hyperbolic functions

A resource aimed at showing how the hyperbolic functions can be derived from their definitions in terms of the area enclosed by a hyperbola and rays from the origin.

Starting with the graph of

we can define the points with coordinates (C, S) and (C, –S) to be those that enclose an area   as shown in the graph below. Use the on-screen slider to show how these points change with the value of .

The first step in finding C and S in terms   is to rotate the hyperbola 45 degrees anticlockwise about the origin. Use the slider below to show the transformed graph.

Task 1

Watch the video here and use the technique outlined to show that the rotated graph above has equation

and determine the transformed coordinates of the points  (C, S) and (C, –S). (The answers are on the graph above, but you should prove that these are correct!)

Task 2

Use integration to find the area  in terms of C and S.

Task 3

Use the fact that (C, S) lies on the original hyperbola to write down a relationship between C and S. Use this, together with your answer to Task 2, to find C and S in terms of  .

Task 4

Use the definitions of C and S to work through these questions exploring some of their properties.