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Graphs of the inverse hyperbolic functions The hyperbolic functions sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = Aθ , from trigonometry, sin θ = O / H and tan θ = O / A , and from the picture, O ≈ s and H ≈ A leads to: sin θ = O H ≈ O A = tan θ = O A ≈ s A = A θ A = θ . {\displaystyle \sin \theta ={\frac ...
Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane. In the descriptions below the constant Gaussian curvature of the plane is −1. Sinh, cosh and tanh are hyperbolic functions.
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). [51] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. [52]
Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [1] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices . Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane.
tanh – hyperbolic tangent function. TFAE – the following are equivalent. tg – tangent function. (Also written as tan, tgn.) tgn – tangent function. (Also written as tan, tg.) Thm – theorem. Tor – Tor functor. Tr – field trace. tr – trace of a matrix or linear transformation. (Also written as Sp.)