STROUD Worked examples and exercises are in the text PROGRAMME 3 HYPERBOLIC FUNCTIONS

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Worked examples and exercises are in the text

PROGRAMME 3

HYPERBOLIC FUNCTIONS

STROUD


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

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Introduction








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Introduction


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

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Introduction


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

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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


2 4 6 3 5 7cosh 1 ... and sinh ...2! 3! 6! 3! 5! 7!

x x x x x xx x x

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The graphs of the hyperbolic sine and the hyperbolic cosine are:

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The graph of the hyperbolic tangent is:

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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

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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

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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

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Similarly:

and

2-1

-1

cosh ln 1

11 tanh ln12

y x x x

xy xx

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Reciprocals


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

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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

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Similarly:

2 2

2 2

sech 1 tanh

cosech coth 1

x x

x x

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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

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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

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Similarly:

And further:

cosh cos

sin sinh

jx x

jx j x

tanh tan

tan tanh

jx j x

jx j x

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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


Documents

STROUD Worked examples and exercises are in the text PROGRAMME 3 HYPERBOLIC FUNCTIONS