2.1.1—Hyperbolic functions

sinh x=(e^x−e^{−x})/2 and cosh x=(e^x+e^{−x})/2. Their quotient is tanh x=sinh x/cosh x. These definitions make exponential methods available for hyperbolic equations.

The key identity is cosh²x−sinh²x=1, analogous to a trigonometric identity with a sign change. sinh is odd, cosh is even, and tanh is odd.

At x=0, sinh0=0, cosh0=1 and tanh0=0. For large positive x, tanh x approaches 1 because e^{−x} becomes negligible relative to e^x.

Hyperbolic functions are not ordinary sine and cosine with a different name; their signs, domains and identities differ.