The number of roots of `x^2-2=[sinx],w h e r e[dot]` stands for the greatest integer function is 0 (b) 1 (c) 2 (d) 3.

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the equation \( x^2 - 2 = [\sin x] \), where \([\cdot]\) denotes the greatest integer function, we will analyze the graphs of the two functions involved: \( y = x^2 - 2 \) and \( y = [\sin x] \).

### Step 1: Analyze the function \( y = x^2 - 2 \)
The function \( y = x^2 - 2 \) is a parabola that opens upwards. The vertex of this parabola is at the point (0, -2). The parabola intersects the y-axis at -2 and approaches infinity as \( x \) moves away from 0 in both directions.

### Step 2: Analyze the function \( y = [\sin x] \)
The function \( y = [\sin x] \) takes the values of the sine function and applies the greatest integer function. The sine function oscillates between -1 and 1. Therefore, the possible values of \( [\sin x] \) are:
- \( [\sin x] = 0 \) when \( 0 \leq \sin x < 1 \) (which occurs in intervals like \( [0, \pi) \))
- \( [\sin x] = 1 \) when \( \sin x = 1 \) (which occurs at \( x = \frac{\pi}{2} + 2k\pi \) for integers \( k \))
- \( [\sin x] = -1 \) when \( -1 < \sin x < 0 \) (which occurs in intervals like \( (\pi, 2\pi) \))

### Step 3: Graph the two functions
1. **Graph of \( y = x^2 - 2 \)**:
- The vertex is at (0, -2).
- The parabola intersects the x-axis at \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

2. **Graph of \( y = [\sin x] \)**:
- The graph oscillates between -1 and 1.
- It is 0 for most of the intervals in \( [0, \pi) \) and \( (2\pi, 3\pi) \), and it is -1 for intervals like \( (\pi, 2\pi) \).

### Step 4: Find intersections
To find the number of roots of \( x^2 - 2 = [\sin x] \), we look for intersections between the two graphs:
- **For \( [\sin x] = 0 \)**: The equation becomes \( x^2 - 2 = 0 \) or \( x^2 = 2 \). This gives two solutions: \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
- **For \( [\sin x] = -1 \)**: The equation becomes \( x^2 - 2 = -1 \) or \( x^2 = 1 \). This gives two solutions: \( x = 1 \) and \( x = -1 \).

### Step 5: Count the total number of roots
From our analysis:
- We have 2 solutions from \( [\sin x] = 0 \) (i.e., \( x = \sqrt{2}, -\sqrt{2} \)).
- We have 2 solutions from \( [\sin x] = -1 \) (i.e., \( x = 1, -1 \)).

Thus, the total number of roots is \( 2 + 2 = 4 \).

### Conclusion
The number of roots of the equation \( x^2 - 2 = [\sin x] \) is **4**.