## Task B-3.5.

Determine the number of real solutions to the equation $\sin x=\frac{x}{2017 \pi}$.
## Solution.

To determine the number of solutions of the given equation, we will use the graphical representation of the functions $f(x) = \sin x$ and $g(x) = \frac{x}{2017 \pi}$.

The number of solutions is equal to the number of points where the graphs of functions $f$ and $g$ intersect.

![](https://cdn.mathpix.com/cropped/2024_05_30_7713364b1a15726862ddg-14.jpg?height=223&width=1471&top_left_y=1459&top_left_x=178)

![](https://cdn.mathpix.com/cropped/2024_05_30_7713364b1a15726862ddg-14.jpg?height=54&width=1545&top_left_y=1818&top_left_x=187)
$g(-2017 \pi) = -1$ and $g(2017 \pi) = 1$.

The line $y = \frac{x}{2017 \pi}$ intersects the graph of function $f$ at 2 points within the basic period of $2 \pi$.

Since $2017 \pi = 1008 \cdot 2 \pi + \pi$, on the interval $[0, 2017 \pi]$, there are $1008 \cdot 2 + 2 = 2018$ intersections. Similarly, on the interval $[-2017 \pi, 0]$, there are also 2018 intersections.

In this process, the origin has been counted twice, so the final number of real solutions to the given equation is $2 \cdot 2018 - 1 = 4035$.