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单词 GoniometricFormulas
释义

goniometric formulas


The goniometric (from Greek γωνíα “angle” and μετϱικóς “measuring”) concerns the trigonometric functionsDlmfMathworldPlanetmath and their mutual connections. There are a great amount of formulas involving these functionsMathworldPlanetmath (usually for real argumentsMathworldPlanetmath).

  1. 1.

    Pythagorean identities

    • sin2x+cos2x=1

    • tan2x+1=sec2x

    • 1+cot2x=csc2x

  2. 2.

    Fractional identities

    • tanx=sinxcosx

    • cotx=cosxsinx

    • cotx=1tanx

    • tanx=1cotx

    • cscx=1sinx

    • secx=1cosx

  3. 3.

    Formulas involving radicalsMathworldPlanetmathPlanetmath (http://planetmath.org/Radical6)

    • sinx=±tanx1+tan2x

    • cosx=±11+tan2x

  4. 4.

    Weierstrass substitution formulas and related formula for tanx

    • sinx=2tan(x2)1+tan2(x2)

    • cosx=1-tan2(x2)1+tan2(x2)

    • tanx=2tan(x2)1-tan2(x2)

  5. 5.

    Trigonometric functions of a purely imaginary numberMathworldPlanetmath

    • sin(ix)=isinhx

    • cos(ix)=coshx

    • tan(ix)=itanhx

    • cot(ix)=icothx

    • csc(ix)=icschx

    • sec(ix)=sechx

  6. 6.

    Addition formulasPlanetmathPlanetmath and subtraction formulas (http://planetmath.org/AdditionFormulasForSineAndCosine)

    • sin(x±y)=sinxcosy±cosxsiny

    • cos(x±y)=cosxcosysinxsiny

    • tan(x±y)=tanx±tany1tanxtany

  7. 7.

    Formulas for trigonometric functions of a complex numberMathworldPlanetmathPlanetmath

    • sin(x+iy)=sinxcoshy+icosxsinhy

    • cos(x+iy)=cosxcoshy-isinxsinhy

    • tan(x+iy)=tanx+itanhy1-itanxtanhy

  8. 8.

    Complement formulas

    • sin(π2-x)=cosx

    • cos(π2-x)=sinx

    • tan(π2-x)=cotx

  9. 9.

    Supplement formulas

    • sin(π-x)=sinx

    • cos(π-x)=-cosx

    • tan(π-x)=-tanx

  10. 10.

    Explement formulas

    • sin(2π-x)=-sinx

    • cos(2π-x)=cosx

    • tan(2π-x)=-tanx

  11. 11.

    angle formulas

    • sin(-x)=-sinx

    • cos(-x)=cosx

    • tan(-x)=-tanx

  12. 12.

    Periodicity (http://planetmath.org/Periodic) formulas

    • sin(x+2π)=sinx

    • cos(x+2π)=cosx

    • tan(x+π)=tanx

  13. 13.

    Double angle formulas

    • sin(2x)=2sinxcosx

    • cos(2x)=cos2x-sin2x=2cos2x-1=1-2sin2x

    • tan(2x)=2tanx1-tan2x

  14. 14.

    Triple angle formulas

    • sin(3x)=3sinx-4sin3x=(4cos2x-1)sinx

    • cos(3x)=4cos3x-3cosx=(1-4sin2x)cosx

    • tan(3x)=3tanx-tan3x1-3tan2x

  15. 15.

    Half angle formulas

    • sin(x2)=±1-cosx2

    • cos(x2)=±1+cosx2

    • tan(x2)=sinx1+cosx=1-cosxsinx=±1-cosx1+cosx

  16. 16.

    Prosthaphaeresis formulasMathworldPlanetmath

    • sinx+siny=2sin(x+y2)cos(x-y2)

    • sinx-siny=2sin(x-y2)cos(x+y2)

    • cosx+cosy=2cos(x+y2)cos(x-y2)

    • cosx-cosy=-2sin(x+y2)sin(x-y2)

  17. 17.

    formulas

    • sinxsiny=cos(x-y)-cos(x+y)2

    • cosxsiny=sin(x+y)-sin(x-y)2

    • cosxcosy=cos(x-y)+cos(x+y)2

  18. 18.

    Other sums and differences

    • tanx±tany=sin(x±y)cosxcosy

    • cotx±coty=sin(y±x)sinxsiny

    • cosx±sinx=2sin(π4±x)=2cos(π4x)

  19. 19.

    formulas

    • Second power

      • *

        sin2x=1-cos(2x)2

      • *

        cos2x=1+cos(2x)2

      • *

        tan2x=1-cos(2x)1+cos(2x)

    • Third power

      • *

        sin3x=3sinx-sin(3x)4

      • *

        cos3x=3cosx+cos(3x)4

      • *

        tan3x=3sinx-sin(3x)3cosx+cos(3x)

    • Fourth power

      • *

        sin4x=cos(4x)-4cos(2x)+38

      • *

        cos4x=cos(4x)+4cos(2x)+38

      • *

        tan4x=cos(4x)-4cos(2x)+3cos(4x)+4cos(2x)+3

  20. 20.

    Recursion formulas

    • sin[(n+1)x]=2cosxsin(nx)-sin[(n-1)x]

    • cos[(n+1)x]=2cosxcos(nx)-cos[(n-1)x]

  21. 21.

    Exponential formulas (http://planetmath.org/ExponentialFunction)

    • eix=cosx+isinx

    • e-ix=cosx-isinx

    • cosx=eix+e-ix2

    • sinx=eix-e-ix2i

    • tanx=eix-e-ixi(eix+e-ix)

  22. 22.

    Some special formulas

    • tan(x+π4)=cosx+sinxcosx-sinx=±1+sin2x1-sin2x

    • tanx+secx=tan(x2+π4)

    • tan(x±y2)=sinx±sinycosx+cosy=cosy-cosxsinxsiny

    • tan(x+y2)tan(x-y2)=cosy-cosxcosy+cosx

    • sin(x+y)sin(x-y)=sin2x-sin2y

Titlegoniometric formulas
Canonical nameGoniometricFormulas
Date of creation2013-03-22 17:00:40
Last modified on2013-03-22 17:00:40
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id39
AuthorWkbj79 (1863)
Entry typeTopic
Classificationmsc 33B10
Classificationmsc 26A09
Synonymtrigonometric identities
Synonymgoniometric formulae
Related topicTrigonometryMathworldPlanetmath
Related topicDefinitionsInTrigonometry
Related topicExampleOnSolvingAFunctionalEquation
Related topicIntegrationOfRationalFunctionOfSineAndCosine
Related topicWeierstrassSubstitutionFormulas
Related topicTangentOfHalvedAngle
Related topicExampleOfTelescopingSum
Related topicDerivativeForParametricForm
Related topicComplementaryAngles
Related topicS
Definessupplement formula
Definescomplement formula
Defineshalf angle formula
Definesproduct formula
DefinesPythagorean identities
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更新时间:2025/5/4 11:21:53