I am looking for a term that would describe the sets of functions that are very closely related, such as:
- the trigonometric functions $\sin$ and $\cos$
- the hyperbolic functions $\sinh$ and $\cosh$
- etc (?)
By "very closely related" I mean that to evaluate one function of the set, one might evaluate one, more or all of the other functions in the set and then could derive the result thanks to some relation of interdependency.
The way I would choose to look at this is that these functions are closely related since they parametrise some curve - and it is the parametrisation that gives the relationship between them.
For example. $\sin x$ and $\cos x$ are related since they parametrically represent a point on the unit circle, $$(\cos x, \sin x), x \in [0, 2 \pi) \\x^2 + y^2 = 1 \ \ \text{(circle equation)}$$
Similary, $\cosh x$ and $\sinh x$ are parametric points of a hyperbola,
$$(\cosh x, \sinh x) \\ x^2 - y^2 = 1 \ \ \text{hyperbola equation}$$
One could argue that there are always parametrizations of any choice of functions, but the functions that you mentioned, which are the trignometric functions are very closely tied to the conics, hence are more "natural" parametrisations that many other choices.