why do we use the hyperbolic functions?

Question

The first time I knew about the hyperbolic function was when I was studying the derivatives. And I know that the derivative of $\sinh x= ( e^x - e^{-x} )/ 2$ , but i still confused with what are they really are? and how did we get them and for what we are using them?

Thanks very much, and i hope that wasn't a long question.

Answer 1

So you know how $\sin$ and $\cos$ relate the side lengths of right triangles on the unit circle to the interior angle? Well, $\sinh$ and $\cosh$ do the same but the instead of having right triangle which lie on the unit circle we have them lie on the unit hyperbola.

The equation for the unit circle is,

$$x^2+y^2 = 1$$

While the equation for the unit hyperbola is,

$$x^2 -y^2 = 1$$

This picture helps depict what I am describing, https://en.wikipedia.org/wiki/Hyperbolic_function#/media/File:Hyperbolic_functions-2.svg

Answer 2

The hyperbolic functions $\sinh$ and $\cosh$ parameterize the hyperbola $x^2-y^2=1$ since $\cosh^2 t-\sinh^2 t=1$ for all $t\in\mathbb{R}$.

Answer 3

The imverse hyperbolic functions are particularly useful in integration, for example when dealing with positive quadratic functions inside square roots. Although such integrals can be done with trig functions, using hyperbolic functions makes them much easier.

Find $\int\sqrt{x^2-1}dx$ for example

Answer 4

Every real functions can be uniquely represented by the sum of the even function and the odd function. $$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$ Now let's put $f(x)=e^x$.

Answer 5

The catenary ( or chaînette) is the shape of the curve assumed by a hanging chain or cable with the two ends fixed, under its own weight. It happens its equation is $$y=a\cosh \frac xa$$ where the constant $a$ depends on physical parameters (tension and mass per unit length). It is used in architecture and engineering for archs, bridges, &c.

You also find a derived curve in the shape of a skipping rope.