# Trigonometric functions

In mathematics, the **trigonometric functions** (also called **circular functions**, **angle functions** or **goniometric functions**[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

Trigonometry |
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Reference |

Laws and theorems |

Calculus |

The most widely used trigonometric functions are the sine, the **cosine**, and the **tangent**. Their reciprocals are respectively the **cosecant**, the **secant**, and the **cotangent**, which are less used in modern mathematics.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. For extending these definitions to functions whose domain is the whole projectively extended real line, one can use geometrical definitions using the standard unit circle (a circle with radius 1 unit). Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of the sine and the cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

## Right-angled triangle definitions

*In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length.*

Given an acute angle A of a right-angled triangle (see figure) the hypotenuse h is the side that connects the two acute angles. The side b *adjacent* to A is the side of the triangle that connects A to the right angle. The third side a is said *opposite* to A.

If the angle A is given, then all sides of the right-angled triangle are well defined up to a scaling factor. This means that the ratio of any two side lengths depends only on A. These six ratios define thus six functions of A, which are the trigonometric functions. More precisely, the six trigonometric functions are:[3]