第五讲 向量与三角函数创新题型的解题技巧（The 5th lecture vector and the trig function innovation problem solving technique）

第五讲向量与三角函数创新题型的解题技巧（The5thlecturevectorandthetrigfunctioninnovationproblemsolvingtechnique）Thefifthvectorandtrigonometricfunctioninnovationproblemsolvingtechnique[propositionaltrend]In2007,wefoundthefollowingtypesandfeaturesofthetrigfunction:1.Thenatureofthetrigonometricfunction,imageanditstransformation,mainly,images,andthepropertiesoftransformation.Examinetheconceptoftrigonometricfunctions,parityandperiodicity,monotonicity,boundedness,imagetranslationandsymmetry,etc.Toselectorfillsupthetopicoranswerquestions,belongstothemedium,thepaperexaminesthenatureoftrigonometricfunctionofthesinglesmall,aproblemoftrigonometricfunctioninvolvedpropertyintwoormorethantwo,examinestheknowledgefromtheteachingmaterial.2.Triangletransform.Themaintestformulaofflexibleuse,transformability,generaltoapplyandAngle,AngleanddoubleAngleformula,especiallyfortheapplicationoftheformulaswiththecomprehensiveusageoftrigonometricfunctionproperties.Intheformofmultiplechoiceorfillsupthetopicoranswerquestions,belongstothemid-range.3.Theapplicationoftrigonometricfunction.Howtoplanevectorandparsingasthecarrier,orwithtriangletoinvestigatestudentsontriangleidenticaldeformationandthetrigonometricfunctionpropertiesandapplicationofcomprehensiveability.Specialattentionshouldbepaidtothetrianglefunctioninthepracticalproblemsandknowledgeapplication,theapplicationnotetrigonometricfunctioninthesolutionaboutfunction,vector,planeandsolidgeometry,analyticgeometryhowfewproblemsinstrumentalrole.Thiskindofproblemgenerallytakestheformofanswerstoquestions,belongstothemid-range.4.Inatestset,trigonometricfunctiongeneralrespectivelyhaveonechoice,onefillsupthetopic,andasolution,ormultiplechoiceandfillsupthetopic1,solution1,scorebetween17-22points.5.Inthehighexam,thetriangulationproblemismainlyinthelowerormiddlegradetopics,whichgenerallywon'tbedifficultordifficult,sothetriangulationisthescorepointofthecollegeentranceexam.[testperspective]1.UnderstandtheconceptofanyAngleandthemeaningofradians,andmaketheconversionofradianandAnglecorrectly.2.TomasteranyAngleofthedefinitionofsine,cosine,tangent,understandthecotangent,secant,thedefinitionofCSC,thesamesolutionoftrigonometricfunction,thebasicformulaofmastercontrolinducedformulaofsine,cosine,understandtheperiodicfunctionandthemeaningoftheleastpositiveperiod.3.Graspthesine,cosineandtangentformulaofthetwoanglesandthetwoangles,andmasterthesine,cosineandtangentformulaofdoubleAngle.4.Canusetrigonometricformulacorrectlytosimplify,evaluate,andidentityofsimpletrigonometricfunctions.5.Understandthesine,cosinefunction,tangentfunctionandnatureoftheimagewiththe"fivepoints"paintedsinefunctionandcosinefunctionandthefunctiony=Asin(x+bitsofomega)diagram,understanding,omega,thephysicalsignificanceofthebitsofA.We'regoingtotaketheAngleofthedeltafunction,andwe'regoingtousethesignarcsinx,arcosx,arctanx.7.Graspthesinusoidaltheorem,thelawofcosines,andusethemtosolvetheobliquetriangle,andcansolvethecalculationproblemofsolvingthetrianglewiththecalculator.8.Thesolutiontothesynthesisofvectorandtrigonometricfunctions.Themethodofsolvingtheproblem1.Basicstrategiesfortheidentityoftrigonometricfunctions.(1)chang-valuesubstitution:especiallywiththesubstitutionof"1",suchas1=cos2theta+sin2theta=tanx?Cotx=tan45°,etc.