A hyperbolic function is related in some way to the hyperbola. Hyperbolic functions such as cosh, sinh, etc. are also related to the function e. They have many curious identities or properties that are analogous to trigonometric functions and their properties. One example is as follows:

sin2x + cos2x = 1

cosh
2x - sinh2x = 1
Hyperbolic functions are called such because the two fundamental hyperbolic functions of sinh(t) and cosh(t) are points on the hyperbola x^2 - y^2 = 1 for some number t.

That is, cosh^2(t) - sinh^2(t) = 1

Where sinh(x) = {e^x - e^(-x)}/2

and cosh(x) = {e^x + e^(-x)}/2

It logically follows that

tanh(x)= sinh(x)/cosh(x) = {e^x - e^(-x)}/{e^x + e^(-x)}

The identities for other hyperbolic functions follow logically from trigonometric ones:

cosech(x) = 1/{sinh(x)}

sech(x) = 1/{cosh(x)}

coth(x) = 1/{tanh(x)}

Derivatives and integrals for the hyperbolic functions are found by expanding the function into its "e^x" format and then differentiating or integrating as normal. Many of these results follow analogously from trigonometric functions. Some are different in terms of sign.

d/dt(sinh(t)) = cosh(t)

d/dt(cosh(t)) = sinh(t)

d/dt(tanh(t)) = sech^2(t)

d/dt(coth(t)) = -cosech^2(t)

d/dt(sech(t)) = -sech(t)*tanh(t)

d/dt(cosech(t) = -cosech(t)*coth(t)

Also, there are various double angle formulae that can be derived by using the definition of cosh and sinh and by the laws of exponents:

cosh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y)

sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y)

cosh(2x) = cosh^2(x) + sinh^2(x) = 1 + 2sinh^2(x) = 2cosh^2(x) - 1

sinh(2x) = 2sinh(x)cosh(x)

Like cos(x), cosh(x) is an even function (reflected in y-axis); and like sin(x), sinh(x) is an odd function (sinh(-x) = - sinh(x)).

The inverse hyperbolic functions are derived by solving quadratic functions in e^y. They are as follows:

arcsinh(x) = ln(x + sqrt(x^2 + 1))

arccosh(x) = ln(x + sqrt(x^2 - 1)), x >= 1

arctanh(x) = 1/2*ln((1 + x)/(1 - x))

The six hyperbolic functions are as follows:

The inverses of these functions are denoted:

NB: The f-1 notation here implies inverse function, as opposed to f raised to the power of -1.

They are defined, in terms of e, in the following way:

ez - e-z
sinh(z) = --------
2

ez + e-z
cosh(z) = --------
2

sinh(z)   ez - e-z    e2z - 1
tanh(z) = ------- = -------- = --------
cosh(z)   ez + e-z    e2z + 1

2
sech(z) = --------
ez + e-z

2
csch(z) = --------
ez - e-z

cosh(z)   ez + e-z    e2z + 1
coth(z) = ------- = -------- = --------
sinh(z)   ez - e-z    e2z - 1

The following relationships exist between the functions:

sinh(z) = -sinh(-z)
cosh(z) = cosh(-z)
For purely imaginary arguments,
sinh(iz) = isin(z)
cosh(iz) = cos(z)

Many of the trigonometric identities can be easily transferred to hyperbolic functions, through the use of Osborne's rule.

Some examples are:

cosh2(x) - sinh2(x) = 1
cosh(x) + sinh(x) = ex
cosh(x) - sinh(x) = e-x

For further manipulation of hyperbolic functions one may use the Half Angle Formulae and Double Angle Formulae.

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