In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.

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, inverse hyperbolic secant, and inverse hyperbolic cotangent.

One of the interesting uses of Hyperbolic Functions is the curve made by suspended cables or chains. A hanging cable forms a curve called a catenary defined using the cosh function: f (x) = a cosh (x/a) Like in this example from the page arc length : From sinh and cosh we can create: tanh (x) = sinh (x) cosh (x) = ex − e-x ex + e-x.

$\text{tanh} (x + k\pi i) = \text{tanh}\ x$ $\text{coth} (x + k\pi i) =\text{coth} x$ Relationship between inverse hyperbolic and inverse trigonometric functions

Hyperbolic Definitions sinh(x) = ( e x - e-x)/2 . csch(x) = 1/sinh(x) = 2/( e x - e-x) . cosh(x) = ( e x + e-x)/2 . sech(x) = 1/cosh(x) = 2/( e x + e-x) . tanh(x ...

The hyperbolic tangent function is also one-to-one and invertible; its inverse, \tanh^{-1} x, is shown in green. It is defined only for -1 x 1 . Just as the hyperbolic functions themselves may be expressed in terms of exponential functions, so their inverses may …

Nov 25, 2024 · Learn the different hyperbolic trigonometric functions, including sine, cosine, and tangent, with their formulas, examples, and diagrams. Also, learn their identities.

This tanh calculator is the ultimate tool you need to deal with the hyperbolic tangent function! In what follows, we will explain what tanh is in math and how to calculate both the inverse and the derivative of tanh. We won't forget to give you extra tips on how to use this hyperbolic tangent calculator most efficiently. Let's go!

Nov 4, 2024 · Inverse hyperbolic tangent (tanh-1 or arctanh) These functions, similar to the trigonometric (or circular) functions, are widely applied in mathematics for solving integrals and differential equations.

Aug 13, 2024 · Inverse hyperbolic functions are the inverse functions of the hyperbolic functions: hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh), hyperbolic cotangent (coth), hyperbolic secant (sech), and hyperbolic cosecant (csch). These functions are important in various fields, including engineering, physics, and mathematics.