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Currently in Calculus II and I was introduced to hyperbolic trigonometric functions and it threw me for a loop. I’m really confused on their MEANING... and what they represent. I can use the formulas for them easily but it doesn’t actually make sense to me. Can someone please help me out? Are there any good books you can recommend as well?

J W
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CalebWilliamsUIC
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2 Answers2

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The first thing you must bear in mind, is that for real functions of a real variable, anything you can do with hyperbolic functions, you can also do in other ways. In particular,an integral that can be worked out by an hyperbolic substitution can also be worked out by a trigonometric substitution. However, it is true that there are some analogues between circles and trigonometric functions and hyperbolas and hyperbolic functions, for which you should read the Wikipedia article on hyperbolic functions. But the real picture only comes into focus when you know about real and complex power series and understand $ e^{i\theta} = cos(\theta) + isin(\theta) $ . Replace $ \theta $ by $ -\theta $ and solve for $ cos(\theta) $ and $ sin(\theta) $. So hyperbolic functions are only a minor curiosity for real calculus but are vitally important for complex analysis.

P. Lawrence
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My own answer to a question like this is to say that just as the ordinary trigonometric functions play a role in spherical trigonometry, so the hyperbolic functions play a role in hyperbolic geometry. For instance the Pythagorean Theorem on the surface of a sphere is $\cos c=\cos a\cos b$, where $a$ and $b$ are the lengths of the legs of a right spherical triangle and $c$ is the length of the hypotenuse. On the hyperbolic plane, the corresponding form of Pythagoras is $\cosh c=\cosh a\cosh b$.

Hardly the most important fact about the hyperbolic functions, but it does show a parallel with the circular functions.

Lubin
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