Inverse hyperbolic functions - Wikipedia
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.
Hyperbolic Functions: Inverses - Imperial College London
Hyperbolic Functions: Inverses. 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.
Hyperbolic Functions - sinh, cosh, tanh, coth, sech, csch - Math10
Inverse hyperbolic functions. 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.
y =cosh−1 x. By definition of an inverse function, we want a function that satisfies the condition x =coshy = e y+e− 2 by definition of coshy = e y+e−y 2 e ey = e2y +1 2ey. 2eyx = e2y +1. e2y −2xey +1 = 0. (ey)2 −2x(ey)+1 = 0. ey = 2x+ √ 4x2 −4 2 = x+ x2 −1. ln(ey)=ln(x+ x2 −1). y =ln(x+ x2 −1). Thus cosh−1 x =ln(x+ x2 ...
calculus - Inverse of cosh(x) - Mathematics Stack Exchange
But if we restrict the domain of $\cosh$ suitably, then there is an inverse. The usual definition of $\cosh^{-1}x$ is that it is the non-negative number whose $\cosh$ is $x$ . Note that for $x\gt 1$ , we have $$x-\sqrt{x^2-1}=\frac{1}{x+\sqrt{x^2-1}}\lt 1,$$ and therefore $\ln(x-\sqrt{x^2-1})\lt 0$ whereas we were looking for the non-negative ...
Inverse Hyperbolic Functions - Formulas, Graphs, & Examples
Sep 19, 2024 · Inverse Hyperbolic Cosecant (cosech -1 or csch -1) The formula of the inverse hyperbolic cosecant is expressed as: cosech-1 x = $ {\ln \left ( \dfrac {1+\sqrt {1+x^ {2}}} {x}\right)}$. Proof. Let csch -1 x = y, where y ∈ ℝ. ⇒ x = csch y. Using the exponential combination of the hyperbolic cosecant function, we get.
Inverse Hyperbolic Functions -- from Wolfram MathWorld
Oct 10, 2024 · The inverse hyperbolic functions, sometimes also called the area hyperbolic functions (Spanier and Oldham 1987, p. 263) are the multivalued function that are the inverse functions of the hyperbolic functions. They are denoted cosh^(-1)z, coth^(-1)z, csch^(-1)z, sech^(-1)z, sinh^(-1)z, and tanh^(-1)z.
Cosh Calculator | Hyperbolic Cosine Function
Jul 23, 2024 · What is the inverse cosh function? The inverse cosh function is called arcosh , and its formula reads: arcosh x = ln ( x + x 2 − 1 ) \small \begin{align*} \text{arcosh}\ x &= \ln \left(x + \sqrt{x^2 - 1}\right)\\[1em] \end{align*} arcosh x = ln ( x + x 2 − 1 )
Inverse Hyperbolic Cosine -- from Wolfram MathWorld
Oct 10, 2024 · The inverse hyperbolic cosine cosh^(-1)z (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic cosine (Harris and Stocker 1998, p. 264) is the multivalued function that is the inverse function of the hyperbolic cosine.
Inverse hyperbolic functions - Encyclopedia of Mathematics
Jan 17, 2024 · The inverse hyperbolic functions are single-valued and continuous at each point of their domain of definition, except for $ \cosh ^ {-1} x $, which is two-valued.