Advantage of using Hyperbolic Trigonometric functions?

$$\DeclareMathOperator{\sech}{sech}\DeclareMathOperator{\csch}{csch}$$Is there any added advantage of using Hyperbolic Trigonometric functions? Since you can always use normal trigonometric functions in all cases: $$\left.\begin{array}{ccc} \sin&\leftrightarrow&\tanh\\ \cos&\leftrightarrow&\sech\\ \end{array} \right\} \sin^2 x + \cos^2 x = 1;\sech^2x+\tanh^2x=1\\ \left.\begin{array}{ccc} \tan&\leftrightarrow&\sinh\\ \sec&\leftrightarrow&\cosh\\ \end{array} \right\} \sec^2x-\tan^2x=1;\cosh^2x-\sinh^2x=1\\ \left.\begin{array}{ccc} \csc&\leftrightarrow&\coth\\ \cot&\leftrightarrow&\csch\\ \end{array} \right\} \csc^2x-\cot^2x=1;\coth^2x-\csch^2x=1$$

Examples:

• For hyperbolas $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ denoted by $$(x,y)$$ both parametrization $$(a\cosh t,b\sinh t)$$ and $$(a\sec t,b\tan t)$$ work.
• For ellipses $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ denoted by $$(x,y)$$ both parametrization $$(a\,{\rm sech} t,b\tanh t)$$ and $$(a\sin t,b\cos t)$$ work.
• and so on.
• What's the use of tan(x)? (tan(x)=sin(x)/cos(x))?
– zoli
Jan 29 '15 at 13:03
• @zoli compactness in writing expressions Jan 29 '15 at 13:04
• Some people have an irrational fear of complex numbers. Strange but true. So, instead of writing $\cos(ix)$ they prefer to write $\cosh x$. Jan 29 '15 at 13:15
In the first place, $\cos(z) = \dfrac{e^{iz}+e^{-iz}}{2} = \cosh(iz)$ and $\sin(z) = \dfrac{e^{iz}-e^{-iz}}{2i}=-i\sinh(iz)$. So why don't people stick to the exponential function and forget about all trigonometric or hyperbolic functions? In some cases it is indeed better to go straight to the complex exponential function, but in other cases one might consider it nice to express something using (what were originally) real functions. For example, vibrations on a string can be modeled as a sum of trigonometric functions. One could have solved the differential equations in full generality and obtain complex exponentials as solutions too, but our string has only real positions (in classical mechanics) and trigonometric functions are sufficient to provide a basis for the real solutions we seek.
These functions, all tightly related to each other, nonetheless have different mathematical applications. In particular, the hyperbolic trigonometric functions have applications to hyperbolic geometry. For instance, in the hyperbolic plane the circumference of a circle of radius $r$ equals $2\sinh(r)$.