Find the number of solution of the following equation `|sin x|=|x/10|`

Answer

To find the number of solutions for the equation \( | \sin x | = \left| \frac{x}{10} \right| \), we will analyze the equation step by step.

### Step 1: Understand the equation
The equation involves absolute values, which means we need to consider both positive and negative cases for \( \sin x \) and \( \frac{x}{10} \).

### Step 2: Set up the cases
We can rewrite the equation as:
1. \( \sin x = \frac{x}{10} \)
2. \( \sin x = -\frac{x}{10} \)
3. \( -\sin x = \frac{x}{10} \)
4. \( -\sin x = -\frac{x}{10} \)

However, the last case simplifies to the first case. Therefore, we only need to consider the first three cases.

### Step 3: Analyze the first case
**Case 1: \( \sin x = \frac{x}{10} \)**

The function \( \sin x \) oscillates between -1 and 1, while \( \frac{x}{10} \) is a straight line with a slope of \( \frac{1}{10} \). For this case, we need to find the intersection points of these two functions.

- The maximum value of \( \frac{x}{10} \) is 1, which occurs when \( x = 10 \). Therefore, we need to find solutions for \( x \) in the interval \( [-10, 10] \).

### Step 4: Analyze the second case
**Case 2: \( \sin x = -\frac{x}{10} \)**

In this case, we again look for intersections. The line \( -\frac{x}{10} \) will also intersect with \( \sin x \) in the same interval \( [-10, 10] \).

### Step 5: Graphical representation
To find the number of solutions, we can visualize the graphs:
- The graph of \( y = \sin x \) oscillates between -1 and 1.
- The lines \( y = \frac{x}{10} \) and \( y = -\frac{x}{10} \) are straight lines crossing the y-axis at the origin.

### Step 6: Count the intersections
1. For \( \sin x = \frac{x}{10} \):
- The line intersects the sine wave multiple times. We can estimate the number of intersections by observing the behavior of the sine function and the line.
- In the interval \( [0, 10] \), we can see that the sine function completes approximately \( 3 \) full cycles (since \( 3\pi \approx 9.42 \) and \( 4\pi \approx 12.56 \)).
- Each cycle will have 2 intersections with the line \( y = \frac{x}{10} \) (one going up and one coming down), giving us about \( 6 \) intersections.

2. For \( \sin x = -\frac{x}{10} \):
- Similarly, in the interval \( [-10, 0] \), we can expect another \( 5 \) intersections (since the sine function is symmetric).

### Step 7: Total solutions
Adding the solutions from both cases:
- From \( \sin x = \frac{x}{10} \): approximately \( 6 \) solutions.
- From \( \sin x = -\frac{x}{10} \): approximately \( 5 \) solutions.

Thus, the total number of solutions is \( 6 + 5 = 11 \).

### Final Answer
The total number of solutions for the equation \( | \sin x | = \left| \frac{x}{10} \right| \) is **11**.

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