(2)theseparationofitemsandthematchingofangles.SoifIdividethisup,sineof2xplus2cosine2xisequaltosineof2xpluscosineof2xpluscosineof2xisequalto1pluscosineof2x;ThematchingAngle:alphaalphaalphaplusbetabetabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetaalphabetabeta(3)descendingandascending.ThedoubleAngleformulaisreducedandthehalfAngleformulaisraised.(4)string(cut)method.Thetrigonometricfunctionsareusedtoformthestring(cut).(5)introductionofauxiliaryAngle.Sinethetathetaplusbcosinethetathetathetathetathetaplusthetatheta,thetathetaplusthetatheta,thetathetaplusthetathetaplusthetatheta,thetathetaplusthetathetaplusthetatheta,thetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplusthetathetaplustheta(6)universalsubstitution.Thereisamagicformulathatcanturntrigonometricfunctionsintotangent.2.Provetheideaandmethodoftrigonometricequation.(1)trainofthought:usethetrigonometricformulaforthealias,theAngle,changetheoperationstructure,sothatthetwosidesareinthesameform.(2)proofmethod:synthesis,analysis,comparativemethod,substitutionmethod,phasedissectionmethodandmathematicalinductionmethod.3.Provethatthemethodoftriangleinequality:,reductiontoabsurdity,analysismethod,comparisonmethod,themethodusingmonotonicityoffunction,usingthesineandcosinefunctionboundedness,usetheunitcircletriangleHanShuXiananddiscriminantmethod,etc.4.Answerthestrategyoftrigonometry.(1)discoveryofdifferences:thedifferencebetweenobservationAngleandfunctionoperation,namelytheso-called"differenceanalysis".(2)findtheconnection:userelevantformulastofindouttheinternalconnectionbetweenthedifferences.(3)reasonabletransformation:choosetherightformulatomakethedifference.[problemanalysis]EvaluatethetrigonometricfunctionandsimplifyitThesetopicsmainlyincludethefollowingtypesofquestions:Thebasicmethodforevaluatingthefunctionoftrigonometricfunctionsandthevalueoftrigonometricfunctionsispresentedinthepaper.Theessayexaminestheproblemofusinginductionformula,doubleAngleformula,sineformulaoftwoangles,andthevalueofusingtheboundednessoftrigonometricfunctions.Thispaperexaminesthebasictransformationmethodsoftrigonometricfunctionsoftrigonometricidentitiesandthebasicknowledgeoftrigonometry.Thefunctionf(x)isknownasf(x)=.(Ⅰ)f(x)domain;If(Ⅱ)AngleinthefirstquadrantandaThesisaims:thisitemismainlyexaminesthetrigonometricfunctiondomainandtwoAngledifferenceformula,withtheAngleofthetrianglefunctionbasicknowledge,suchastherelationshipbetweenthetestoperationandreasoningability,andthebasicknowledgeforAngle..Solution:(Ⅰ)bySothedomainoffofxis(Ⅱ)bytheknownconditionsThus===2.(anhuivolume2006)(Ⅰ)values;(Ⅱ).Thepurposeofthethesisistostudytherelationformulaofthetrigonometricfunctions,theformulaofthetwoangles,thedoubleAngleformulaandotherbasicknowledge,tochecktheoperationandreasoningability.Answer:(Ⅰ)process,Tosolveortosolve..(II),==.Example3(sichuancurlyli2007)Known<<,Value(Ⅰ).(Ⅱ).Objective:basicknowledgeandbasiccomputingskillsoftrigonometricfunctionsandtrigonometricfunctions.Solution:(Ⅰ)by,too∴,so(Ⅱ)by,∴∵again,By:soThevalueofthetaisknown.Thesisaims:thisitemismainlyexaminestheinductionformula,withtherelationbetweentheAngleofthetrianglefunction,twoAngledifferenceformula,andthebasicknowledge,suchasdoubleAngleformulatestarithmeticandreasoningability,andthebasicknowledgeforAngle..Solution:givenbyknownconditions.Thatis.Thetaislessthantheta.TestthetriangleSuchtopictoinvestigatesinetheorem,cosinetheorem,twoAngledifferenceformulaofsinewiththerelationbetweentheAngleofthetrianglefunctionandinducedformulaandotherbasicknowledge,andexaminesthebasicoperationasthemaincharacteristics.Suchquestionsshouldpayattentiontointegratedapplicationoftheknowledge.AtypicalexampleTheknownperimeteris,and.