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 …

The two basic hyperbolic functions are sinh and cosh: sinh (x) = ex - e-x2. (pronounced shine or sinch). cosh (x) = ex + e-x2.

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

Definition 4.11.1 The hyperbolic cosine is the function cosh x = e x + e x 2, and the hyperbolic sine is the function sinh x = e x e x 2 . Notice that cosh is even (that is, cosh (x) = cosh (x)) …

There are six hyperbolic functions are sinh x, cosh x, tanh x, coth x, sech x, csch x. In this article, we will define these hyperbolic functions and their properties, graphs, identities, derivatives, …

This shape, defined as the graph of the function y = λ cosh x λ y = λcosh λx, is also referred to as a catenary. The shape of a dangling chain is a hyperbolic cosine.

Cosh, short for hyperbolic cosine, is a mathematical function that is part of the family of hyperbolic functions. It is closely related to the hyperbolic sine function and is used extensively in the …

Sep 25, 2020 · There are results very similar to those for circular functions; they are easily proved directly from the definitions of cosh and sinh: If y = sinh (x), we can define the inverse function …

Jun 20, 2025 · The cosh function is a hyperbolic function that describes the ratio of the sum of two exponential functions to 2. It has numerous practical applications in physics, engineering, and …

The identity cosh 2 t sinh 2 t, shown in Figure 7, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from …