Section Summary: 7.6 - Hyperbolic functions

1. Definitions

   • sinh:

     

   • arcsinh: the inverse function of sinh. <img src=7.6gr7.gif">

   • cosh:

     

   • arccosh: the inverse function of cosh (usually taken on the interval by default). <img src=7.6gr13.gif">

   • tanh:

     

   • arctanh: the inverse function of tanh. <img src=7.6gr23.gif">

2. Theorems

   • 

   • 

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     Derivatives of these hyperbolic functions' inverses can be found by implicit differentiation, as shown in Example 4.

3. Properties/Tricks/Hints/Etc.
   • Symmetry:

     

     

     

   • 

4. Summary

   The cosh and sinh functions are important from the standpoint of the sciences. Their definitions are considered by many to be mere conveniences, but these functions have some important actual applications. The graph of the cosh function is known as the catenary, and is an important structural tool for engineers. The <A href=http://www.nku.edu/~longa/classes/mat120/images/cid_gateway_arch_001.150.jpg > Saint Louis </A> arch is in the shape of a catenary. A hanging chain is in the form of a catenary.

   These functions are named after the trigonometric functions because they share many properties of the trig functions (or at least the properties resemble those of the trig functions).

Problems to consider: pp. 483-485, #8, 22, 34, 38, 49, 52, 54; on the board: 7, 23