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 calledhyper-
bolic functions. Applications of hyperbolic functions
include transmissionline theory and catenary problems.
By definition:
(i) Hyperbolic sine ofx,
sinhx=
ex−e−x
2
(1)
‘sinhx’ is often abbreviated to ‘shx’andis
pronounced as ‘shinex’
(ii) Hyperbolic cosine ofx,
coshx=
ex+e−x
2
(2)
‘coshx’ is often abbreviated to ‘chx’andis
pronounced as ‘koshx’
(iii) Hyperbolic tangent ofx,
tanhx=
sinhx
coshx
=
ex−e−x
ex+e−x
(3)
‘tanhx’ is often abbreviated to ‘thx’andis
pronounced as ‘thanx’
(iv) Hyperbolic cosecant ofx,
cosechx=
1
sinhx
=
2
ex−e−x
(4)
‘cosechx’ is pronounced as ‘coshecx’
(v) Hyperbolic secant ofx,
sechx=
1
coshx
=
2
ex+e−x
(5)
‘sechx’ is pronounced as ‘shecx’
(vi) Hyperbolic cotangent ofx,
cothx=
1
tanhx
=
ex+e−x
ex−e−x
(6)
‘cothx’ is pronounced as ‘kothx’
Some properties of hyperbolic functions
Replacingxby 0 in equation (1) gives:
sinh0=
e^0 −e−^0
2
=
1 − 1
2
= 0
Replacingxby 0 in equation (2) gives:
cosh 0=
e^0 +e−^0
2
=
1 + 1
2
= 1
If a function ofx,f(−x)=−f(x),thenf(x)is called
anodd functionofx. Replacingxby−xinequation(1)
gives:
sinh(−x)=
e−x−e−(−x)
2
=
e−x−ex
2
=−
(
ex−e−x
2
)
=−sinhx
Replacingxby−xin equation (3) gives:
tanh(−x)=
e−x−e−(−x)
e−x+e−(−x)
=
e−x−ex
e−x+ex
=−
(
ex−e−x
ex+e−x
)
=−tanhx