cyclometric functions

cyclometric functions

The

trigonometric functions

(

http://planetmath.org/DefinitionsInTrigonometry

) are periodic, and thus get all their values infinitely many times.  Therefore their

inverse functions

, the

cyclometric functions

, are multivalued, but the values within suitable chosen

intervals

are unique; they form

single-valued functions

, called the

branches

of the

multivalued functions

.

The of the most used cyclometric functions are as follows:

•

arcsin

⁡

x

is the angle

y

satisfying

sin

⁡

y

=

x

and

-

π

2

<

y

≦

π

2

(defined for

-

1

≦

x

≦

1

)

•

arccos

⁡

x

is the angle

y

satisfying

cos

⁡

y

=

x

and

0

≦

y

<

π

(defined for

-

1

≦

x

≦

1

)

•

arctan

⁡

x

is the angle

y

satisfying

tan

⁡

y

=

x

and

-

π

2

<

y

<

π

2

(defined in the whole

ℝ

)

•

arccot

⁡

x

is the angle

y

satisfying

cot

⁡

y

=

x

and

0

<

y

<

π

(defined in the whole

ℝ

)

Those

functions

are denoted also

sin

-

1

⁡

x

,

cos

-

1

⁡

x

,

tan

-

1

⁡

x

and

cot

-

1

⁡

x

.  We here use these notations temporarily for giving the corresponding multivalued functions (

n

=

0

,

±

1

,

±

2

,

…

):

sin

-

1

⁡

x

=

n

⁢

π

+

(

-

1

)

n

⁢

arcsin

⁡

x

cos

-

1

⁡

x

=

2

⁢

n

⁢

π

±

arccos

⁡

x

tan

-

1

⁡

x

=

n

⁢

π

+

arctan

⁡

x

cot

-

1

⁡

x

=

n

⁢

π

+

arccot

⁡

x

Some formulae

arcsin

⁡

x

+

arccos

⁡

x

=

π

2

arctan

⁡

x

+

arccot

⁡

x

=

π

2

arcsin

⁡

x

=

∫

0

x

d

⁢

t

1

-

t

2

⁢

𝑑

t

arctan

⁡

x

=

∫

0

x

d

⁢

t

1

+

t

2

⁢

𝑑

t

arcsin

x

=

x

+

1

2

⋅

x

3

3

+

1

⋅

3

2

⋅

4

⋅

x

5

5

+

1

⋅

3

⋅

5

2

⋅

4

⋅

6

⋅

x

7

7

+

…

(

|

x

|

≦

1

)

arctan

x

=

x

-

x

3

3

+

x

5

5

-

x

7

7

+

-

…

(

|

x

|

≦

1

)

d

d

⁢

x

arccos

x

=

-

1

1

-

x

2

(

|

x

|

<

1

)

d

d

⁢

x

arccot

x

=

-

1

1

+

x

2

(

∀

x

∈

ℝ

)

The classic abbreviations of the cyclometric functions are usually explained as follows.  The values of the trigonometric functions are got from the

unit circle

by means of its arc (in Latin

arcus

) with starting point
(1, 0).  The arc the angle (which may be thought as a

central angle

of the circle), and its

end point

(

ξ

,

η

)

is achieved when the starting point has circulated along the

circumference

anticlockwise for

positive

angle (and clockwise for negative angle).  Then the cosine of the arc (i.e. angle) is the

abscissa

ξ

of the end point, the sine of the arc is the ordinate

η

of it.  Correspondingly, one can get the

tangent

and

cotangent

of the arc by using certain points on the

tangent lines

x

=

1

and

y

=

1

of the unit circle.

The word cosine is in Latin

cosinus

, its genitive form is

cosini

.  So e.g. “

arccos

” of a given

real number

x

means the ‘arc of the cosine value

x

’, in Latin

arcus cosini x

.

Title

cyclometric functions

Canonical

name

CyclometricFunctions

Date of creation

2013-03-22 14:36:00

Last modified on

2013-03-22 14:36:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

34

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A09

Synonym

arc functions

Synonym

arcus functions

Synonym

inverse trigonometric functions

Related topic

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ComplexSineAndCosine

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TaylorSeriesOfArcusSine

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TaylorSeriesOfArcusTangent

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Related topic

I

Defines

branch

Defines

principal branch

Defines

sine

Defines

cosine

Defines

arc sine

Defines

arc cosine

Defines

arc tangent

Defines

arc cotangent

Defines

inverse sine

Defines

inverse tangent

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