Chapter 5
Hyperbolic functions
5.1 Introduction to hyperbolic
functions
Functions which are associated with the geometry of
the conic section called a hyperbola are called hyperbolic functions. Applications of hyperbolic functions
include transmission line theory and catenary problems.
By definition:
(i) Hyperbolic sine of x,
sinh x =
e x − e −x
2
(1)
‘sinh x’ is often abbreviated to ‘sh x’ and is
pronounced as ‘shine x’
(ii) Hyperbolic cosine of x,
cosh x =
e x + e −x
2
(2)
‘cosh x’ is often abbreviated to ‘ch x’ and is
pronounced as ‘kosh x’
(iii) Hyperbolic tangent of x,
tanh x =
sinh x
cosh x
=
e x − e −x
e x + e −x
(3)
‘tanh x’ is often abbreviated to ‘th x’ and is
pronounced as ‘than x’
(iv) Hyperbolic cosecant of x,
cosech x =
1
sinh x
=
2
e x − e −x
(4)
‘cosech x’ is pronounced as ‘coshec x’
(v) Hyperbolic secant of x,
sech x =
1
cosh x
=
2
e x + e −x
(5)
‘sech x’ is pronounced as ‘shec x’
(vi) Hyperbolic cotangent of x,
coth x =
1
tanh x
=
e
x
+ e
−x
e x − e −x
(6)
‘coth x’ is pronounced as ‘koth x’
Some properties of hyperbolic functions
Replacing x by 0 in equation (1) gives:
sinh 0 =
e 0 − e −0
2
=
1 − 1
2
= 0
Replacing x by 0 in equation (2) gives:
cosh 0 =
e 0 + e −0
2
=
1 + 1
2
= 1
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
2
= −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