ABSTRACT
If a function of x , f (−x)=− f (x), then f (x) is called an odd function of x . Replacing x by−x in equation (1) gives:
sinh(−x) = e −x − e−(−x)
2 = e
−x − e x 2
= − ( e x − e−x
) = −sinh x
Replacing x by −x in equation (3) gives:
tanh(−x) = e −x − e−(−x)
e−x + e−(−x) = e−x − e x e−x + e x
= − ( e x − e−x e x + e−x
) = −tanh x
x are odd functions
(see Section 5.1), as also are cosech x ( = 1
sinh x
) and
coth x ( = 1
tanh x
) If a function of x , f (−x) = f (x), then f (x) is called an even function of x . Replacing x by −x in equation (2) gives:
cosh(−x) = e −x + e−(−x)
2 = e
−x + e x 2
= cosh x Hence coshx is an even function (see Section 5.2), as also is sech x
( = 1
cosh x
) Hyperbolic functionsmaybeevaluated easiest using a
calculator. Many scientific notation calculators actually possess sinh and cosh functions; however, if a calculator does not contain these functions, then the definitions given above may be used.
