Find the number of solutions of the equation $\sin x=\frac{x}{10}$ in the interval $(-\pi ,\pi )$.

Solution

1. Let $f(x)=\sin x$ and $g(x)=\frac{x}{10}$.
2. At $x=0$, $f(0)=0$ and $g(0)=0$, so $x=0$ is one solution.
3. Since the slope of $\sin x$ at $x=0$ is $1$ and the slope of $\frac{x}{10}$ is $0.1$, the line intersects the curve at $x=0$ and two other points.
4. For $x>0$, $\sin x$ decreases from $1$ to $0$ while $\frac{x}{10}$ increases, leading to one intersection in $(0,\pi )$.
5. By symmetry, there is one intersection in $(-\pi ,0)$.
6. Hence the answer is (C).