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:

There are two common approaches to this material: 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
What are these hyperbolic trig(onmetric) functions anyway?
Hyperbolic trigonometric functions are defined in terms of the natural exponential function ex. (At least, natural in "stratosphere" math). Observe that sinh(x) and cosh(x) are the even and odd components of ex, by definition. The following equations relating sinh(x), cosh(x), and ex are special instances of equations relating even and odd parts of functions to the function itself: Notation for powers and inverses of hyperbolic trigonmetric functions is similar to that of trigonometric functions:
Algebraically even and odd hyperbolic trig(onmetric) functions
EXERCISE: Verify that:
Hyperbolic Pythagorean identities

EXERCISE: Verify that 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".
Hyperbolic double-argument formulae

EXERCISE: Verify that
Hyperbolic half-argument formulae

EXERCISE: Verify that 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).


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