These functions are surprisingly similar to trigonometric functions, although they do not have anything to do with triangles. The similarity follows from the similarity of definitions. At the end of this section we mention another reason why trigonometric and hyperbolic functions might be close.

Definition.

Thehyperbolic sineandhyperbolic cosineare defined byThe

hyperbolic tangentandhyperbolic cotangentare defined by

**The hyperbolic sine.** The domain:

*D*(sinh) = .

The graph:

The function is continuous on its domain, unbounded, and symmetric, namely
odd, since we have sinh(-*x*) = -sinh(*x*).

There is one zero point, namely *x* = 0, which is also a point of
inflection. There are no local extrema, limits at endpoints of the domain are

The derivative:

[sinh(*x*)]' = cosh(*x*).

**The hyperbolic cosine.** The domain:

*D*(cosh) = .

The graph:

The function is continuous on its domain, bounded from below, and symmetric,
namely even, since we have cosh(-*x*) = cosh(*x*).

There is no zero point, but a local minimum at *x* = 0, the
function is always concave up. Limits at endpoints of the domain are

The derivative:

[cosh(*x*)]' = sinh(*x*).

**The hyperbolic tangent.** The domain:

*D*(tanh) = .

The graph:

The function is continuous on its domain, bounded, and symmetric,
namely odd, since we have tanh(-*x*) = -tanh(*x*).

There is a zero point, namely *x* = 0, which is also a point of
inflection. There are no local extrema, limits at endpoints of the domain are

The derivative:

[tanh(*x*)]' = 1/cosh^{2}(*x*).

**The hyperbolic cotangent.** The domain:

*D*(coth) = − {0} =
(-,0) (0,).

The graph:

The function is continuous on its domain, unbounded, and symmetric,
namely odd, since we have coth(-*x*) = -coth(*x*).

There is no zero point and no point of inflection, there are no local extrema. Limits at endpoints of the domain are

The derivative:

[cotgh(*x*)]' = -1/sinh^{2}(*x*).

Note that we often write sinh^{n}(*x*) instead of the
correct [sinh(*x*)]^{n}, similarly for the other
hyperbolic functions.

The following identities are very similar to trig identities, but they are tricky, since once in a while a sign is the other way around, which can mislead an unwary student.

The identities for hyperbolic tangent and cotangent are also similar.

Here the situation is much better than with trig functions. Apart from the
hyperbolic cosine, all other hyperbolic functions are 1-1 and therefore they
have inverses. To get the inverse of cosh(*x*), we restrict it to the
interval [0,). The
inverse functions are called *argument of hyperbolic sine*, denoted
argsinh(*x*), *argument of hyperbolic cosine*, denoted
argcosh(*x*), *argument of hyperbolic tangent*, denoted
argtanh(*x*), and *argument of hyperbolic cotangent*, denoted
argcoth(*x*). Their graphs are

Basic properties:

Now we come to another advantage of hyperbolic functions over trigonometric functions. We actually have "nice" formulas for the inverses:

**Note:** The inverse functions are also sometimes called "area hyperbolic
functions". There are two alternative notations, instead of argsinh(*x*)
some would write arcsinh(*x*) or sinh^{-1}(*x*). The first
notation is probably inspired by inverse trig functions, the second one is
unfortunately quite prevalent, but it is extremely misleading. The reason is
that many students see an admitedly existing similarity between
sinh^{-1}(*x*) and sinh^{2}(*x*), so they think
that sinh^{-1}(*x*) is actually 1/sinh(*x*). We talked
about some justification for this misleading notation when we introduced inverse functions in Theory -
Real functions. Still it is very unfortunate, especially since there is a
perfectly adequate arg-notation that we introduced above. We will stick to it
here in Math Tutor.

One interesting property of trig functions is that they provide a nice
description of a circle. Indeed, the circle of radius *r* centred at the
origin (given by
*x*^{2} + *y*^{2}=*r*^{2} in
Cartesian coordinates) is given by the parametric equations
*x*=*r*·cos(*t*),
*y*=*r*·sin(*t*).

What happens if we replace these functions with their hyperbolic cousins? The
equations
*x*=*r*·cosh(*t*),
*y*=*r*·sinh(*t*) describe exactly the right branch of
the rectangular hyperbola
*x*^{2} - *y*^{2}=*r*^{2}.