(I)thelengthoftheedge;(II)iftheareais,thedegreeoftheAngleisobtained.Objective:toinvestigatethebasicknowledgeofsinetheorem,lawofcosinesandtrigonometry,andtheabilitytoanalyzeandsolveproblems.Solution:(I)bythemeaningandthesinusoidaltheorem,Solet'ssubtract.(II)byarea,Bythelawofcosines,so.(tianjinvolume,2006))See,inthemiddle,in.(1)thevalueoftherequest;(2)thevalue.Purposeofproposition:thisitemtestwiththeAngleofthetrianglefunctionrelation,twohornsandformulas,doubleAngleformula,theoremofsineandcosinetheoremandotherbasicknowledge,examinesthebasiccomputingabilityandanalyticalproblemsolvingskills.Solveprocess:(Ⅰ)bycosinetheorem,tooSo,(Ⅱ)byandbysinetheorem,tooSo,...ThedoubleAngleformula,And,therefore,.Inthemiddle,inthemiddle,inthemiddle.ThesizeoftheAngle(Ⅰ)o;(Ⅱ)iftheedgeofthelongfor,theedgeofthelong.Propositionalobjective:thistopicmainlyexaminestheinductionformula,sinusoidaltheoremandthetwoanglesandformulasoftrigonometricfunctions,andteststheabilityofoperation.Solution:(Ⅰ).,again.(Ⅱ)and,,,.3.Determinethedomain,range,orvalueoftrigfunctionsThesetopicsmainlyincludethefollowingtypesofquestions:Inthispaper,thefunctionofthetwoanglesandthesineformulationsofthetrigonometricfunctionsofthetwoanglesandtheboundsoftrigonometricfunctionsareusedtoevaluatetheabilityoftherange.Theessayexaminesthepropertiesoftrigfunctions,theformulaofinduction,theformulaofthetrigonometricfunctions,theformulaofthetwoangles,thebasicknowledgeofthedoubleAngleformula,andtheabilitytocheckthecomputationandreasoning.Theabilitytousetheboundednessoftrigonometricfunctionstomaximizethemaximumandminimumvalue.AtypicalexampleKnownfunctions,therangeis()A.b.c.D.Thepurposeofthethesisistousetwoanglesandthesineformulaofthesineformula,andtheboundednessoftrigonometricfunctionstoevaluatetheabilityoftherange.9.(shaanxipaper17,2007)Let'ssetthefunction.Thenumberof(Ⅰ)realisticvalue;Theminimumvalueof(afunctionorⅡ).Objective:tostudythesimplifiedtrigonometricfunctionofsineoftwoanglesandsine,Andwecanusetheboundednessoftrigfunctionstofindthebestpossible.Solution:(Ⅰ),too.(Ⅱ)by(Ⅰ),whentheminimumvalueis.Knownfunctions,(Ⅰ)domain;(Ⅱ)isthefourthquadrantAngle,and,forthevalue.Purposeofproposition:thenatureofthesubjecttestusingtrigonometricfunctions,inductionofformula,andtherelationbetweentheAngleofthetrianglefunction,twoAngledifferenceformula,andthebasicknowledge,suchasdoubleAngleformulatestarithmeticandreasoningskills.Solveprocess(Ⅰ)byhaveto.Sothedomainis,Because(Ⅱ)andtheAngleofthefourthquadrant,soTherefore,Example11setperiod,maximumvalue,(1)thevalueofthevalue;(2).Propositionalpurpose:theideaofthesystemisthebasicthoughtmethodwhichisoftenusedintheproblemsolving.Don'tforgettheperiodicityofthetrigfunctionswhenyou'resolvingproblems.Solution:(1),themaximumvalue,and,Let'ssolveforaisequalto2,andbisequalto3.(2),,,or,Namely(collinear),or,.(inthecaseofchongqingvolumein2006),thegraphofthefirsthighestpointontherightoftheaxisisthex-coordinate.(I)value;(II)iftheminimumvalueisevaluatedontheinterval.Thepurposeofthisthesisistoinvestigatethebasicknowledgeoftrigonometricfunctionsandthesineformulaoftwoangles.Answer:(Ⅰ)process,Solveit.