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))