The name itself of "hyperbolic functions" is due to the fact that the functions $cosh(t)$ and $sinh(t)$ play, in the parametric representation of the equilateral hyperbola $x^{2}-y^{2}= a^{2}$ the same role as the functions $cos(t)$ and $sin(t)$ for the circle $x^{2}+y^{2}=a^{2}$
The parametric representation of circle is: $x=a\cos(t) , y= a \sin(t)$
and, for the hyperbola $x=a\cosh(t), y=a\sinh(t)$
as it is easy to see using the relation: $cosh(t)^{2}-sinh(t)^{2}=1$
The geometrical significance of the parameter $t$ in both cases, for the circle and for the hyperbola, is identical.
If we designate $S$ the area of ​​a portion of the circle and by $S_{0}$ the area of ​​the entire circle ($S_{0}=\pi a^{2}$), we have: $t= 2\pi \frac{S}{S_{0}}$
Let us now assume that S denotes the area of ​​an analogous sector of the equilateral hyperbola. We have $S= area\, OMN - area\, AMN$, M a point on the heperbolus and N its projection on the axis (Ox) and A the point of intersection of the hyperbola with the axis (Ox).
$S=\frac{1}{2}xy-\int_{a}^{x} ydx=\frac{1}{2}x\sqrt{x^{2}-a^{2}}-\int_{a}^{x}\sqrt{x^{2}-a^{2}}\,dx$ $S=\frac{1}{2}a^{2}\ln(\frac{x}{a}+\sqrt{\frac{x^{2}}{a^{2}}-1}\,)$
If now we put, denoting again by $S_{0}$ the area of ​​the circle $t= 2\pi \frac{S}{S_{0}}=\ln(\frac{x}{a}+\sqrt{\frac{x^{2}}{a^{2}}-1}\,) $
we easily find that:
$e^{t}=\frac{x}{a}+\sqrt{\frac{x^{2}}{a^{2}}-1}\,\,\,\,\;, e^{-t}=\frac{x}{a}-\sqrt{\frac{x^{2}}{a^{2}}-1}$
hence, adding term to term and multiplying by $\frac{a}{2}$:
$x=\frac{a}{2}(e^{t}+e^{-t})=a\cosh(t)$
$y=\sqrt{x^{2}-a^{2}}=\sqrt{a^{2}\cosh^{2}(t)-a^{2}}=a\sinh(t)$
Reference: Higher Mathematics Course,Volume I, V.Smirnov 521