The number of solutions of the equation `sinx. Sin2x.sin3x. Sin4x.sin5x=0` in `[0,pi]` is equal to

Answer

To find the number of solutions for the equation \( \sin x \cdot \sin 2x \cdot \sin 3x \cdot \sin 4x \cdot \sin 5x = 0 \) in the interval \( [0, \pi] \), we need to determine when each sine function is equal to zero.

### Step-by-Step Solution:

1. **Identify when each sine function is zero:**
- For \( \sin x = 0 \):
- \( x = 0, \pi \) (within the interval \( [0, \pi] \))
- For \( \sin 2x = 0 \):
- \( 2x = n\pi \) where \( n \) is an integer.
- Thus, \( x = 0, \frac{\pi}{2}, \pi \) (within the interval \( [0, \pi] \))
- For \( \sin 3x = 0 \):
- \( 3x = n\pi \)
- Thus, \( x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi \) (within the interval \( [0, \pi] \))
- For \( \sin 4x = 0 \):
- \( 4x = n\pi \)
- Thus, \( x = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \) (within the interval \( [0, \pi] \))
- For \( \sin 5x = 0 \):
- \( 5x = n\pi \)
- Thus, \( x = 0, \frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}, \pi \) (within the interval \( [0, \pi] \))

2. **Collect all unique solutions:**
- From \( \sin x \): \( 0, \pi \)
- From \( \sin 2x \): \( 0, \frac{\pi}{2}, \pi \)
- From \( \sin 3x \): \( 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi \)
- From \( \sin 4x \): \( 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \)
- From \( \sin 5x \): \( 0, \frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}, \pi \)

3. **List all unique values:**
- Combining all the values, we have:
- \( 0, \frac{\pi}{5}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{5}, \frac{\pi}{2}, \frac{3\pi}{5}, \frac{2\pi}{3}, \frac{3\pi}{4}, \pi \)

4. **Count the unique solutions:**
- The unique solutions are:
- \( 0, \frac{\pi}{5}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{2\pi}{5}, \frac{\pi}{2}, \frac{3\pi}{5}, \frac{2\pi}{3}, \frac{3\pi}{4}, \pi \)
- This gives us a total of **10 unique solutions**.

### Final Answer:
The number of solutions of the equation \( \sin x \cdot \sin 2x \cdot \sin 3x \cdot \sin 4x \cdot \sin 5x = 0 \) in the interval \( [0, \pi] \) is **10**.