The basic hyperbolic trigonometric functions are defined as
sinh x =
ex − e−x
2
cosh x =
ex + e−x
2
Note that sinh x and cosh x are both linear combinations of ex , e−x , and they are linearly
independent. Thus {sinh x, cosh x} is a fundamental set for y 00 − y = 0.
Other hyperbolic trig functions are then defined by exact analogy to the ordinary trig
functions
cosh x
sinh x
coth x =
tanh x =
cosh x
sinh x
1
1
sech x =
csch x =
cosh x
sinh x
These functions have algebra and calculus properties which are very close to those for
the ordinary trig functions, except that there is sometimes a sign difference. For example
the following identities may be verified directly from the definitions:
cosh2 x − sinh2 x = 1
(sinh x)0 = cosh x
(cosh x)0 = sinh x
(tanh x)0 = sech 2 x
Z
tanh x dx = ln (cosh x) + C
Substitutions using inverse hyperbolic trig functions may often be used in a way
similar to the way inverses of the ordinary trig functions are used, for example
Z
1
√
dx = cosh−1 x + C
x2 − 1
Here cosh−1 is the inverse hyperbolic cosine (not the reciprocal of the hyperbolic cosine).