(Ⅱ)by(Ⅰ)know,,Andwhenit'stimeagain,Soyougettheminimum.Therefore,itiswellknown.Thefunctionisknown(Ⅰ)theleastpositiveperiod;(Ⅱ)andthemaximumandtheminimum;If(Ⅲ),forvalue.Purposeofproposition:thenatureofthesubjecttestusingtrigonometricfunctions,inductionofformula,andtherelationbetweentheAngleofthetrianglefunction,twohornsandformulas,thebasicknowledge,suchasdoubleAngleformulatestarithmeticandreasoningskills.Solutionprocess:(Ⅰ)theleastpositiveperiodfor;Formaximumandminimumvalue(Ⅱ);(Ⅲ)because,thatis,Namely.TheimageandnatureofthetrigonometricfunctionsExaminationofimagesoftrigonometricfunctionandnatureofthesubject,isthekeyoftheuniversityentranceexamquestions.Suchquestionsrequiretheexamineeinmasteringtrigonometricfunctionimagetothenatureoftrigonometricfunctiononthebasisofflexibleuse.Usethenumberformcombiningideastoproblemsolving.AtypicalexampleKnownfunctions.Amaximumof(afunctionorⅠ)andthecollectionofthemaximumvalueoftheindependentvariable;(Ⅱ)functionsofmonotoneincreasingrange.Objective:thispaperexaminestrigonometricformulas,thepropertiesoftrigonometricfunctionsandthebasicknowledgeoftrigonometricfunctions.Solution:(I)1..Whenthatis,themaximumisachieved.Asaresult,Thesetoftheindependentvariablexthatgetsthemaximumvalueis.Method2:.Whenthatis,themaximumisachieved.Sothesetoftheindependentvariablethatgetsthemaximumvalueofx.(Ⅱ)solution:Fromthepointofquestion,thatis.Sothemonotonicallyincreasingintervalis.(in2007,hunanscroll16).Weknowwhatthefunctionis.(I)settingisanaxisofsymmetryofthefunctionimage,thevalueofit.(II)evaluatethemonotonicallyincreasingintervalofthefunction.Objective:thistopicmainlyexaminesthebasicknowledgeoftrigonometricfunctionsandthebasicknowledgeofmonotonicityandparity,aswellastheanalyticalandreasoningability.Solution:(I)bythequestion.Becauseit'sanaxisofsymmetryofafunctionimage,so,The().So.Whenit'seven,Whenit'sodd.(II).When,when(),Thefunctionistheaugmentedfunction,Themonotonicallyincreasingintervalofthefunctionis()Thefunctionisknown(I)theminimumpositiveperiodicandmonotoneintervalofthefunction;(II)theimageofthefunctioncanbetransformedbytheimageofthefunction.Objective:thebasicknowledgeoftrigonometricfunctions,trigonometricidentities,functionsoftrigonometricfunctions,andtheabilityofreasoningandcomputingarethemaintasksofthistopic.Solution:(I)TheminimumpositiveperiodThequestionbynamelyThemonotonegrowthintervaliszero(II)method1:theimageisobtainedbytranslatingallthepointsontheimagetotheleftbythelengthoftheunit,resultingintheimage,andthentranslatingallthepointsintheimageuptothelengthoftheunit.Method2:shiftallthedotsontheimagebythevector,andgettheimage.Knownfunctions(cirrus2006)(I)theminimumpositiveperiodofthefunction;(II)findthesetofmaximumvaluesofthefunction.Propositionalobjective:thispaperexaminestrigonometricformulas,periodicfunctionsoftrigonometricfunctions,andthebasicknowledgeoftrigonometricfunctionvalues,etc.,toinvestigatetheabilityofusingtrigonometricfunctionsoftrigonometricfunctions.Solveprocess:(Ⅰ)f(x)=3sin(2x-PI(6)+1-cos2(x-12)ofPI=2[32sin2(x-PI12)-12cos2(x-PI12)]+1That's2sineof2timesxminusPI12minusPI6plus12sineof2xminusPI3plus1.∴T=2=2PIPI.