Chapter 5, Section 10   Hyperbolic Function

Hyperbolic Function 

In this section you look briefly at a special class of exponential functions called hyperbolic functions.  The name hyperbolic function arose from comparison of the area of a semicircle region, with the area of a region under a hyperbola.  The integral for the semicircular region involves an inverse trigonometric (circular) function.

One of the many ways in which a hyperbolic function is similar to the trigonometric functions:

Circle: x2 + y2 = 1                     Hyperbola: - x2 + y2 = 1 


Definition of the Hyperbolic Functions  

(There are two basic functions and the rest are variations using those two)

tanh x = (sinh x / cosh x)

coth x = (1/ tanh x)

sech x = (1 / cosh x)

csch x = (1 / sinh x),   0

Link to: Hyperbolic Functions  

which gives the formulas and 

shows the graphs of the functions.



Integrals with "Hyperbolic-Sin" Integrals with "Hyperbolic-Cos"
Integrals with "Hyperbolic-Tan" Integrals with "Hyperbolic-Cot"

 Integrals with "Hyperbolic-Sec"

Integrals with "Hyperbolic-Csc"

Integrals with "Hyperbolic-Sin" and "Hyperbolic-Cos"

The Inverse Hyperbolic Functions

Integrals with Inverse 

Hyperbolic Trigonometric Functions


Tractrix or Pursuit Curve

The inverse Hyperbolic secant can be used to define a curve called a tractrix or pursuit curve.


The tractrix is sometimes called a tractory or equitangential curve. It was first studied by Huygens in 1692 who gave it its name. Later Leibniz, Johann Bernoulli and others studied the curve.

The study of the tractrix started with the following problem being posed to Leibniz:

What is the path of an object dragged along a horizontal plane by a string of constant length when the end of the string not joined to the object moves along a straight line in the plane?

He solved this using the fact that the axis is an asymptote to the tractrix.

The highlighted sentences are  from this link:

A link to a program for calculating Tractrix: Tractrix Contour Calculator















For homework examples, 

see Assignment #1