4040.11 – When Does sin(x) = x/100?

This problem is great for a graphing calculator.

Because $|\sin(x)| \leq 1$, we need to look for values of $x$ such that $\left|\dfrac{x}{100}\right| \leq 1$, that is, $|x| \leq 100$. In each period of $2\pi$, from 0 to 100, there are two intersections of $f(x) = \sin(x)$ and $g(x) = \dfrac{x}{100}$, including the one at $x = 0$.

We know $\dfrac{100}{2\pi}$ is about $15.9$, so there are 16 pairs of intersections between 0 and 100. The 16th cycle, though not complete, does include the 2 intersections. Likewise, there are 16 pairs of intersections between -100 and 0, so that makes 32 pairs of intersections, or 64 solutions. Wait, we can't count 0 twice, so there are 63 solutions.