Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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 at 22:42
3  
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 at 22:42
1  
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 at 22:43
1  
@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 at 22:45
4  
Ah, the useless identity function. Throw it away, together with all those useless identity elements of groups. –  Lee Mosher Jul 8 at 22:46

4 Answers 4

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.

share|improve this answer

$\cos(\theta)$ 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.

share|improve this answer

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.

share|improve this answer

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*}

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.