Trigonometric functions

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are <a href="/facts/Real_function/bbNG3DnN">real functions</a> which relate an angle of a <a href="/facts/Right-angled_triangle/dD6M9GAn">right-angled triangle</a> to ratios of two side lengths. They are widely used in all sciences that are related to <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, such as <a href="/facts/Navigation/sGGAJ0AQ">navigation</a>, <a href="/facts/Solid_mechanics/e1Yq50IS">solid mechanics</a>, <a href="/facts/Celestial_mechanics/FA3uR78J">celestial mechanics</a>, <a href="/facts/Geodesy/EJH0Te0m">geodesy</a>, and many others. They are among the simplest <a href="/facts/Periodic_function/Sm8lJQsF">periodic functions</a>, and as such are also widely used for studying periodic phenomena through <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a>.

The trigonometric functions most widely used in modern mathematics are the <a href="/facts/Sine/gQ6LXK8z">sine</a>, the <a href="/facts/Cosine/czej7ur0">cosine</a>, and the tangent functions. Their <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocals</a> are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding <a href="/facts/Inverse_trigonometric_functions/bVwU9FPG">inverse function</a>, and an analog among the <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic functions</a>.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for <a href="/facts/Acute_angle/gFUTEQYq">acute angles</a>. To extend the sine and cosine functions to functions whose <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> is the whole <a href="/facts/Real_line/6eq42xX5">real line</a>, geometrical definitions using the standard <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> (i.e., a circle with <a href="/facts/Radius/9Gr9deYD">radius</a> 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 <a href="/facts/Series_(mathematics)/yDnlsgIe">infinite series</a> or as solutions of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>. This allows extending the domain of sine and cosine functions to the whole <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.

Trigonometric functions open-in-new

Trigonometric functions