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

Explanation

To solve the equation \( \sin\{x\} = \cos\{x\} \) in the interval \([0, 2\pi]\), where \(\{x\}\) denotes the fractional part of \(x\), we can follow these steps:

### Step 1: Understand the fractional part
The fractional part \(\{x\}\) is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \(x\). The fractional part \(\{x\}\) is always in the range \(0 \leq \{x\} < 1\).

### Step 2: Analyze the equation
Given the equation:
\[ \sin\{x\} = \cos\{x\} \]
Since \(\{x\} < 1\), we can replace \(\{x\}\) with \(x\) when \(x\) is in the interval \([0, 1)\). Therefore, in this interval:
\[ \sin x = \cos x \]
This implies:
\[ \tan x = 1 \]
The solutions for this equation in the interval \([0, 1)\) are:
\[ x = \frac{\pi}{4} \]

### Step 3: Consider the next interval
For the interval \([1, 2)\):
\[ \{x\} = x - 1 \]
So, we rewrite the equation:
\[ \sin(x - 1) = \cos(x - 1) \]
This leads to:
\[ \tan(x - 1) = 1 \]
The solutions in this interval are:
\[ x - 1 = \frac{\pi}{4} \implies x = \frac{\pi}{4} + 1 \]

### Step 4: Continue for the remaining intervals
We repeat this process for the intervals \([2, 3)\), \([3, 4)\), \([4, 5)\), and \([5, 6)\):
- For \([2, 3)\):
\[ \{x\} = x - 2 \implies \tan(x - 2) = 1 \implies x - 2 = \frac{\pi}{4} \implies x = \frac{\pi}{4} + 2 \]
- For \([3, 4)\):
\[ \{x\} = x - 3 \implies \tan(x - 3) = 1 \implies x - 3 = \frac{\pi}{4} \implies x = \frac{\pi}{4} + 3 \]
- For \([4, 5)\):
\[ \{x\} = x - 4 \implies \tan(x - 4) = 1 \implies x - 4 = \frac{\pi}{4} \implies x = \frac{\pi}{4} + 4 \]
- For \([5, 6)\):
\[ \{x\} = x - 5 \implies \tan(x - 5) = 1 \implies x - 5 = \frac{\pi}{4} \implies x = \frac{\pi}{4} + 5 \]

### Step 5: Count the solutions
Now we count the number of solutions:
1. In \([0, 1)\): \(x = \frac{\pi}{4}\)
2. In \([1, 2)\): \(x = \frac{\pi}{4} + 1\)
3. In \([2, 3)\): \(x = \frac{\pi}{4} + 2\)
4. In \([3, 4)\): \(x = \frac{\pi}{4} + 3\)
5. In \([4, 5)\): \(x = \frac{\pi}{4} + 4\)
6. In \([5, 6)\): \(x = \frac{\pi}{4} + 5\)

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

### Final Answer:
The total number of solutions is **6**.