If `[sinx]+[x/(2pi)]+[(2x)/(5pi)]=(9x)/(10pi)`, where `[*]` denotes the greatest integer function, the number of solutions in the interval (30,40) is ………… .

Answer

To solve the equation \([ \sin x ] + [ \frac{x}{2\pi} ] + [ \frac{2x}{5\pi} ] = \frac{9x}{10\pi}\) where \([*]\) denotes the greatest integer function, we will analyze each term and find the number of solutions in the interval \( (30, 40) \).

### Step 1: Analyze \([ \sin x ]\)
The sine function oscillates between -1 and 1. Therefore, \([ \sin x ]\) can take values of -1 or 0:
- \([ \sin x ] = 0\) when \( \sin x \) is in the range \( [0, 1) \).
- \([ \sin x ] = -1\) when \( \sin x \) is in the range \( [-1, 0) \).

### Step 2: Analyze \([ \frac{x}{2\pi} ]\)
For \( x \) in the interval \( (30, 40) \):
- \( \frac{30}{2\pi} \approx 4.77 \) and \( \frac{40}{2\pi} \approx 6.37 \).
- Therefore, \([ \frac{x}{2\pi} ]\) can take values 4, 5, or 6.

### Step 3: Analyze \([ \frac{2x}{5\pi} ]\)
For \( x \) in the interval \( (30, 40) \):
- \( \frac{2 \times 30}{5\pi} \approx 3.82 \) and \( \frac{2 \times 40}{5\pi} \approx 5.09 \).
- Therefore, \([ \frac{2x}{5\pi} ]\) can take values 3, 4, or 5.

### Step 4: Set up the equation
We can rewrite the equation as:
\[ [ \sin x ] + [ \frac{x}{2\pi} ] + [ \frac{2x}{5\pi} ] = \frac{9x}{10\pi} \]

### Step 5: Consider cases for \([ \sin x ]\)

#### Case 1: \([ \sin x ] = 0\)
The equation becomes:
\[ 0 + [ \frac{x}{2\pi} ] + [ \frac{2x}{5\pi} ] = \frac{9x}{10\pi} \]
This simplifies to:
\[ [ \frac{x}{2\pi} ] + [ \frac{2x}{5\pi} ] = \frac{9x}{10\pi} \]
Substituting possible values for \([ \frac{x}{2\pi} ]\) and \([ \frac{2x}{5\pi} ]\), we find that this does not yield any integer solutions.

#### Case 2: \([ \sin x ] = -1\)
The equation becomes:
\[ -1 + [ \frac{x}{2\pi} ] + [ \frac{2x}{5\pi} ] = \frac{9x}{10\pi} \]
This simplifies to:
\[ [ \frac{x}{2\pi} ] + [ \frac{2x}{5\pi} ] = \frac{9x}{10\pi} + 1 \]
Again, substituting possible values for \([ \frac{x}{2\pi} ]\) and \([ \frac{2x}{5\pi} ]\) shows that this does not yield any integer solutions either.

### Conclusion
After analyzing both cases, we find that there are no integer solutions for the equation in the interval \( (30, 40) \).

### Final Answer
The number of solutions in the interval \( (30, 40) \) is **0**.