A reader asked me about the hyperbolic trig functions, sinh(x) and cosh(x). What are they for, and do they have an intuitive interpretation in physics?
That’s a pretty good question. After all, most of the time you first meet the hyperbolic trig functions in intro calculus, where their rather odd definitions are presented and then used as test beds for blindly applying newly-learned differentiation rules. Ok, great. But what are they really?
To answer the question, we should start off with Euler’s identity, which relates the exponential function with the regular trig functions. Proving this identity would take us a little far afield for this post, so for now we’ll just take it for granted:
Now, replace x with -x and write down the equation again. But remember that cos(-x) = cos(x) and sin(-x) = -sin(x) because of the even/odd properties of those functions. With that in mind, Euler’s identity is just as well written:
Now we have two equations, and by adding the first equation to the second we can cancel out the sin(x) terms. Or by subtracting the second equation from the first we can cancel out the cos(x) terms. We might as well do both, an we end up with:
This is kind of neat – we’ve taken functions that have their origins in ancient people studying triangles and we’ve written them in terms of the modern language of exponential functions and imaginary numbers. Pythagoras and crew would have no idea at all what something like cos(iπ) would be, but now we’re in position to answer those sorts of questions. In the equations we’ve just derived, substitute ix in place of just x. Since i*i = -1 by definition, the complex exponentials become purely real, like this:
Thus if we want to know what cos(iπ) is, we just plug in (1/2)*(e^π + e^-π) into our calculators, and it turns out to be about 11.592.
Which brings us to the hyperbolic trig functions. Instead of the strange ex cathedra definition in intro calc books, we see that they’re simply defined as the regular old trig functions when you plug imaginary numbers into them:
Which is actually sort of a nice little connection. For completeness, here’s their graphs:
Now what’s the physical interpretation of sinh and cosh? To be honest there isn’t much of one – really they’re just sort of a change of basis, and anything you can write in terms of sinh and cosh can usually be written more clearly in terms of exponential or ordinary trig functions. They do crop up in various differential equations, including the Laplace equation in classical E&M, and potential steps in the Schrodinger equation. But at least now we know where they come from.