calculuspowerup.com

Hyperbolic Identities and Osborne’s Rule

April 29th, 2009 |

Author: Merrin

The hyperbolic functions are defined analagously to the trigonometric functions except they follow a slightly different sort of geometry. But unlike how the trigonometric functions are built around a right triangle inscribed in a circle. The hyperbolic functions obey a different sort of rule which might be compared to an imaginary hyperbolic reference triangle. The basis for this triangle would be $latex x^2-y^2=h^2&s=1$. With this in mind all hyperbolic functions can be defined analogously to the trigonometric functions with the same combinations.

$\sinh x = (1/2)(e^x - e^{-x})\qquad \cosh x = (1/2)(e^x + e^{-x})$

$\tanh x = \cfrac{\sinh x}{\cosh x}\qquad \text{csch}\, x = \cfrac{1}{\sinh x}$

$\text{sech}\, x = \cfrac{1}{\cosh x} \qquad \coth x = \cfrac{1}{\tanh x}$

The identities regarding the hyperbolic functions will be different from the trigonometric identities because they follow a different sort of geometry. One way to approach the problem is to ignore geometry and just use the relationships between functions. You may notice that if you add the first two equations you get something similar to Euler’s formula

$\sinh x + \cosh x = e^x$

With the algebraic formulas then all the analagous hyperbolic identities to the trigonometric identities can be derived amongst relations regarding exponentials. Instead of going through all that algebra, I will show you a handy rule which gives all the answers provided you know the analagous trigonometric identity.

Osborne’s Rule: In a valid trigonometric identity anywhere you see a sine or cosine, replace either with $latex \sinh x&s=1$ or $latex \cosh x&s=1$ respectively and anytime you would multiply two $latex \sinh x&s=1$ functions together you multiply that term by a minus sign.

Pretty simple right? There should be a Corollary to Osborne’s Rule that state you can turn any hyperbolic identity into a trigonometric identity by replacing $latex \cosh x&s=1$ with $latex \cos x&s=1$ and $latex \sinh x&s=1$ and $latex \sin x&s=1$. Every time you would multiply two $latex \sin x&s=1$ functions together you must multiply that term by a minus sign.

Let’s review some of our trigonometric identities one more time and write down the analogous hyperbolic identities.

These are the Pythagorean type identities

$\cos^2 x + \sin^2 x = 1 \quad \to \quad \cosh^2 w - \sinh^2 w = 1$