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单词 CyclometricFunctions
释义

cyclometric functions


The trigonometric functionsDlmfMathworldPlanetmath (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 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,):

sin-1x=nπ+(-1)narcsinx
cos-1x=2nπ±arccosx
tan-1x=nπ+arctanx
cot-1x=nπ+arccotx

Some formulae

arcsinx+arccosx=π2
arctanx+arccotx=π2
arcsinx=0xdt1-t2𝑑t
arctanx=0xdt1+t2𝑑t
arcsinx=x+12x33+1324x55+135246x77+(|x|1)
arctanx=x-x33+x55-x77+-(|x|1)
ddxarccosx=-11-x2(|x|<1)
ddxarccotx=-11+x2(x)

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 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.

Titlecyclometric functions
Canonical nameCyclometricFunctions
Date of creation2013-03-22 14:36:00
Last modified on2013-03-22 14:36:00
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id34
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 26A09
Synonymarc functions
Synonymarcus functions
Synonyminverse trigonometric functionsDlmfMathworld
Related topicTrigonometry
Related topicComplexSineAndCosine
Related topicTaylorSeriesOfArcusSine
Related topicTaylorSeriesOfArcusTangent
Related topicAreaFunctions
Related topicRamanujansFormulaForPi
Related topicSawBladeFunction
Related topicTerminalRay
Related topicDerivativeOfInverseFunction
Related topicLaplaceTransformOfFracftt
Related topicOstensiblyDiscontinuousAntiderivative
Related topicI
Definesbranch
Definesprincipal branch
Definessine
Definescosine
Definesarc sine
Definesarc cosine
Definesarc tangent
Definesarc cotangent
Definesinverse sine
Definesinverse tangent
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更新时间:2025/5/4 6:07:24