cyclometric functions

The trigonometric functionsDlmfMathworldPlanetmath ( 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 intervalsMathworldPlanetmathPlanetmath are unique; they form single-valued functions, called the branches of the multivalued functions.

The of the most used cyclometric functions are as follows:

  • arcsinx  is the angle y satisfying  siny=x  and  -π2<yπ2  (defined for -1x1)

  • arccosx  is the angle y satisfying  cosy=x  and  0y<π  (defined for -1x1)

  • arctanx  is the angle y satisfying  tany=x  and  -π2<y<π2  (defined in the whole )

  • arccotx  is the angle y satisfying  coty=x  and  0<y<π  (defined in the whole )

Those functionsMathworldPlanetmath are denoted also sin-1x, cos-1x, tan-1x and cot-1x.  We here use these notations temporarily for giving the corresponding multivalued functions (n=0,±1,±2,):


Some formulae


The classic abbreviations of the cyclometric functions are usually explained as follows.  The values of the trigonometric functions are got from the unit circleMathworldPlanetmath by means of its arc (in Latin arcus) with starting point  (1, 0).  The arc the angle (which may be thought as a central angleMathworldPlanetmath 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 abscissaMathworldPlanetmath ξ of the end point, the sine of the arc is the ordinate η of it.  Correspondingly, one can get the tangentPlanetmathPlanetmath and cotangent of the arc by using certain points on the tangent linesx=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 functionsDlmfMathworld
Related topic Trigonometry
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Related topic TaylorSeriesOfArcusSine
Related topic TaylorSeriesOfArcusTangent
Related topic AreaFunctions
Related topic RamanujansFormulaForPi
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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