Inverse trigonometric function

  • Arcsin ( x)
  • Arccos (x )

The arcsine - written or - and the arccosine (or arccosine ) - written or - are inverse functions of (suitable ) restricted sine or cosine function: Since sine and cosine are periodic functions, becomes its reversal of the domain of the sine to the interval and the cosine of the regions on the interval. Sine and cosine are on these intervals strictly monotonic.

Together with the arctangent as an inverse function of the (of course also suitable restricted ) form of the tangent arc sine and arc cosine of the core of the class of Inverse Trigonometric Functions. Due to the recently commonly used for inverse functions spelling begin notably widespread on calculators spellings and to displace the classical spelling or what possibly may lead to confusion with the reciprocals of the sine and cosine ( cosecant and secant ).

Definitions

The sine function is periodic, not injective and within a period. Therefore, their scope should be suitably limited in order to obtain a reversible - valued function. There are several possibilities for this limitation, one speaks of branches of the inverse sine. Most of the main branch (or main value)

The inverse of the restriction of the sine function on the interval considered.

Analogous to the arcsine is the main branch of the arc cosine as the inverse function of defined. This results with

Also a bijective function. by means of

Can be converted into each other, these two functions.

Properties

Formulas for negative arguments

Due to the symmetry properties of the following applies:

Series expansions

The Taylor series of the arcsine are obtained by applying the binomial series to the discharge, it is given by:

The term denotes the double factorial.

The Taylor series of the inverse cosine is due to the relationship:

Both series have radius of convergence 1

Integral representations

The integral representations of the arcsine or arccosine are given by:

Linkages with sine and cosine

For the Inverse Trigonometric Functions, among others, the following formulas apply:

Relationship with the arctangent

Of particular importance in older programming languages ​​implemented without arc sine and arc cosine function following relations which make it possible to calculate the inverse sine and inverse cosine of the arctangent are perhaps implemented. Applies Based on the above formulas

For one defines these two equations are also correct. Alternatively, you can also

Use what is clear from the above, by applying the functional equation of the arc tangent and apply. For the latter approach can also be

Simplify.

Derivations

Integrals

Complex arguments

To function arcosh see Areakosinus hyperbolic, applies to the signum function

Comments

Special values

Continued fraction representation of the arcsine

HS Wall took place following the arc sine for presentation in 1948 as a continued fraction:

Others

One can express arcsine and arccosine by the main branch of the complex logarithm:

Asymptote Hyperbolic function Inverse hyperbolic function
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