The parameter for the normal trigonometric functions represents the length of the opposite and adjacent sides of a triangle in a unit circle. The parameter is the angle of the triangle that is located at the radius. The vertex that touches the circle has the coordinates of $(\cos{\theta},\sin{\theta})$.
From what I understand, the hyperbolic trig functions represent a triangle that touches a unit hyperbola ($x^2-y^2=1$). The coordinates of the vertex that touches the hyperbola is $(\cosh{t},\sinh{t})$, but what does the parameter represent here?
Best Answer
The meaning of the parameter $a$ in $\sinh(a)$ and in the case $\cosh(a)$ is the same as in the case of $\sin(a)$ and $\cos(a)$. Take a look at the figure below.
In both cases the parameter equals the half area of the red region.