The period of the function $y = \tan(x)$ is $\pi$ radians.

To elaborate, the period of a function refers to the length of the smallest interval over which the function exhibits repetitive behavior. For the tangent function $y = \tan(x)$, this interval is $\pi$ radians. This implies that for any value of $x$, if you add $\pi$ to it, the value of $\tan(x)$ remains unchanged. This relationship can be mathematically expressed as:

The periodicity of the tangent function arises from its definition. In the context of a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. When considering the unit circle, the tangent function can be interpreted as the ratio of the y-coordinate to the x-coordinate of a point located on the circle. As one traverses the circle, the values of the tangent function repeat every $\pi$ radians due to the symmetry of the circle, which results in the same ratios reappearing.

Grasping the period of the tangent function is essential for solving trigonometric equations and analyzing graphs. For instance, if tasked with finding all solutions to the equation $\tan(x) = 1$, understanding the period indicates that solutions will occur at intervals of $\pi$ radians, such as $x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}$, and so forth.

In conclusion, the period of $y = \tan(x)$ is $\pi$ radians, signifying that the function repeats its values every $\pi$ radians. This characteristic is fundamental for comprehending and working with the tangent function in trigonometry.