a-level数学三角函数知识点学习与介绍

 

　　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

　　Fromourship-mastexamplebefore,therelationshipbetweenanangleanditstangentcanbedeterminedfromitsgraph,shownbelow.Thegraphsofsineandcosineareincludedaswell.

　　▎下为三张三角函数公式表：

　　

a-level数学