## Tangent

### Tangent is a trigonometric function that relates the ratio of the lengths of two sides of a right triangle

Tangent is a trigonometric function that relates the ratio of the lengths of two sides of a right triangle. Specifically, tangent is defined as the ratio of the length of the side opposite to an acute angle to the length of the adjacent side. In other words, tangent is the ratio of the sine of an angle to the cosine of that angle.

The tangent function is commonly abbreviated as “tan” and is denoted as tan(x), where x represents the angle in question. It is important to note that tangent is only defined for acute angles, as the tangent of a right angle (90 degrees) is undefined.

Mathematically, the tangent of an angle can be calculated using the following formula:

tan(x) = opposite / adjacent

or in terms of the sine and cosine functions:

tan(x) = sin(x) / cos(x)

The tangent function has a periodicity of pi radians or 180 degrees, meaning that tan(x) = tan(x + n * pi), where n is an integer. It is also worth mentioning that the tangent function has asymptotes at every value of x that satisfies cos(x) = 0.

Tangent has various applications in mathematics, physics, and engineering. It is commonly used in trigonometry to solve for missing sides or angles of a right triangle. Additionally, tangent can be used to model and analyze periodic phenomena, such as oscillating motion or waveforms.

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