11.17. How many roots does the equation $\sin x=\frac{x}{100}$ have?

## 11.4. Sums of Sines and Cosines Related to Regular Polygons

In solving the problems of this section, the following geometric problem is useful.
11.17. Answer: 63. First, note that the number of positive roots is equal to the number of negative roots, and there is also a root 0. Therefore, it is sufficient to verify that the number of positive roots is 31. If $\sin x = x / 100$, then $|x| = 100|\sin x| \leqslant 100$. Consider the graphs of the functions $y = x / 100$ and $y = \sin x$. The segment of the $O x$ axis from 0 to 100 contains 15 segments of length $2 \pi$ and one segment of length less than $2 \pi$. By examining the specified graphs, it is easy to see that on the first segment of length $2 \pi$ there is one root of the given equation, and on each of the other 14 segments of length $2 \pi$ there are two roots. Calculations show that the length of the last segment is greater than $\pi$, so there are also two roots on it. In total, we get 31 positive roots.