]

### Step 5: Exclude the points where \(\sin x = 1\) or \(\sin x = 0\)
From the list, we exclude:
- \(x = \frac{\pi}{2}\) (where \(\sin x = 1\))
- \(x = \frac{5\pi}{2}\) (where \(\sin x = 1\))
- \(x = 0\) (where \(\sin x = 0\))
- \(x = \pi\) (where \(\sin x = 0\))
- \(x = 4\pi\) (where \(\sin x = 0\))

### Step 6: Count the valid values
The valid integral multiples of \(\frac{\pi}{4}\) in the interval \([0, 4\pi]\) are:
- \(\frac{\pi}{4}\)
- \(\frac{3\pi}{4}\)
- \(\frac{5\pi}{4}\)
- \(\frac{7\pi}{4}\)
- \(\frac{9\pi}{4}\)
- \(\frac{11\pi}{4}\)
- \(\frac{13\pi}{4}\)
- \(\frac{15\pi}{4}\)

Counting these gives us a total of **8 valid values**.

### Final Answer
The number of values of \(x\) which are integral multiples of \(\frac{\pi}{4}\) and satisfy the given condition is **8**.