The Hyperbolic Cosine is pronounced Cosh.
The Hyperbolic Sine is pronounced Shine.
The Hyperbolic Tangent is pronounced Than with a soft Th.

At least that's the way I've been doing it for the last 55 years.

They don't seem to come up very often in conversation although they are just
as fundamental in mathematics as are the trigonometrical functions. They
crop up all over the place especially in transmission lines where they
appear in complex form such as Tanh(A+jB).