addition and subtraction formulas for hyperbolic functions

addition and subtraction formulas for hyperbolic functions

The

addition formulas

for

hyperbolic sine

,

hyperbolic cosine

, and

hyperbolic tangent

will be achieved via brute .

sinh

⁡

(

x

+

y

)

=

e

x

+

y

-

e

-

(

x

+

y

)

2

=

e

x

⁢

e

y

-

e

x

⁢

e

-

y

+

e

x

⁢

e

-

y

-

e

-

x

⁢

e

-

y

2

=

e

x

⁢

(

e

y

-

e

-

y

2

)

+

e

-

y

⁢

(

e

x

-

e

-

x

2

)

=

(

cosh

⁡

x

+

sinh

⁡

x

)

⁢

sinh

⁡

y

+

(

cosh

⁡

y

-

sinh

⁡

y

)

⁢

sinh

⁡

x

=

cosh

⁡

x

⁢

sinh

⁡

y

+

sinh

⁡

x

⁢

sinh

⁡

y

+

sinh

⁡

x

⁢

cosh

⁡

y

-

sinh

⁡

x

⁢

sinh

⁡

y

=

sinh

⁡

x

⁢

cosh

⁡

y

+

cosh

⁡

x

⁢

sinh

⁡

y

cosh

⁡

(

x

+

y

)

=

e

x

+

y

+

e

-

(

x

+

y

)

2

=

e

x

⁢

e

y

-

e

x

⁢

e

-

y

+

e

x

⁢

e

-

y

+

e

-

x

⁢

e

-

y

2

=

e

x

⁢

(

e

y

-

e

-

y

2

)

+

e

-

y

⁢

(

e

x

+

e

-

x

2

)

=

(

cosh

⁡

x

+

sinh

⁡

x

)

⁢

sinh

⁡

y

+

(

cosh

⁡

y

-

sinh

⁡

y

)

⁢

cosh

⁡

x

=

cosh

⁡

x

⁢

sinh

⁡

y

+

sinh

⁡

x

⁢

sinh

⁡

y

+

cosh

⁡

x

⁢

cosh

⁡

y

-

cosh

⁡

x

⁢

sinh

⁡

y

=

cosh

⁡

x

⁢

cosh

⁡

y

+

sinh

⁡

x

⁢

sinh

⁡

y

tanh

⁡

(

x

+

y

)

=

sinh

⁡

(

x

+

y

)

cosh

⁡

(

x

+

y

)

=

sinh

⁡

x

⁢

cosh

⁡

y

+

cosh

⁡

x

⁢

sinh

⁡

y

cosh

⁡

x

⁢

cosh

⁡

y

+

sinh

⁡

x

⁢

sinh

⁡

y

=

sinh

⁡

x

cosh

⁡

x

⋅

cosh

⁡

y

cosh

⁡

y

+

cosh

⁡

x

cosh

⁡

x

⋅

sinh

⁡

y

cosh

⁡

y

cosh

⁡

x

cosh

⁡

x

⋅

cosh

⁡

y

cosh

⁡

y

+

sinh

⁡

x

cosh

⁡

x

⋅

sinh

⁡

y

cosh

⁡

y

=

tanh

⁡

x

+

tanh

⁡

y

1

+

tanh

⁡

x

⁢

tanh

⁡

y

Note that

sinh

and

tanh

are

odd functions

and

cosh

is an

even function

, i.e. (

http://planetmath.org/Ie

)

sinh

⁡

(

-

t

)

=

-

sinh

⁡

t

,

tanh

⁡

(

-

t

)

=

-

tanh

⁡

t

, and

cosh

⁡

(

-

t

)

=

cosh

⁡

t

. These facts enable us to obtain the subtraction formulas.

sinh

⁡

(

x

-

y

)

=

sinh

⁡

(

x

+

(

-

y

)

)

=

sinh

⁡

x

⁢

cosh

⁡

(

-

y

)

+

cosh

⁡

x

⁢

sinh

⁡

(

-

y

)

=

sinh

⁡

x

⁢

cosh

⁡

y

-

cosh

⁡

x

⁢

sinh

⁡

y

cosh

⁡

(

x

-

y

)

=

cosh

⁡

(

x

+

(

-

y

)

)

=

cosh

⁡

x

⁢

cosh

⁡

(

-

y

)

+

sinh

⁡

x

⁢

sinh

⁡

(

-

y

)

=

cosh

⁡

x

⁢

cosh

⁡

y

-

sinh

⁡

x

⁢

sinh

⁡

y

tanh

⁡

(

x

-

y

)

=

tanh

⁡

(

x

+

(

-

y

)

)

=

tanh

⁡

x

+

tanh

⁡

(

-

y

)

1

+

tanh

⁡

x

⁢

tanh

⁡

(

-

y

)

=

tanh

⁡

x

-

tanh

⁡

y

1

-

tanh

⁡

x

⁢

tanh

⁡

y

Title

addition and subtraction formulas for hyperbolic functions

Canonical name

AdditionAndSubtractionFormulasForHyperbolicFunctions

Date of creation

2013-03-22 17:50:45

Last modified on

2013-03-22 17:50:45

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Derivation

Classification

msc 26A09

Classification

msc 33B10

Synonym

addition and subtraction formulae for hyperbolic functions

Synonym

addition formulas for hyperbolic functions

Synonym

addition formulae for hyperbolic functions

Synonym

subtraction formulas for hyperbolic functions

Synonym

subtraction formulae for hyperbolic functions

Synonym

addition form

Related topic

AdditionFormula

Related topic

HyperbolicIdentities

Related topic

AdditionFormulas

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