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On the Wikipedia page for hyperbolic functions, https://en.wikipedia.org/wiki/Hyperbolic_function, is the variable "$a$" from the second image identical to the variable "$x$" from the definition section? In other words, if $\sinh(x) = (e^x - e^{-x})/2$ and $\cosh(x) = (e^x + e^{-x})/2$, is the area of the red region in the second image $x/2$? If not, what is the relationship between $a$ and $x$, and do the other functions, $\tanh$, $\rm sech$, $\rm csch$, and $\coth$, have a place in the image as the analogous trig functions do in the geometry of a unit circle?

John B
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user10478
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  • Yes. (and more than enough more characters to reach the 15 minimum for a comment) – Ethan Bolker Mar 17 '18 at 18:50
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    Your question "[D]o the other functions, $\tanh$, $\operatorname{sech}$, $\operatorname{csch}$, $\coth$, have a place in the image as the analogous trig functions do in the geometry of a unit circle?" is answered a few times on this site; for example, there's my answer. You might also find this answer helpful. – Blue Mar 17 '18 at 19:04

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Yes, we can find many interesting relations and graphs with hyperbolic functions as well as trigonometric functions.

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

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

We find out that $$\cosh^2(x) - \sinh^2(x) =1$$

Which explains the word " hyperbolic " in hyperbolic functions.

In comparison to $$\cos^2(x) +\sin^2(x) =1$$

Which are the normal trigonometric functions.

The graph that you referred to in your question is simply showing the hyperbolic nature of these functions in comparison with the circular nature of the trig functions. Of course any other relation could be demonstrated by a graph.

For example we can say $$ sec^2(x) - tan^2(x) =1 $$ is a hyperbola made with trig functions.

On the other hand $$ sech ^2(x) + tanh^2(x) =1 $$ is a circle made with hyperbolic functions.

Mohammad Riazi-Kermani
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As labelled on that graph, the indicated point on the unit hyperbola has horizontal coordinate $\sinh a$ (and vertical coordinate $\cosh a$). This is the evaluation of $\sinh x$ where $x$ has been assigned the value $a$, i.e., this $a$ is "that $x$". As explained in the caption, $a$ is twice the area of the red region, so "that $x$" is twice the area of the red region. (I don't just say "$x$" in either usage because in the diagram $x$ is bound as the independent variable of the function being graphed, which is not the input to the hyperbolic sine function.)

Much like this graph for circular functions, the hyperbolic functions can also be graphed. The "Comparison with circular functions" section of the article you cite has a graph showing how to relate this graph with the hyperbolic sine and cosine. More of the functions are shown in this answer.

Eric Towers
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