The total number of solution of `sin{x}=cos{x}` (where `{}` denotes the fractional part) in `[0,2pi]` is equal to 5 (b) 6 (c) 8 (d) none of these

Options

A. 5
B. 6
C. 8
D. None of these

Answer

To solve the equation \( \sin\{x\} = \cos\{x\} \) where \(\{x\}\) denotes the fractional part of \(x\), we need to analyze the behavior of the sine and cosine functions over the interval \([0, 2\pi]\).

### Step-by-step Solution:

1. **Understanding the Fractional Part**:
The fractional part of \(x\), denoted as \(\{x\}\), is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
This means \(\{x\}\) is always in the range \([0, 1)\).

2. **Setting Up the Equation**:
We need to solve:
\[ \sin\{x\} = \cos\{x\} \]
This can be rewritten using the identity \(\sin\theta = \cos\theta\) which holds true when:
\[ \theta = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]

3. **Finding Solutions in the Interval**:
Since \(\{x\}\) is in the interval \([0, 1)\), we need to find values of \(\theta\) in this range:
- For \(n = 0\):
\[ \{x\} = \frac{\pi}{4} \approx 0.785 \]
- For \(n = 1\):
\[ \{x\} = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \quad (\text{not in } [0, 1)) \]

Thus, the only solution for \(\{x\}\) in \([0, 1)\) is:
\[ \{x\} = \frac{\pi}{4} \]

4. **Finding Corresponding Values of \(x\)**:
Since \(\{x\} = x - \lfloor x \rfloor\), we can express \(x\) as:
\[ x = n + \frac{\pi}{4} \quad (n \in \mathbb{Z}) \]
We need to find \(n\) such that:
\[ 0 \leq n + \frac{\pi}{4} < 2\pi \]
This gives:
\[ -\frac{\pi}{4} \leq n < 2\pi - \frac{\pi}{4} \]
Approximating \(\frac{\pi}{4} \approx 0.785\) and \(2\pi \approx 6.283\):
\[ -0.785 \leq n < 5.498 \]
Therefore, \(n\) can take values \(0, 1, 2, 3, 4, 5\).

5. **Counting the Solutions**:
For each integer \(n\) from \(0\) to \(5\), we have a corresponding \(x\):
- \(n = 0 \Rightarrow x = \frac{\pi}{4}\)
- \(n = 1 \Rightarrow x = 1 + \frac{\pi}{4}\)
- \(n = 2 \Rightarrow x = 2 + \frac{\pi}{4}\)
- \(n = 3 \Rightarrow x = 3 + \frac{\pi}{4}\)
- \(n = 4 \Rightarrow x = 4 + \frac{\pi}{4}\)
- \(n = 5 \Rightarrow x = 5 + \frac{\pi}{4}\)

Thus, we have a total of **6 solutions** in the interval \([0, 2\pi]\).

### Final Answer:
The total number of solutions of \( \sin\{x\} = \cos\{x\} \) in \([0, 2\pi]\) is **6**.