I'm currently covering Stewart's Early Transcendentals, and there is a whole section dedicated to defining and differentiating hyperbolic functions. The same amount of space is used to cover other types of functions like trigonometric and exponential functions.
My question is, what is the significance of these hyperbolic functions? What are they used for?
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One nice application of hyperbolic functions occurs in special relativity. Forgetting about two of the spatial dimensions, we have so-called 1+1 spacetime: points $(t,x)$ with the metric $d\tau^2=dt^2-dx^2$. (Here we use units where the speed of light is 1.) Then the Lorentz transformation can be written as $$\begin{align*}t'&=t\cosh\phi-x\sinh\phi\\x'&=-t\sinh\phi+x\cosh\phi\end{align*}$$where $\phi$ satisfies $\tanh\phi=v$, where $v$ is the relative velocity. Now compare this with the formula for rotating an $(x,y)$ coordinate system through an angle $\phi$:$$\begin{align*}x'&=x\cos\phi-y\sin\phi\\y'&=x\sin\phi+y\cos\phi\end{align*}$$where the relative slope is given by $\tan\phi$.
If we use the imaginary form $\cosh(\phi)=\cos(i\phi)$, $\sinh(\phi)=-i\sin(i\phi)$, we can rewrite the Lorentz formula as a "rotation through an imaginary angle" $i\phi$.
They are the complexifications of the trig functions. For example, $\cosh(x) = \cos(ix)$, and $i \sinh(x) = \sin(ix)$. In this way, you can calculate $\cos(z)$ for any complex number $z$.
Trig functions arise naturally as solutions to the ODE $y''(x) = k y(x)$ when $k$ is negative. Hyperbolic functions then arise as solutions when $k$ is positive.