The basic ones are:

sine, cosine, tangent, cosecant, secant, cotangent;

Less common ones are:

arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent;

hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, hyperbolic cotangent;

hyperbolic arcsine, hyperbolic arccosine, hyperbolic arctangent, hyperbolic arccosecant, hyperbolic arcsecant, hyperbolic arccotangent.

An arc-hyperbolic function is an inverse hyperbolic function.

Elliptic and Hyperbolic geometry.

Yes sure

It works in Euclidean geometry, but not in hyperbolic.