### Hyperbolic Trigonometry: A Crash Review

This is one of the less-practiced subjects in U.S. education. All too often, it makes its first
appearance in the middle of the Calculus series, with very little context. Many of the identities are
similar (but often have sign differences) to those in Trigonometry. Much of this can be understood with only
some familiarity with College Algebra, however.
Two physical applications (there are undoubtedly more) are:

- Hanging an inelastic rope between two points. [The actual arc of the rope turns out to be from the hyperbolic trig function cosh.
- The emulation of a uniform gravity field by a uniform acceleration, in General Relativity.

There are two common approaches to this material:
- Always reduce to definitions immediately
- Learn both trigonometry and hyperbolic trigonmetry so well that the identities cannot be confused.

It's your choice...but be advised that the first option is often much quicker to implement.

##### Hyperbolic Trigonometry Survival 101

What are these hyperbolic trig(onmetric) functions anyway?

##### Formula lists

Definitions of hyperbolic trig(onmetric) functions

Algebraically even and odd hyperbolic trig(onmetric) functions

Hyperbolic Pythagorean identities

Hyperbolic double-argument formulae

Hyperbolic half-argument formulae

Hyperbolic trigonometric functions are defined in terms of the natural exponential function e^{x}.
(At least, natural in "stratosphere" math).
- sinh(x):=[e
^{x}-e^{-x}]/2
- cosh(x):=[e
^{x}+e^{-x}]/2
- tanh(x):=sinh(x)/cosh(x)=[e
^{x}-e^{-x}]/[e^{x}+e^{-x}]
- coth(x):=cosh(x)/sinh(x)=[e
^{x}+e^{-x}]/[e^{x}-e^{-x}]
- sech(x):=1/cosh(x)=2/[e
^{x}+e^{-x}]
- csch(x):=1/sinh(x)=2/[e
^{x}-e^{-x}]

Observe that sinh(x) and cosh(x) are the even and odd components of e^{x}, by definition.
The following equations relating sinh(x), cosh(x), and e^{x} are special instances of equations
relating even and odd parts of functions to the function itself:
- cosh(x)+sinh(x)=e
^{x}
- cosh(x)-sinh(x)=e
^{-x}

Notation for powers and inverses of hyperbolic trigonmetric functions is similar to that of trigonometric functions:
- cosh(cosh(x))!=cosh
^{2}(x)=cosh(x)^{.}cosh(x), but
- 1/cosh(x)!=cosh
^{-1}(x), and cosh^{-1}(cosh(x))=1 for all real numbers x.

EXERCISE: Verify that:
- sinh(x), tanh(x), coth(x), csc(x) are all odd functions.
- cosh(x), sech(x) are all even functions.

EXERCISE: Verify that
- cosh
^{2}(x)-sinh^{2}(x)=1
- 1-tanh
^{2}(x)=sech^{2}(x)
- coth
^{2}(x)-1=csch^{2}(x)

If you remember coordinate geometry, then the first equation should look **very** familiar:
it is an equation for a unit hyperbola in the Cartesian coordinate plane. [Rename (x,y) to (cosh(x),sinh(x))] This is what motivated
the notation for hyperbolic trigonometry, and indeed the name "hyperbolic trigonometry".

EXERCISE: Verify that
- sinh(2x)=2sinh(x)cosh(x)
- cosh(2x)=sinh
^{2}(x)+cosh^{2}(x)

EXERCISE: Verify that
- sinh
^{2}(x/2)=[cosh(x)-1]/2
- cosh
^{2}(x/2)=[cosh(x)+1]/2

To isolate, take square roots. Note that cosh(x) is always positive, and the sign of sinh(x) is
the same as the sign of x, so we can always tell **which** square root we want.
One way to do this is a minor variation of the derivation of the trigonometric half-angle formulae for sin(x) and cos(x).

Opinions, comments, criticism, etc.? Let me know about it.
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