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.

If x = sinh y, then y = sinh-1 a is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions we restrict ourselves to principal values for which they can be considered as single-valued.

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

5 days ago · The hyperbolic sine is defined as sinhz=1/2(e^z-e^(-z)). (1) The notation shz is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Sinh[z].

cosh(z) = cos(iz) and sinh(z) = −isin(iz). It follows that sinh(−z) = −isin(−iz) = isin(iz) = −sinh(z), cosh(−z) = cos(−iz) = cos(iz) = cosh(z), cosh2(z)−sinh2(z) = cos2(iz)−(−isin(iz))2 = cos2(iz)+sin2(iz) = 1, sinh(z 1 +z 2) = −isin(i(z 1 +z 2) = −isin(iz 1)cos(iz 2)−icos(iz 1)sin(iz 2) = sinh(z 1)cosh(z 2)+cosh(z 1 ...

Feb 8, 2024 · Let $z \in \C$ be a complex number. Then: $i \sinh z = \map \sin {i z}$ where: $\sin$ denotes the complex sine $\sinh$ denotes the hyperbolic sine $i$ is the imaginary unit: $i^2 = -1$. Proof

cosh 2 (x) - sinh 2 (x) = 1 tanh 2 (x) + sech 2 (x) = 1 coth 2 (x) - csch 2 (x) = 1 Inverse Hyperbolic Defintions. arcsinh(z) = ln( z + (z 2 + 1) ) arccosh(z) = ln( z (z 2 - 1) ) arctanh(z) = 1/2 ln( (1+z)/(1-z) ) arccsch(z) = ln( (1+ (1+z 2) )/z ) arcsech(z) = ln( (1 (1-z 2) )/z ) arccoth(z) = 1/2 ln( (z+1)/(z-1) ) Relations to Trigonometric ...

Sinh. Elementary Functions Sinh: Introduction to the Hyperbolic Sine Function : Defining the hyperbolic sine function : A quick look at the hyperbolic sine function : Representation through more general functions : Definition of the hyperbolic sine function for a complex argument :

This sinh calculator allows you to quickly determine the values of the hyperbolic sine function.

In the following description, \(z\) stands for the complex number. \(x\) stands for the real value \(Re\) and \(y\) for the imaginary value \(Im\). \(sinh(z) = sinh(x) · cos(y) + cosh(x) · sin(y)\)