An angle can't be imaginary! But the "trig functions", sine, cosine, etc. are not always related to angles. Sine and cosine, in particular are the "ideal" periodic functions and are often used to model periodic phenomena that have nothing to do with angles (although engineers will still insist upon talking about the "phase angle" of an electrical current where there is no actual angle involved).
The easiest thing to do with complex numbers is to multiply and add them so for more complicated functions it is best to work with power series expansions:
If you replace x by "ix" in the first, you get
But , , and then , etc.
so all of the "odd" terms have "i" while the "even" terms do not. Separating "real" and "imaginary" parts,
or . Replacing x by -x, , since cos(-x)= cos(x) and sin(-x)= -sin(x).
Adding those two equations, so . In particular, if we now replace x by ix, we have .
(Which is why you titled this "hyperbolics", right?)