Exploring Hyperbolic Functions

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Hyperbolic function are less known as trigonometric function. But at some level, in Complex Analysis, for example, they become necessary. Let's explore them with help of Math Center Level 2 . The notation for hyperbolic functions resembles notation for trigonometric functions: sinh, cosh, tanh, ctgh, sech, csch. Where h stands for hyperbolic.

Consider y = sinh(x):

y = sinh(x)

Visually sinh(x) reassembles sin(x) only at the origin.

Consider y = cosh(x) :

y = cosh(x)


Consider y = tanh(x) :

y = tanh(x)


Consider y = ctgh(x) :

y = ctgh(x)


We see that hyperbolic functions are not periodic and their graph are not very similar to corresponding trigonometric functions. Let's explore numeric derivatives of hyperbolic functions.

y = sinh(x):

 y = sinh(x) with first derivative

We see that the graph of first derivative (green graph) of sinh coincide with graph of cosh.

y= sinh(x) with first and second derivative

The graph of second derivative of sinh (light blue) coincides with the graph of sinh itself.


Similarly for y = cosh(x) :

y =  cosh(x) with first derivative

y = cosh( with first and second derivative

So, sinh' = cosh and cosh' = sinh. Recall sin' = cos and cos' = -sin .


Let's illustrate some identities.

sinh(x) = (ex - e-x)/2 and   cosh = (ex + e-x)/2 :

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


tanh(x) = sinh(x)/cosh(x)  and ctgh(x) =cosh(x)/sinh(x):

tanh(x) = sinh(x)/cosh(x)  and ctgh(x) =cosh(x)/sinh(x)


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

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


It's time to explore inverse hyperbolic functions. Read first "Exploring Inverse Functions", if you haven't.

Similarly to trigonometric functions, there are a few different notations for inverse hyperbolic functions:  Arcsinh, arcsinh, asinh, sinh-1, Arccosh, arccosh, acosh, cosh-1, Arctanh, arctanh, atanh, atngh, atanh-1, Arcctg, arcctgh, actgh, ctgh-1, Arcsech, arcsech, asech, ascnh, sech-1, Arccsch, arccsch, acsch, scsh-1 .

Consider y = asinh(x) :

y = asinh(x)


y = acosh(x) :

y = acosh(x)


y = atanh(x) :

y = atanh(x)


y = actgh(x) :

y = actgh(x)


y = ascnh(x) :

y = ascnh(x)


y = acsch(x) :

y = acsch(x)



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