# Hyperbolic Trigonometric Functions

Just as trigonometry can be performed on the unit circle ,
it can also be performed on the unit hyperbola :

x^{2} − y^{2} = 1.

Define the hyperbolic cosine function by

and the hyperbolic sine function by

A simple computation then shows that

cosh^{2}(x) − sinh^{2}(x) = 1,

which is the hyperbolic trigonometric identity analogous to the Pythagorean
identity

cos^{2}(x) + sin^{2}(x) = 1

from ordinary (circular) trigonometry.

Next, as in the case of ordinary trigonometry , we define the remaining four
hyperbolic trigonometric functions

from these first two :

Define the hyperbolic tangent function by

and the hyperbolic cotangent function by

Define the hyperbolic secant function by

and the hyperbolic cosecant function by

**Hyperbolic Pythagorean Identities
**

In addition to the identity derived earlier,

cosh

^{2}(x) − sinh

^{2}(x) = 1,

there are two other hyperbolic identities obtained by dividing this one by sinh

^{2}(x) and cosh

^{2}(x), respectively:

We summarize these here:

**Derivatives of Hyperbolic Functions**

By using the definition of the hyperbolic sine function, we have

Similarly,

The derivatives of the remaining hyperbolic trigonometric
functions are obtained from their definitions using the

Pythagorean hyperbolic trigonometric identities , the definitions of the
hyperbolic functions, and the quotient rule :

Similarly,

The remaining two hyperbolic trigonometric functions are handled similarly:

and

**Inverse Hyperbolic Trigonometric Functions
**

The four functions

are one-to-one on their domains, while the functions

are one-to-one on the interval x ≥ 0. So, we can define six inverse hyperbolic trigonometric functions .

Since the hyperbolic functions are defined in terms of the exponential function, it is only natural to expect that

the inverse hyperbolic functions can be defined in terms of the natual logarithm function . We derive these six formulas

next:

Prev | Next |