The hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. They are:
sinh, cosh and tanh
csch, sech and coth

 (hyperbolic sine, pronounced "shine" or "sinch")

 (hyperbolic cosine, pronounced "cosh")

 (hyperbolic tangent, pronounced "tanch")

 (hyperbolic cotangent, pronounced "coth")

 (hyperbolic secant, pronounced "sech")

 (hyperbolic cosecant, pronounced "cosech")
 (hyperbolic cosecant, pronounced "cosech")
Relationship to regular trigonometric functions
Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the hyperbola x²  y² = 1 for . This is based on the easily verified identity
 (cosh t)^{2}  (sinh t)^{2} = 1.
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborne's rule states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of two sinh's. This yields for example the addition theorems
Inverse hyperbolic functions
The inversess of the hyperbolic functions are