Conversion between trig functions and hyperbolic trig functions

Question

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first graph produces a semicircle when $-\frac \pi2 \le t \le \frac \pi2$ and the second graph when $-\infty \lt t \lt \infty$. My question is if it is possible to convert between the graphs; e.g. if there is a function $f$ where $(\text{sech} t, \tanh t) = (\cos f(t), \sin f(t))$.

Answer 1

You want the Gudermannian function: it is defined by $$ \operatorname{gd}{x} = \int_0^x \operatorname{sech}{y} \, dy $$ In particular, this can be expressed, by changing variables, as $$ \operatorname{gd}{x} = (\operatorname{sgn}{x}) \arccos{(\operatorname{sech}{x})} = \arcsin{(\tanh{x})}, $$ so $$ \cos{(\operatorname{gd}{x})} = \operatorname{sech}{x},\\ \sin{(\operatorname{gd}{x})} = \tanh{x}, $$ and then everything follows as you wish, since $\operatorname{gd}:(-\infty,\infty) \to (-1,1)$ bijectively.