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Hyperbolic Functions and Identities

April 8th, 2009 |

Author: Merrin

The hyperbolic functions are defined analagously to the trigonometric functions except for a triangle that obeys a different rule for the relation between the sides. I call this a hyperbolic reference triangle. Hyperbolic functions are very useful in calculus and are closely related to the equations for trigonometric functions. As your studies progress you will see more and more links between the two.

$x^2-y^2=h^2$

$\sinh w = \cfrac{\text{opposite}}{\text{hypotenuse}}$
$\cosh w = \cfrac{\text{adjacent}}{\text{hypotenuse}}$
$\tanh w = \cfrac{\text{opposite}}{\text{adjacent}}$
$\text{csch}\, w = \cfrac{1}{\sinh x}$
$\text{sech}\, w = \cfrac{1}{\cosh x}$
$\text{coth}\, w = \cfrac{1}{\tanh x}$

The sides of a hyperbolic reference triangle obey the rule

$x^2 - y^2 = z^2$

The geometry of hyperbolics is kind of weird so instead of these definitions we can use an alternative algebraic defintion

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

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

You may notice that if you add these 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. 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 hyperbolic sine or hyperbolic cosine respectively and anytime you would multiply two hyperbolic sine 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 hyperbolic cosine with cosine and hyperbolic sine with sine. Every time you would multiply two sine 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 analagous hyperbolic identities.

These are the Pythagorean type identities

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

$1 + \tan^2 x = \sec^2 x \quad \to \quad 1-\tanh^2 w = \text{sech}\,^2 w$

$1 + \cot^2 x = \csc^2 x \quad \to \quad 1-\text{coth}\,^2 w = -\text{csch}\,^2 w$

For the double angle formula we have

$\sin(2x) = 2\sin x\cos x \quad \to \quad \sinh(2w) = 2 \sinh w \cosh w$

$\cos(2x) = 2\cos^2x-1 x\quad \to \quad \cosh(2w) = 2\cosh^2w-1$

$\tan (A+B) = \cfrac{\tan A+\tan B}{1-\tan A\tan B} \quad \to \quad \tanh(u+v)= \cfrac{\tanh u + \tanh v}{1+\tanh u \tanh v}$

From these examples you should be able to get a handle of it. Just remember the minus signs for each double power of sine multiplied together.

Example 1. Recall we derived that

$\sin ^{7} \theta =\cfrac{1}{64}(-\sin 7\theta +7\sin 5\theta -21\sin 3\theta +35\sin \theta )$

What is the corresponding hyperbolic identity?

Solution 1. There are three negative signs for the seventh power of hyperbolic sine.

$(\sinh x)^{7} =\cfrac{1}{64} (\sinh 7x-7\sinh 5x+21\sinh 3x-35\sinh x)$

We can plot the hyperbolic functions, they look much different from the trigonometric functions. For one, they are not periodic functions. It is not that much more difficult to plot the inverse hyperbolic functions. Compare the two graphs to see how the domain has been restricted so we can have actual inverses.

\includegraphics[width= 4in]{../images/Hyperbolics.pdf}
\caption{Graphs of the six hyperbolics functions near the origin}

\includegraphics[width= 4in]{../images/hyperinv.pdf}
\caption{Graphs of the six inverse hyperbolics functions near the origin}

We have gone through a wide variety of functions. These functions comprise the basis for the elementary functions. The elementary functions are a subset of a larger set of functions called special functions. Usually introductions to calculus are limited to the elementary functions. The elementary functions can be sumarized as power functions, roots, polynomials, exponentials, logarithms, trigonometric functions, trigonometric inverses, hyperbolic function, and hyperbolic inverses. Also any combination or composition between the elementary functions is also consider an elementary function. The elementary functions are important players in calculus and without them there would be no calculus.

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Hyperbolic Functions and Identities

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