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a-level数学三角函数知识点学习与介绍

本文出处:alevel培训课程 发布时间:2019-05-23 10:38:41 字体大小: A+ A-

  WhatisTrigonometry?


  本文为全英叙述,可以充分引领你了解三角函数的前世今生,来龙去脉。在本文结尾处,附上三张三角函数公式表,帮助你自如地应付A-Level数学考试


  Trigonometryisabranchofmathematicsthatstudiesrelationshipsbetweenthesidesandanglesoftriangles.Trigonometryisfoundallthroughoutgeometry,aseverystraight-sidedshapemaybebrokenintoasacollectionoftriangles.


  Furtherstill,trigonometryhasastoundinglyintricaterelationshipstootherbranchesofmathematics,inparticularcomplexnumbers,infiniteseries,logarithmsandcalculus.


  Thewordtrigonometryisa16th-centuryLatinderivativefromtheGreekwordsfortriangle(trigōnon)andmeasure(metron).ThoughthefieldemergedinGreeceduringthethirdcenturyB.C.,someofthemostimportantcontributions(suchasthesinefunction)camefromIndiainthefifthcenturyA.D.


  BecauseearlytrigonometricworksofAncientGreecehavebeenlost,itisnotknownwhetherIndianscholarsdevelopedtrigonometryindependentlyorafterGreekinfluence.AccordingtoVictorKatzin“AHistoryofMathematics3rdEdition)”(Pearson,2008),trigonometrydevelopedprimarilyfromtheneedsofGreekandIndianastronomers.


  Anexample:Heightofasailboatmast


  Supposeyouneedtoknowtheheightofasailboatmast,butareunabletoclimbittomeasure.Ifthemastisperpendiculartothedeckandtopofthemastisriggedtothedeck,thenthemast,deckandriggingropeformarighttriangle.


  Ifweknowhowfartheropeisriggedfromthemast,andtheslantatwhichtheropemeetsthedeck,thenallweneedtodeterminethemast’sheightistrigonometry.


  Forthisdemonstration,weneedtoexamineacouplewaysofdescribing“slant.”Firstisslope,whichisaratiothatcompareshowmanyunitsalineincreasesvertically(itsrise)comparedtohowmanyunitsitincreaseshorizontally(itsrun).Slopeisthereforecalculatedasrisedividedbyrun.


  Supposewemeasuretheriggingpointas30feet(9.1meters)fromthebaseofthemast(therun).Bymultiplyingtherunbytheslope,wewouldgettherise—themastheight.Unfortunately,wedon’tknowtheslope.Wecan,however,findtheangleoftheriggingrope,anduseittofindtheslope.


  Anangleissomeportionofafullcircle,whichisdefinedashaving360degrees.Thisiseasilymeasuredwithaprotractor.Let’ssupposetheanglebetweentheriggingropeandthedeckis71/360ofacircle,or71degrees.


  Wewanttheslope,butallwehaveistheangle.Whatweneedisarelationshipthatrelatesthetwo.Thisrelationshipisknownasthe“tangentfunction,”writtenastan(x).Thetangentofananglegivesitsslope.Forourdemo,theequationis:tan(71°)=2.90.(We'llexplainhowwegotthatanswerlater.)


  Thismeanstheslopeofourriggingropeis2.90.Sincetheriggingpointis30feetfromthebaseofthemast,themastmustbe2.90×30feet,or87feettall.(Itworksthesameinthemetricsystem:2.90x9.1meters=26.4meters.)


  ▎Sine,cosineandtangent.


  Dependingonwhatisknownaboutvarioussidelengthsandanglesofarighttriangle,therearetwoothertrigonometricfunctionsthatmaybemoreuseful:the“sinefunction”writtenassin(x),andthe“cosinefunction”writtenascos(x).


  Beforeweexplainthosefunctions,someadditionalterminologyisneeded.Sidesandanglesthattoucharedescribedasadjacent.Everysidehastwoadjacentangles.Sidesandanglesthatdon’ttoucharedescribedasopposite.Forarighttriangle,thesideoppositetotherightangleiscalledthehypotenuse(fromGreekfor“stretchingunder”).Thetworemainingsidesarecalledlegs.


  Usuallyweareinterested(asintheexampleabove)inanangleotherthantherightangle.Whatwecalled“rise”intheaboveexampleistakenaslengthoftheoppositelegtotheangleofinterest;likewise,the“run”istakenasthelengthoftheadjacentleg.Whenappliedtoananglemeasure,thethreetrigonometricfunctionsproducethevariouscombinationsofratiosofsidelengths.


  ▎Inotherwords:


  ◆ThetangentofangleA=thelengthoftheoppositesidedividedbythelengthoftheadjacentside


  ◆ThesineofangleA=thelengthoftheoppositesidedividedbythelengthofthehypotenuse


  ◆ThecosineofangleA=thelengthoftheadjacentsidedividedbythelengthofthehypotenuse


a-level

a-level


  Fromourship-mastexamplebefore,therelationshipbetweenanangleanditstangentcanbedeterminedfromitsgraph,shownbelow.Thegraphsofsineandcosineareincludedaswell.


  ▎下为三张三角函数公式表:


  

a-level数学

a-level数学


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