Question

what's the relation between hyperbola and hyperbolic function?

Answer 1

Normal trigonometric functions are defined in terms of angles and measurements in the unit circle (eg, sine = y coordinate of where the radius intersects with the circumference). The hyperbolic functions are defined in the same way, except that they use the unit hyperbola, not the unit circle.

Answer 2

could you elaborate?

Answer 3

This is the best drawing I could find (Wikipedia):

Answer 4

... how to explicitly express alpha/2 as function of x and y??

Answer 5

You mean the definitions of sinh and cosh?
\[\sinh(\alpha)=\frac{e^\alpha-e^{-\alpha}}{2}\]
\[\cosh(\alpha)=\frac{e^\alpha+e^{-\alpha}}{2}\]

Answer 6

all right thanks!