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 …

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, …

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)) …

Illustrated definition of Sinh: The hyperbolic sine function sinh (x) = (ex minus; eminus;x) /...

One of the key characteristics that motivates the hyperbolic trigonometric functions is the striking similarity to trigonometric functions, which can be seen from Euler's formula:

The properties of the sinh function, such as its connection to the exponential function and its derivative and integral relationships, are fundamental to the mathematical formulations and …

These functions are most conveniently defined in terms of the exponential function, with sinh z = 1/2 (ez − e−z) and cosh z = 1/2 (ez + e−z) and with the other hyperbolic trigonometric …

May 17, 2025 · Dive into the properties, graphs, and applications of the hyperbolic sine function sinh in trigonometry, complete with step-by-step examples.

Sep 25, 2020 · If y = sinh (x), we can define the inverse function x = sinh -1 y, and similarly for cosh and tanh. The inverses of sinh and tanh are uniquely defined for all x.

Since sinh and cosh were de ned in terms of the exponential function that we know and love, proving all the properties and identities above was no big deal. On the other hand, you spent a …