How many solutions does the following equation have:

$$
x=3142 \sin x+157 .
$$
We understand the angle in radians, as is generally the case with trigonometric equations. For solving such equations, where trigonometric functions and the angle itself are involved, approximate methods usually lead to the goal. For determining an initial approximation, a graphical representation is often well-suited. Here, too, we determine the number of solutions based on the graphical representation of functions.

$\sin x$ only takes values between -1 and 1, so only those $x$ values are relevant for which

$$
x \leqq 3142+157=3299 \quad \text { and } \quad x \geqq-3142+157=-2985
$$

For better clarity, we write the equation as

$$
\sin x=\frac{x-157}{3142}
$$

We denote the linear expression on the right side by $e(x)$. Its graph is the line $e$. As long as $e(x) \leqq 0$, i.e., $x \leqq 157$, the line can only intersect the parts of the $\sin x$ curve that lie below the $X$-axis, i.e., where

$$
(2 k-1) \pi \leqq x \leqq 2 k \pi \quad (k \text { is an integer })
$$

When $e(x) \geqq 0$, i.e., $x \geqq 157$, it can only intersect the parts of the $\sin x$ curve that lie above the $X$-axis, i.e., where

$$
2 k \pi \leqq x \leqq(2 k+1) \pi
$$

Every part of the curve below the $X$-axis is intersected by the line as long as

$$
\begin{aligned}
&-2985 \leqq (2 k-1) \pi \quad \text { and } \quad 2 k \pi \leqq 157, \quad \text { i.e., } \\
&-\frac{2985}{2 \pi}+\frac{1}{2} \leqq k \leqq \frac{157}{2 \pi}, \quad -474.57 \ldots \leqq k \leqq 24.98 \ldots
\end{aligned}
$$

because in this case, at the endpoints of (3), $\sin x=0 > e(x)$, while at the midpoint

$$
\sin \left(2 k-\frac{1}{2}\right) \pi=-1=e(-2985) \quad e\left(\left[2 k+\frac{1}{2}\right] \pi\right)>0
$$

and there are 2 intersection points on every arc because the interval (4) and the arc above it form a convex domain. (6) is satisfied by the values $k=25, 26, \ldots, 524$, thus we get 500 arcs and 1000 intersection points (Figure 4).

(2) is also satisfied by the part of the interval (4) corresponding to $k=525$ (1050 \pi=3298.57 \ldots, 3299) (Figure 5), which is shorter than $\pi / 2$, so $\sin x$ does not reach the value $1=e(3299)$ within it, and there is no intersection point.

![](https://cdn.mathpix.com/cropped/2024_05_02_b9ce2be96d61c106b784g-2.jpg?height=282&width=353&top_left_y=1504&top_left_x=802)

Figure 5

According to all this, the line $e$ intersects the $\sin x$ curve at 1999 points, which is the number of solutions to equation (1). Fencsik Gábor (Budapest, Berzsenyi D. g. III. o. t.) Dabóczi Ákos (Budapest, Fáy A. g. III. o. t.)