Significance of Hyperbolic functions: Hyperbolic functions are analogs of the ordinary trigonometric functions defined for the hyperbola rather than on the circle: just as the points $(\cos t, \sin t)$ form a circle with a unit radius $~(x^2+y^2=1)~$, the points $(\cosh t, \sinh t)$ form the right half of the equilateral hyperbola $~(x^2-y^2=1)~$. Hyperbolic functions also have important physical applications. These functions occur in the solutions of many linear differential equations [for example, the hyperbolic cosine function may be used to describe the shape of the curve formed by a high-voltage line suspended between two towers (catenary)], of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. Hyperbolic functions may also be used to define a measure of distance in certain kinds of non-Euclidean geometry.
How is hyperbolic function related to trigonometry ?
The followings are the graphical representation between geometric and hyperbolic functions (especially for $\sin x,~~\cos x,~~\tan x,~~\sinh x,~~\cosh x,~~\tanh x$ ).



- Relation between Hyperbolic function and Trigonometric function:
Hyperbolic sine: $~\sinh x=-i\sin(ix)~$
Hyperbolic cosine: $~\cosh x=\cos(ix)~$
Hyperbolic tangent: $~\tanh x=-i\tan(ix)~$
Hyperbolic cotangent: $~\coth x=i\cot(ix)~$
Hyperbolic secant:
$~\operatorname {sech} x=\sec(ix)~$
Hyperbolic cosecant: $~\operatorname {csch} x=i\csc(ix)~$
where $~i~$ is the imaginary unit with the property that $~i^2 = −1~$.
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For more details find the following:
https://brilliant.org/wiki/hyperbolic-trigonometric-functions/
https://en.wikipedia.org/wiki/Hyperbolic_functions