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.

The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh(x) = e x − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh(x) = e x + e-x 2 (pronounced "cosh") They use the natural exponential function e x. And are not the same as sin(x) and cos(x), but a little bit similar: sinh vs sin. cosh vs cos. Catenary

Figure \(\PageIndex{2}\): Geometric definitions of sin, cos, sinh, cosh: \(t\) is twice the shaded area in each figure. Given the definitions of the hyperbolic functions, finding their derivatives is straightforward.

Geometric definitions of sin, cos, sinh, cosh: t t is twice the shaded area in each figure. Given the definitions of the hyperbolic functions, finding their derivatives is straightforward. Here again we see similarities to the trigonometric functions.

$\sinh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{2}}$ [+ if x > 0, - if x . 0] $\cosh \frac{x}{2} = \sqrt{\frac{\cosh x + 1}{2}}$ $\tanh \frac{x}{2} = \pm \sqrt{\frac{\cosh x - 1}{\cosh x + 1}}$ [+ if x > 0, - if x . 0] $=\frac{\text{sinh}(x)}{1 + \text{cosh}(x)} = \frac{\text{cosh}(x) - …

Sep 18, 2023 · In this article we will look at the hyperbolic functions sinh and cosh. We will see why they are called hyperbolic functions, how they relate to sine and cosine, and why the parameter of the sinh and cosh functions can be considered to represent an angle. The sinh function is defined as: The cosh function is defined as:

Aug 29, 2023 · All six hyperbolic functions can now be defined in general, analogous to the trigonometric (circular) functions: The graphs of the hyperbolic functions are shown below: The graph of y = coshx in Figure [fig:hyperfcns] (a) might look familiar: a catenary —a uniform cable hanging from two fixed points—has the shape of a hyperbolic cosine function.

The hyperbolic cosine (cosh) is one of the hyperbolic functions in trigonometry. $$ \cosh(x) $$ Its domain is all real numbers (-∞, ∞), and its range is [1, ∞) Unlike the cosine, the hyperbolic cosine is not calculated using the unit circle, but instead on the hyperbola x 2 - y 2 = 1.

The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are presented. The six hyperbolic functions are defined as follows: Hyperbolic Sine Function : \( \sinh(x) = \dfrac{e^x - e^{-x}}{2} \)

This applet explores the relation between the area under a region of a hyperbolic cosine (cosh) function, and the length of that function's graph over the same region. It can be shown that this ratio is constant, and the constant is the parameter of the cosh function.