Editing Trigonometric functions (section)

{{short description|Functions of an angle}}
{{redir-multi|2|Logarithmic sine|Logarithmic cosine|the Clausen-related functions|log cosine function|and|log sine function}}
{{Use dmy dates|date=September 2021|cs1-dates=y}}
[[File:Academ Base of trigonometry.svg|thumb|upright=1.35|Basis of trigonometry: if two [[right triangle]]s have equal [[acute angle]]s, they are [[Similarity (geometry)|similar]], so their corresponding side lengths are [[Proportionality (mathematics)|proportional]].]]
In [[mathematics]], the '''trigonometric functions''' (also called '''circular functions''', '''angle functions''' or '''goniometric functions'''){{r|klein}} are [[real function]]s 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 function]]s, and as such are also widely used for studying periodic phenomena through [[Fourier analysis]].
{{Trigonometry}}

The trigonometric functions most widely used in modern mathematics are the [[sine]], the [[cosine]], and the '''tangent''' functions. Their [[multiplicative inverse|reciprocal]]s are respectively the '''cosecant''', the '''secant''', and the '''cotangent''' functions, which are less used. Each of these six trigonometric functions has a corresponding [[Inverse trigonometric functions|inverse function]], and an analog among the [[hyperbolic functions]].

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for [[acute angle]]s. To extend the sine and cosine functions to functions whose [[domain of a function|domain]] is the whole [[real line]], geometrical definitions using the standard [[unit circle]] (i.e., a circle with [[radius]] 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as [[Series (mathematics)|infinite series]] or as solutions of [[differential equation]]s. This allows extending the domain of sine and cosine functions to the whole [[complex plane]], and the domain of the other trigonometric functions to the complex plane with some isolated points removed.