Higher Engineering Mathematics

A

Number and Algebra

5

Hyperbolic functions

5.1 Introduction to hyperbolic

functions

Functions which are associated with the geom-
etry of the conic section called a hyperbola are
calledhyperbolic functions. Applications of hyper-
bolic functions include transmission line theory and
catenary problems. By definition:

(i) Hyperbolic sine ofx,

sinhx=

ex−e−x

2

(1)

‘sinhx’ is often abbreviated to ‘shx’ and is

pronounced as ‘shinex’

(ii) Hyperbolic cosine ofx,

coshx=

ex+e−x

2

(2)

‘coshx’ is often abbreviated to ‘chx’ and is

pronounced as ‘koshx’

(iii) Hyperbolic tangent ofx,

tanhx=

sinhx

coshx

=

ex−e−x

ex+e−x

(3)

‘tanhx’ is often abbreviated to ‘thx’ and is

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:

sinh 0=

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 of x,f(−x)=−f(x), then f(x)is

called anodd functionofx. Replacingxby−xin

equation (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

Hencesinhxand tanhxare both odd functions

(see Section 5.2), as also are cosechx

(

=

1

sinhx

)

and cothx

(

=

1

tanhx

)

If a function ofx,f(−x)=f(x), thenf(x) is called

an even function ofx. Replacing x by −x in

equation (2) gives:

cosh(−x)=

e−x+e−(−x)

2

=

e−x+ex

2

=coshx

Hencecoshxis an even function(see Section 5.2),

as also is sechx

(

=

1

coshx

)