In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. All of these approaches will be presented below.
In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations. A few other functions were common historically (and appeared in the earliest tables), but are now seldom used, such as the versine (1 − cos θ) and the exsecant (sec θ − 1). Many more relations between these functions are listed in the article about trigonometric identities.
| Function | Abbreviation | Relation |
| Sine | sin |  |
| Cosine | cos |  |
| Tangent | tan |  |
| Cotangent | cot |  |
| Secant | sec |  |
| Cosecant | csc (or cosec) |
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