How many roots does the equation $\sin x = x / 100$ have?

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The problem is easily solved graphically. Since the graphs of the sine function and the function $y = x / 100$ are symmetric with respect to the origin, it is sufficient to consider the right part of the graphs. The maximum value of the sine function is 1. Therefore, the points of intersection of the graphs will be within those values of $x$ for which $x / 100$ does not exceed 1, i.e., within the range from 0 to 100. In this interval, there are $100 / 2 \pi$ periods of $\sin x$, $100 / 2 \pi \approx 15.9$. In each period of $\sin x$, the sine wave and the graph of the line $y = x / 100$ have two points of intersection, and in the first half of the period (see figure). Therefore, within 15.5 periods, there will be 32 points of intersection of the graphs. The same number of points of intersection of the graphs will be located to the left of the origin, but it is necessary to take into account that the origin is counted by us twice. Therefore, the equation has a total of 63 roots.

![](https://cdn.mathpix.com/cropped/2024_05_06_ed128c2d5b4d71db92ebg-44.jpg?height=592&width=918&top_left_y=2059&top_left_x=570)

## Answer

63 roots.