Inverse Hyperbolic FunctionsInverse Hyperbolic FunctionsIn mathematics, the inverse hyperbolic functions provide a hyperbolic angle
corresponding to a given value of a hyperbolic function.
The size of the hyperbolic angle is equal to the area of the corresponding
hyperbolic sector of the hyperbola x y = 1, while the area of a circular sector of
the unit circle is one-half the corresponding circular angle.
Some authors have called inverse hyperbolic functions "area functions" to realize
the hyperbolic angles.
The abbreviations arcsinh, arccosh, etc., are commonly used, even though they
are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix
ar stands for area.
Other authors prefer to use the notation argsinh, argcosh, argtanh, and so on. In
computer science this is often shortened to asinh.
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The notation sinh-1(x), cosh-1(x), etc., is also used, despite the fact that care
must be taken to avoid misinterpretations of the superscript -1 as a power as
opposed to a shorthand for inverse (e.g., cosh-1(x) versus cosh(x)-1).
hyperbolic inverse Functions are used to find the area of Hyperbolic Functions. As
we know, hyperbolic functions are similar to the Trigonometric Functions and they
are represented in terms of the Exponential Function.
Here we are going to define three main inverse hyperbolic functions. We will also
talk about some inverse hyperbolic functions identities which involve these
functions, with their inverse functions and reciprocal functions.
The functions f (x) = cosh x and f (x) = sinh x in terms of the exponential function.
The function f (x) = tanh x in terms of cosh x and sinh x.
The Inverse Function means sinh-1 s, cosh-1 s and tanh-1 s and specifies their
domains.
The hyperbolic Trigonometry involves different type of hyperbolic functions. The
trigonometric functions expressed in the form of ex are hyperbolic trigonometric
functions.
Let's talk about the different types of inverse hyperbolic functions:
Hyperbolic of sine h (arc cos h s) = s2 - 1 for | s | > 1,
Hyperbolic of sine h (arc tan h s) = x / 1 - s2 for -1 < s < 1,
Hyperbolic of cosine h (arc sin h s) = s2 + 1,
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Hyperbolic of cosine h (arc tan h s) = 1 / 1- s2, for -1 < s < 1,
Hyperbolic of tan h = (arc sin h s) = s / 1 + s2,
Hyperbolic of tan h = (arc cos h s) = s2 - 1 / s for | s | > 1,
Now we will talk about the logarithmic inverse hyperbolic function:
Arc Sin h s = ln (s + s2 + 1),
Arc Cos h s = ln (s + s + 1 s - 1),
Arc Tan h s = 1/2 ln (1 + s) - ln (1 - z)),
Arc Cosec h s = ln (1/s + (1/s2 + 1),
Arc Sec h s = ln (1/s + (1/s + 1 1/s - 1),
Arc Cot h s = 1/2 ln s + 1/s - 1,
This is all about hyperbolic functions.
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