The solutions to the equation $\sin(x) = 0$ are given by the expression $x = n\pi$, where $n$ is any integer.

To solve the equation $\sin(x) = 0$, we need to identify the values of $x$ for which the sine function equals zero. The sine function, denoted as $\sin(x)$, is periodic with a period of $2\pi$, indicating that it repeats its values every $2\pi$ radians. Within each period, the sine function takes the value of zero at specific angles.

In one complete cycle (or period), the sine function equals zero at the points $x = 0$, $x = \pi$, and $x = 2\pi$. More broadly, $\sin(x)$ is zero at every integer multiple of $\pi$. Thus, the solutions to the equation $\sin(x) = 0$ can be expressed as $x = n\pi$, where $n$ can be any integer (including positive, negative, or zero). This notation encompasses the values such as $\ldots, -2\pi, -\pi, 0, \pi, 2\pi, 3\pi$, and so on.

To grasp why this occurs, consider the unit circle. On this circle, the sine of an angle corresponds to the y-coordinate of the point that represents that angle. The y-coordinate is zero at the points $(1, 0)$ and $(-1, 0)$, which correspond to the angles $0$, $\pi$, $2\pi$, and so on. These points recur every $\pi$ radians, leading us to the general solution $x = n\pi$.

In conclusion, the equation $\sin(x) = 0$ has an infinite number of solutions, all of which are integer multiples of $\pi$. Recognizing this pattern is essential for understanding the behavior of trigonometric functions and for solving more complex trigonometric equations.