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Trigonometric functions are pretty much similar to hyperbolic functions. They could be used in a plenty of linear differential equations. If we take the example of cubic equations then angles or distances calculation in hyperbolic geometry is performed through hyperbolic function formula. It is important in electromagnetic theory too to calculate the heat transfer or special relativity etc.

A list of basic hyperbolic functions includes sinh, cosh, tanhetc and it is shown below how are they written in mathematics.

\[\large e^{x}= cosh\;x + sinh\;x\]

\[\large sinh\;x=\frac{e^{x}-e^{-x}}{2}\]

\[\large cosh\;x=\frac{e^{x}+e^{-x}}{2}\]

\[\large tanh\;x= \frac{\large sinh\;x}{\large cosh\;x} =\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\]



Following is the relationship among hyperbolic function :

\[\large tanh\;x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\]

\[\large coth\;x=\frac{1}{tanh\;x}= \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\]

\[\large sech\;x=\frac{1}{cosh\;x}= \frac{2}{e^{x}+e^{-x}}\]

\[\large csch\;x=\frac{1}{sinh\;x}= \frac{2}{e^{x}-e^{-x}}\]

The real argument associated with hyperbolic functions is hyperbolic angle and the size of a hyperbolic angle is always the twice of the area of hyperbolic sector. They could be defined in terms of right angles too covering the sector completely.

The applications of hyperbolic functions are endless and few of them include linear differential equations, cubic equations, calculation of distances or angles, Laplace equation calculations, electromagnetic theory, heat transfer, physics, fluid dynamics, special relativity and more. The highly complex hyperbolic functions in mathematics are derived from imaginary part of sine and the cosine.

This is a not a new concept but it was proposed in the year 1760s independently to refer circular functions and there are special abbreviations too for a range of different hyperbolic functions. The use of these functions depends on the personal preference not the local language.

Further, these functions are able to satisfy various trigonometric identities that can be further expanded in terms of integrals or more. These are little complex topic to understand if not practiced well on regular intervals.

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