STROUD
Worked examples and exercises are in the text
PROGRAMME 3
HYPERBOLIC FUNCTIONS
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Introduction
Programme 3: Hyperbolic functions
Given that:
then:
and so, if
This is the even part of the exponential function and is defined to be the hyperbolic cosine:
cos sin and cos sinj jj e j e
cos 2j je e
cos 2 2jjx jjx x xe e e ejx
cosh 2x xe ex
jx
STROUD
Worked examples and exercises are in the text
Introduction
Programme 3: Hyperbolic functions
The odd part of the exponential function and is defined to be the hyperbolic sine:
The ratio of the hyperbolic sine to the hyperbolic cosine is the hyperbolic tangent
sinh 2x xe ex
sinhtanhcosh
x x
x xx e ex
e ex
STROUD
Worked examples and exercises are in the text
Introduction
The power series expansions of the exponential function are:
and so:
2 3 4 2 3 41 ... and 1 ...2! 3! 3! 2! 3! 3!
x xx x x x x xe x e x
Programme 3: Hyperbolic functions
2 4 6 3 5 7cosh 1 ... and sinh ...2! 3! 6! 3! 5! 7!
x x x x x xx x x
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Graphs of hyperbolic functions
Programme 3: Hyperbolic functions
The graphs of the hyperbolic sine and the hyperbolic cosine are:
STROUD
Worked examples and exercises are in the text
Graphs of hyperbolic functions
Programme 3: Hyperbolic functions
The graph of the hyperbolic tangent is:
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Evaluation of hyperbolic functions
Programme 3: 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 1.275 3.579 0.279sinh1.275 1.65 to 2dp2 2e e
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Inverse hyperbolic functions
Programme 3: 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:
Hence:
21 2.950 so that 2.950 1 0x x xx
e e ee
3.257 or 0.307 so 1.1808xe x
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Log form of the inverse hyperbolic functions
Programme 3: Hyperbolic functions
If y = sinh-1x then x = sinh y. That is:
therefore:
So that
22 so that 2 1 0y y y ye e x e xe
2
2-1
1
sinh ln 1
ye x x
y x x x
STROUD
Worked examples and exercises are in the text
Log form of the inverse hyperbolic functions
Programme 3: Hyperbolic functions
Similarly:
and
2-1
-1
cosh ln 1
11 tanh ln12
y x x x
xy xx
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Hyperbolic identities
Reciprocals
Programme 3: Hyperbolic functions
Just like the circular trigonometric ratios, the hyperbolic functions also have their reciprocals:
1 coth tanh
1 sech cosh
1cosech sinh
x x
x x
x x
STROUD
Worked examples and exercises are in the text
Hyperbolic identities
Programme 3: Hyperbolic functions
From the definitions of coshx and sinhx:
So:
2 2
2 2
2 2 2 2
cosh sinh2 2
2 24 4
1
x x x x
x x x x
e e e ex x
e e e e
2 2 cosh sinh 1x x
STROUD
Worked examples and exercises are in the text
Hyperbolic identities
Programme 3: Hyperbolic functions
Similarly:
2 2
2 2
sech 1 tanh
cosech coth 1
x x
x x
STROUD
Worked examples and exercises are in the text
Hyperbolic identities
Programme 3: Hyperbolic functions
And:
A clear similarity with the circular trigonometric identities.
2 2
2
2
2
sinh2 2sinh cosh
cosh2 cosh sinh1 2sinh2cosh 1
2 tanhtanh21 tanh
x x x
x x xx
x
xxx
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions
STROUD
Worked examples and exercises are in the text
Relationship between trigonometric and hyperbolic functions
Programme 3: Hyperbolic functions
Since:
it is clear that for
cos and sin2 2j j j je e e ej
cos cosh
sin sinh
jx x
j x jx
jx
STROUD
Worked examples and exercises are in the text
Relationship between trigonometric and hyperbolic functions
Similarly:
And further:
cosh cos
sin sinh
jx x
jx j x
tanh tan
tan tanh
jx j x
jx j x
STROUD
Worked examples and exercises are in the text
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
Programme 3: Hyperbolic functions