PROGRAMME 3
HYPERBOLIC
FUNCTIONS
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Given that:
cos  j sin  e j and cos  j sin  e j
then:
and so, if   jx
j
 j
e

e
cos 
2
jjx
 jjx
x
x
cos jx  e  e  e  e
2
2
This is the even part of the exponential function and is defined to be the
hyperbolic cosine:
x
x
cosh x  e  e
2
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
The odd part of the exponential function and is defined to be the hyperbolic
sine:
x  e x
e
sinh x 
2
The ratio of the hyperbolic sine to the hyperbolic cosine is the hyperbolic
tangent
tanh x 
STROUD
sinh x ex  e x
 x x
cosh x e  e
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
The power series expansions of the exponential function are:
2
3
4
2
3
4
ex 1 x  x  x  x  ... and e x 1 x  x  x  x ...
2! 3! 3!
2! 3! 3!
and so:
2
4
6
3
5
7
cosh x 1 x  x  x  ... and sinh x  x  x  x  x  ...
2! 3! 6!
3! 5! 7!
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Graphs of hyperbolic functions
The graphs of the hyperbolic sine and the hyperbolic cosine are:
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Graphs of hyperbolic functions
The graph of the hyperbolic tangent is:
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Evaluation of hyperbolic functions
The values of the hyperbolic sine, cosine and tangent can be found using a
calculator.
If your calculator does not possess these facilities then their values can be
found using the exponential key instead.
For example:
1.275  e1.275
sinh1.275  e
STROUD
2

3.5790.279
1.65 to 2dp
2
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Inverse hyperbolic functions
To find the value of an inverse hyperbolic function using a calculator
without that facility requires the use of the exponential function.
For example, to find the value of sinh-11.475 it is required to find the value
of x such that sinh x = 1.475. That is:
ex 
Hence:
1
2 x  2.950e x 1  0

2.950
so
that
e
ex
ex  3.257 or  0.307 so x 1.1808
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Log form of the inverse hyperbolic functions
If y = sinh-1x then x = sinh y. That is:
e y  e y  2x so that e2 y  2xe y 1 0
therefore:
e y  x  x 2 1
So that


y  sinh-1 x  ln x  x 2 1
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Log form of the inverse hyperbolic functions
Similarly:


y  cosh -1 x   ln x  x 2 1
and
1 1 x 
y  tanh-1 x  ln 

2 1 x 
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Hyperbolic identities
Reciprocals
Just like the circular trigonometric ratios, the hyperbolic functions also
have their reciprocals:
STROUD
coth x 
1
tanh x
sechx 
1
cosh x
cosechx 
1
sinh x
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Hyperbolic identities
From the definitions of coshx and sinhx:
2
So:
2
x
x
x
x




e

e
e

e
cosh 2 x  sinh 2 x  

 2   2 

 

 e2 x  2 e2 x   e2 x  2 e2 x 

  


4
4

 

1
cosh2 x  sinh2 x 1
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Hyperbolic identities
Similarly:
sech 2 x 1 tanh 2 x
cosech 2 x  coth 2 x 1
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Hyperbolic identities
And:
sinh 2 x  2sinh x cosh x
cosh 2 x  cosh 2 x  sinh 2 x
 1 2sinh 2 x
 2cosh 2 x 1
tanh 2 x 
2 tanh x
1 tanh 2 x
A clear similarity with the circular trigonometric identities.
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Introduction
Graphs of hyperbolic functions
Evaluation of hyperbolic functions
Inverse hyperbolic functions
Log form of the inverse hyperbolic functions
Hyperbolic identities
Relationship between trigonometric and hyperbolic functions
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Relationship between trigonometric and hyperbolic functions
Since:
j
 j
j
 j
cos  e  e
and j sin  e  e
2
2
it is clear that for   jx
cos jx  cosh x
j sin x  sinh jx
STROUD
Worked examples and exercises are in the text
Relationship between trigonometric and hyperbolic functions
Similarly:
cosh jx  cos x
sin jx  j sinh x
And further:
tanh jx  j tan x
tan jx  j tanh x
STROUD
Worked examples and exercises are in the text
Programme 3: Hyperbolic functions
Learning outcomes
Define the hyperbolic functions in terms of the exponential function
Express the hyperbolic functions as power series
Recognize the graphs of the hyperbolic functions
Evaluate hyperbolic functions and their inverses
Determine the logarithmic form of the inverse hyperbolic functions
Prove hyperbolic identities
Understand the relationship between the circular and the hyperbolic trigonometric
ssfunctions
STROUD
Worked examples and exercises are in the text