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Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them.
The argument to the hyperbolic functions is a hyperbolic angle measure. In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant ...
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.
A few functions were common historically, but are now seldom used, such as the chord, versine (which appeared in the earliest tables [30]), haversine, coversine, [39] half-tangent (tangent of half an angle), and exsecant. List of trigonometric identities shows more relations between these functions.
The Gudermannian function z ↦ gd z is a conformal map from an infinite strip to an infinite strip. It can be broken into two parts: a map z ↦ tanh 1 / 2 z from one infinite strip to the complex unit disk and a map ζ ↦ 2 arctan ζ from the disk to the other infinite strip.
A Riemann surface for the argument of the relation tan z = x. The orange sheet in the middle is the principal sheet representing arctan x. The blue sheet above and green sheet below are displaced by 2π and −2π respectively. Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the ...
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
Here, is the Hyperbolic tangent and the superscript of -1, above it, is shorthand notation for the inverse function of the Trigonometric function, tangent. For k > 1 {\displaystyle k>1} , a smooth approximation of the sign function is sgn x ≈ tanh k x . {\displaystyle \operatorname {sgn} x\approx \tanh kx\,.}