In mathematics, the tangent function, often abbreviated as "tan," is a trigonometric function that relates the angle of a right triangle to the ratio of the length of the side opposite that angle to the length of the adjacent side.

The tangent of an angle θ, denoted as tan(θ), is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. This can be expressed as:

tan(θ) = opposite/adjacent

In a right triangle, the opposite side refers to the side opposite to the angle θ, and the adjacent side refers to the side adjacent to the angle θ. The tangent function allows us to calculate the ratio of these sides for a given angle.

The tangent function is periodic, with a period of π radians or 180 degrees. It takes on various values depending on the angle. For example, when the angle is 0 degrees or 0 radians, the tangent is 0. As the angle approaches 90 degrees or π/2 radians, the tangent tends towards positive infinity. Similarly, as the angle approaches -90 degrees or -π/2 radians, the tangent tends towards negative infinity.

The tangent function has various applications in mathematics, physics, engineering, and other fields where angles and ratios are involved. It is commonly used in trigonometry to solve problems related to triangles and circular functions.