Mathematics 162                                 Laboratory 5                           Week of Feb. 22, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

Hyperbolic Trigonometric Functions
and Their Inverses

We know the trigonometric functions that we learned about in high school are useful for describing circles; for this reason they are occasionally called the circular  trigonometric functions.  There are similar functions useful for describing hyperbolas; they are called hyperbolic  trigonometric functions.  In today's lab, we investigate these functions.

Part 1:  Graphs and derivatives

The hyperbolic cosine is abbreviated cosh (and pronounced to rhyme with posh), and the hyperbolic sine is abbreviated sinh (pronounced "cinch").  The definitions in Mathematica syntax are given in the commands below.  Enter them, and write the definitions in usual notation next to them.

cosh[x_] := (E^x + E^(-x))/2

sinh[x_] := (E^x - E^(-x))/2

(Mathematica already knows these functions as Cosh[x], and Sinh[x], but for now use these "homemade" functions.)  The command below will plot the graph of coshx over –2 < x < 2.  Enter it, and examine the graph.  Modify it to plot the graph of sinhx (you can eliminate the PlotRange option), and examine the graph.

Plot[cosh[x],{x,-2,2},PlotRange->{0,4}]

Briefly the describe these two graphs.  Include where they are positive and negative, where they increase and decrease, their concavity, and their limits at infinity (positive and negative).

How are the derivatives of sinhx and coshx related?  Compare their relationship with the relationship between sinx and cosx and their derivatives.

Part 2: A New Identity

Recall the fundamental trigonometric identity: sin2x + cos2x = 1.  Find a similar  identity for sinhx and coshx.  Explain how you found the relationship; use another sheet of paper, if necessary.

The command below generates a plot of all the points in the plane whose x coordinates are cost and y coordinates are sint.  What shape is this graph?  What does this have to do with the fundamental trigonometric identity?  (Hint:  What is the equation for this graph?)

ParametricPlot[{Sin[t],Cos[t]},{t,-Pi,Pi},
    AspectRatio->Automatic]

Modify the command above to plot points in the plane whose x coordinates are cosht and y coordinates are sinht.  What shape is this graph?  What does this have to do with the identity you found above?

Part 3:  Inverse Functions

Mathematica already knows the inverses of coshx and sinhx as ArcCosh[x], and ArcSinh[x], respectively.  Plot the graphs of these functions over representative domains.

In class we found that the basic trigonometric functions have inverses.  For instance we defined arcsinx using a statement like:

arcsinx is the inverse of sin x restricted to –π/2 ≤ x ≤ π/2.

Write similar definitions for arccoshx, and arcsinhx below.

Part 4:  Derivatives of Inverse Functions

Use Mathematica to find the derivatives of arccoshx and arcsinhx.

Your "essay" for this report is a derivation, that shows Mathematica has found the right derivative for arccoshx.  As a guide, use the derivation of the derivative of arcsinx given in class, or the proof of theorem 8.8 in Swokowski.  Your derivation should include justification for every step, and be written in full sentences.