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.

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 hyperbolic sine function, \sinh x, is one-to-one, and therefore has a well-defined inverse, \sinh^{-1} x, shown in blue in the figure. In order to invert the hyperbolic cosine function, however, we need (as with square root) to restrict its domain.

Apr 8, 2025 · The inverse hyperbolic sine sinh^ (-1)z (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264) is the multivalued function that is the inverse function of the hyperbolic sine.

Oct 6, 2020 · I am trying to formulate $\sinh^{-1} x$. I know that $$\sinh x = \dfrac{\exp(x)-\exp(-x)}{2}$$ but how do we translate $\sinh^{-1} x$? Please excuse my lack of knowledge in math.

Nov 4, 2024 · The formula of the inverse hyperbolic sine is given by: sinh-1 x = ln (x + 1 + x 2) Proof. Let sinh -1 x = y, where y ∈ ℝ. ⇒ x = sinh y. Using the exponential combination of the hyperbolic sine function, we get. ⇒ sinh y = e y − e − y 2. ⇒ x = e y − e − y 2. ⇒ 2x = e y – e -y. ⇒ e 2y – 2xe y – 1 = 0 ….. (i)

The two basic hyperbolic functions are "sinh" and "cosh": sinh (x) = ex − e-x 2. cosh (x) = ex + e-x 2. They use the natural exponential function ex. And are not the same as sin (x) and cos (x), but a little bit similar: One of the interesting uses of Hyperbolic Functions is the curve made by suspended cables or chains.

sinh (x)/cosh (x), valid input values range from -1 to 1. 1/cosh (x), valid input values range from 0 to 1. This sinh calculator allows you to quickly determine the values of the hyperbolic sine function.

The calculator will find the inverse hyperbolic sine of the given value. The inverse hyperbolic sine y=\sinh^ {-1} (x) y = sinh−1(x) or y=\operatorname {asinh} (x) y = asinh(x) or y=\operatorname {arcsinh} (x) y = arcsinh(x) is such a function that \sinh (y)=x sinh(y) = x.