# What is the use of hyperbolic trigonometric functions if they are easily expressible algebraically?

I get that there are uses for $\sin(x)$ and $\cos(x)$ because they are defined with imaginary exponents which aren't as easily worked with but the hyperbolic functions are simply $\frac12(e^x\pm e^{-x})$. I don't see why it would be necessary to define a function to represent this. It is almost like defining a function to represent the value of $2$.

The only other time I can think of where there is a "useless" function is for pedagogical purposes such as the identity function.

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They have particular uses in a lot of areas including hyperbolic geometry. You should look things up before calling functions "useless" that are quite useful. en.wikipedia.org/wiki/Hyperbolic_function – Adam Hughes Jul 8 '14 at 22:42
The identity is not useless, it's the identity. It's useful to name it because it's used all the time. As for your question, do you know about complex trigonometric functions? – Git Gud Jul 8 '14 at 22:42
It's shorthand. Imagine having to write that over and over, especially when you need to divide one by the other or combine them in some other way. Would you rather see $tanh(x)$, or a mess of exponential functions everywhere? – Kaj Hansen Jul 8 '14 at 22:43
@KajHansen I think you missed the point of the question. You can introduce an abbreviation for $x\mapsto 2x+3-\sin(x)$, but you don't do it. The question is "Why is it worth it to introduce this notation for these functions?" – Git Gud Jul 8 '14 at 22:45
Ah, the useless identity function. Throw it away, together with all those useless identity elements of groups. – Lee Mosher Jul 8 '14 at 22:46

## 4 Answers

In the same way that the point $(\cos\theta,\sin\theta)$ is on the circle, the point $(\cosh\theta, \sinh\theta)$ is on a hyperbola (hence the hyperbolic part). They show up all the time in solutions to the heat equation, and a hanging rope is actually a hyperbolic cosine function. Many many reasons why we would want to have a notation for it.

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$\cos(\theta)$ and $\sin(\theta)$ are essential for understanding geometry on the sphere, as ocean navigators have known for a couple of millennia. $\cosh(t)$ and $\sinh(t)$ are similarly essential for understanding geometry on the hyperbolic plane.

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consider $y''+y=0$ with initial conditions $y(0)=0,y'(0)=1$ and $y(0)=1, y'(0)=1$. the solutions are $\sin x$ and $\cos x$.

similarly, $y''-y=0$ with initial conditions $y(0)=0,y'(0)=1$ and $y(0)=1, y'(0)=1$ give $\sinh x$ and $\cosh x$.

(transcendental solutions of differential equations often get names when they are used often: $e^x$, bessel functions, weierstrass $\wp$, etc.)

just as $x=\cos t, y=\sin t$ gives a unit speed parameterization of the unit circle, $x=\cosh t, y=\sinh t$ gives a 'unit speed' parameterization of the unit hyperbola.

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But $\sin x = \frac{\mathrm{e}^{\mathrm{i}x} - \mathrm{e}^{-\mathrm{i}x}}{2\mathrm{i}}$ and $\cos x = \frac{\mathrm{e}^{\mathrm{i}x} + \mathrm{e}^{-\mathrm{i}x}}{2}$. So we don't need sine or cosine either.

The hyperbolic functions are handy for computing sine and cosine with complex arguments... \begin{align*} \sin(x+\mathrm{i}y) &= \sin x \cosh y + \mathrm{i} \cos x \sinh y \\ \cos(x+\mathrm{i}y) &= \cos x \cosh y - \mathrm{i} \sin x \sinh y \end{align*}

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