257. a) How many roots does the equation

$$
\sin x=\frac{x}{100} ?
$$

b) How many roots does the equation

$$
\sin x=\lg x ?
$$
257. a) First of all, note that if $x_{0}$ is a root of the equation, then $-x_{0}$ is also a root. Therefore, the number of negative roots is the same as the number of positive roots. Furthermore, the number 0 is a root of the equation. Thus, it is sufficient to find the number of positive roots. Now, let's note that if $\frac{x}{100}=\sin x$, then

$$
|x|=100|\sin x| \leqslant 100 \cdot 1=100
$$

We obtain that the root of the equation cannot have an absolute value greater than 100.

Divide the segment of the $O x$ axis from $x=0$ to $x=100$ into segments of length $2 \pi$ (the last segment may be shorter) and count the number of roots of our equation on each of these segments. Refer to Fig. 29.

On the first segment (from 0 to $2 \pi$), there is one positive root (and one root zero), on each subsequent segment, except the last one, there are two roots. To determine the number of roots on the last segment, let's estimate its size: $\frac{100}{2 \pi}$ is obviously between 15 and $16\left(\frac{100}{15}=6.66 \ldots>2 \pi ; \quad \frac{100}{16}=6.255>\pi \right)$; therefore, on this segment, a complete positive half-wave of the sine wave fits and, thus, there are also two roots on it.

Thus, we have $1+14 \cdot 2+2=31$ positive roots of our equation. There are as many negative roots, and one root is zero.

Finally, the total number of roots is $31 \cdot 2+1=63$.

![](https://cdn.mathpix.com/cropped/2024_05_21_b937612abb2773f7abffg-282.jpg?height=199&width=908&top_left_y=365&top_left_x=120)

Fig. 29.

b) The solution to this problem is analogous to the solution of part a). First of all, it is obvious that if $\sin x=\lg x$, then $x \leqslant 10$ (otherwise, the left side of our equality is no more than 1, while the right side is greater). Since $2 \cdot 2 \pi>10$, on the segment of the $O x$ axis from $x=0$ to $x=10$, one complete wave of the sine curve $y=\sin x$ and part of the next wave fit (Fig. 30). The graph of $y=\lg x$,

![](https://cdn.mathpix.com/cropped/2024_05_21_b937612abb2773f7abffg-282.jpg?height=296&width=948&top_left_y=891&top_left_x=85)

Fig. 30.

obviously intersects the first wave of the sine curve at one point. Further, since $2 \pi+\frac{\pi}{2} \lg x$ and, thus, the graph of $y=\lg x$ intersects the first half of the second positive half-wave; since $\lg x=1>\sin x$ at the point $x=10$, it must intersect the second half of this half-wave as well. Thus, we conclude that the equation $\sin x=\lg x$ has a total of three roots.