If [x] dnote the greatest integer less than or equal to x then the equation ` sin x=[1+sin x ]+[1-cos x ][` has no solution in

Options

A. `[-(pi)/(2),(pi)/(2)]`
B. `[(pi)/(2),pi]`
C. `[pi,(3pi)/(2)]`
D. R

Answer

To solve the equation \( \sin x = [1 + \sin x] + [1 - \cos x] \) where \([x]\) denotes the greatest integer less than or equal to \(x\), we will analyze the possible values of \(\sin x\) and \(\cos x\).

### Step 1: Analyze the equation
The equation can be rewritten as:
\[ \sin x = [1 + \sin x] + [1 - \cos x] \]
We know that \(\sin x\) can take values in the range \([-1, 1]\).

### Step 2: Consider the possible values of \(\sin x\)
The possible integer values for \(\sin x\) are \(-1\), \(0\), and \(1\). We will check each case one by one.

#### Case 1: \( \sin x = -1 \)
If \(\sin x = -1\), then:
- \(x = -\frac{\pi}{2} + 2k\pi\) or \(x = \frac{3\pi}{2} + 2k\pi\) for integer \(k\).
- Now, calculate:
- \( [1 + \sin x] = [1 - 1] = [0] = 0 \)
- \( [1 - \cos x] = [1 - 0] = [1] = 1 \) (since \(\cos(-\frac{\pi}{2}) = 0\) and \(\cos(\frac{3\pi}{2}) = 0\))

Thus, the equation becomes:
\[ -1 = 0 + 1 \implies -1 = 1 \quad \text{(not true)} \]

#### Case 2: \( \sin x = 0 \)
If \(\sin x = 0\), then:
- \(x = k\pi\) for integer \(k\).
- Now, calculate:
- \( [1 + \sin x] = [1 + 0] = [1] = 1 \)
- \( [1 - \cos x] = [1 - 1] = [0] = 0 \) (since \(\cos(k\pi) = (-1)^k\))

Thus, the equation becomes:
\[ 0 = 1 + 0 \implies 0 = 1 \quad \text{(not true)} \]

#### Case 3: \( \sin x = 1 \)
If \(\sin x = 1\), then:
- \(x = \frac{\pi}{2} + 2k\pi\) for integer \(k\).
- Now, calculate:
- \( [1 + \sin x] = [1 + 1] = [2] = 2 \)
- \( [1 - \cos x] = [1 - 0] = [1] = 1 \) (since \(\cos(\frac{\pi}{2}) = 0\))

Thus, the equation becomes:
\[ 1 = 2 + 1 \implies 1 = 3 \quad \text{(not true)} \]

### Step 3: Conclusion
In all three cases, we found that the left-hand side does not equal the right-hand side. Therefore, the equation has no solutions.

### Final Answer
The equation \( \sin x = [1 + \sin x] + [1 - \cos x] \) has no solution in the real numbers \(\mathbb{R}\).

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