The number of intersecting points on the graph for `sinx=x/10` for `x in [-pi,pi]` is

Options

A. 3
B. 4
C. 2
D. 1

Answer

To find the number of intersecting points on the graph of the equation \( \sin x = \frac{x}{10} \) for \( x \) in the interval \([-π, π]\), we can follow these steps:

### Step 1: Understand the Functions
We have two functions:
1. \( y = \sin x \)
2. \( y = \frac{x}{10} \)

### Step 2: Analyze the Range of \( \sin x \)
The sine function oscillates between -1 and 1. Therefore, we have:
\[ -1 \leq \sin x \leq 1 \]

### Step 3: Determine the Range of \( \frac{x}{10} \)
For \( x \) in the interval \([-π, π]\):
- The minimum value of \( \frac{x}{10} \) occurs at \( x = -π \):
\[ \frac{-π}{10} \approx -0.314 \]
- The maximum value of \( \frac{x}{10} \) occurs at \( x = π \):
\[ \frac{π}{10} \approx 0.314 \]

### Step 4: Set Up the Inequality
Since \( \sin x \) must equal \( \frac{x}{10} \), we need to find where:
\[ -1 \leq \frac{x}{10} \leq 1 \]
This leads to:
\[ -10 \leq x \leq 10 \]
However, since we are only considering the interval \([-π, π]\), we restrict our analysis to this interval.

### Step 5: Graph the Functions
Now, we can graph both functions:
- The graph of \( y = \sin x \) oscillates between -1 and 1.
- The line \( y = \frac{x}{10} \) is a straight line passing through the origin with a slope of \( \frac{1}{10} \).

### Step 6: Identify Intersections
We need to find the points where these two graphs intersect within the interval \([-π, π]\).

1. **At \( x = 0 \)**:
\[ \sin(0) = 0 \quad \text{and} \quad \frac{0}{10} = 0 \quad \Rightarrow \text{Intersection at } (0, 0) \]

2. **For \( x < 0 \)**:
The sine function decreases from 0 to -1 and then increases back to 0. The line \( y = \frac{x}{10} \) will intersect the sine curve twice (once while decreasing and once while increasing).

3. **For \( x > 0 \)**:
The sine function increases from 0 to 1 and then decreases back to 0. Similarly, the line will intersect the sine curve twice (once while increasing and once while decreasing).

### Step 7: Count the Intersections
From the analysis:
- One intersection at \( x = 0 \)
- Two intersections for \( x < 0 \)
- Two intersections for \( x > 0 \)

Thus, the total number of intersections is:
\[ 1 + 2 + 2 = 5 \]

### Final Answer
The number of intersecting points on the graph for \( \sin x = \frac{x}{10} \) for \( x \) in \([-π, π]\) is **5**.

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