(Ⅱ)whenmaximumf(x),sin(2x-PI(3)=1,2x-PIPIPI2+3=2k,PIPI+512x=k(k∈Z)xexpects∴collectionfor{x∈R|x=kPIPI+5,12k∈Z}.ThepictureandpropertiesoftheplanevectorandtrigfunctionsExaminationofplanevectorandtrigonometricfunctionimageandnatureofthecombinationofsubject,isahotspotoftheuniversityentranceexamquestions.Suchquestionsrequiretheexamineeinmasteringplanevectorandtrigonometricfunctionimagetoplanevectorandtrigonometricfunctiononthebasisofthenatureoftheflexibleuse.Usethenumberformcombiningideastoproblemsolving.AtypicalexampleIn2006,theimageofthefunctionwasshiftedbythevector,andtheimagewasshownasshowninthepicture.Theanalyticexpressionofthecorrespondingfunctionoftheshiftedimageis().A.B.C.D.Objective:thispaperexaminestheapplicationofplanevectortranslationimagesandapplicationNumbersTheabilitytosolvetheproblemofthought.Solutionprocess:shifttheimageofthefunctiontothevector,andthecorrespondinganalyticalformulaforthetranslatedimageisknownastheimage.Therefore,Cisselected.In19(2006nationalⅡroll)wasknownvectorIf(Ⅰ),o;Themaximumvalueofdemand(Ⅱ).Propositionalobjective:thistopicmainlyexaminestheapplicationofplanevector,trigonometricfunctionknowledgeanalysisandcalculationability.Solution:(Ⅰ)Sotanso(Ⅱ)bywhenItisknownasthetriangletriangle,thevector,and(Ⅰ)Angle;If(Ⅱ),please.Purposeofproposition:subjectexaminestheplanevector,trigonometricfunctionconcept,relationswithAngletrigonometricfunctions,twoAngleandwithdifferenceformulaoftrigonometricfunctionanddoubleAngleformulaandotherknowledge.Testapplication,analysisandcalculationability.Solveprocess:Ⅰ∵,∴,namely.,.∵,∴∴.(Ⅱ)bytopic,tidy∴∴.∴or.Andmake,abandoned.∴.∴.[specialtrainingandcollegeentranceexaminationforecast]One.Choicequestion1.Theimageofthefunctionisshowninthefigure,andtheanalyticalformulamaybe().(A)(B)(C)(D)2.Known,and()(A)(B)(C)(D)3.ThedistancebetweenAandBontheothersideoftherivershouldbemeasured,andCandDof40metersareselectedalongtheriverbank<<ACB=60°,45°BCD=,<=60°,ADB<ADC=30°,thedistanceofABis().(A)20(B)20(C)40(D)204.Setisthefunctionofthedepthy(m)ofthewaterataport.Thefollowingtableistherelationshipbetweentimetandwaterdepthofacertaindayfrom0to24.T0,3,6,9,9,12,15,18,21,24Y1215.1,12.1,11.9,11.9,11.9,11.9,12.1Theimageofafunctioncanbeapproximatedastheimageofafunctioninthelongtermobservation.Inthefollowingfunction,themostapproximatefunctionofthecorrespondingrelationbetweendatainthetableis().(A)(B)(C)(D)5.Itisknownthat,inthefollowingfouranswers,itmaybetruethat()(A)(B)3or(C)(D)orFillintheblanks.6.Asshowninfigure,aradiusof10metersofthehydraulicpressanticlockwiseturnfourlapsperminute.RememberthedistanceofthepointPonthewheeltothesurfaceofdm(Pdnegative)whenthesurfaceofthewater,thed(m)andsatisfytherelationbetweentimet(inseconds):,andwhenthepointPfromsurfacecurrenttime.Therearefourconclusions:(1)A=10;(2);(3);(4)=5k.Theserialnumberofallthecorrectconclusions.7.Weknowthatsineof3alphapluscosineof3alphais1,sinealphapluscosinealpha;Sin4alpha+cos4alpha;Sineof6alphapluscosineof6alpha.Three.Problemsolving8.Findtheminimumpositiveperiodandminimumvalueofthefunction;Andwritethemonotonicallyincreasingintervalofthefunction.9.Findtheminimumpositiveperiod,maximumandminimumvalueofthefunction.WeknowthatalphaistheacuteAngle,andtheevaluatedvalue.Weknowthat0<alpha<,tan+cot=,andIwanttoevaluatethesineofx.12..13.Knownvalues.14.Asshowninfigure,AandBisAmomentofOEFGdifferentpointsontheborder,and<AOB=45°,OE=1,EF=,<AOE=alpha.(1)writetheareaofthedeltaAOBintermsofthefunctionrelationf(alpha)ofalpha.(2)writetherangeofthefunctionf(x).Soweknowthatyisequaltocosineof2xplussinex,right?Cosineofxplus1,(1)whenthefunctionyobtainsthemaximumvalue,thesetoftheindependentvariablex;(2)thegraphofthisfunctioncanbeobtainedbythetranslationandscalingtransformationoftheimageofy=sinx(x).【reference】A.inb.inc.ind.inc.ind.inTwo.Six.7.Solution1:sinealphapluscosinealphaequalst,sineofalpha?Thecosineofalpha=,∴sin3alpha+cos3alpha=(sinealpha+cosinealpha)(sin2alphasinealpha?Cosinealpha+cos2alpha)=t?(1-)=1.T3-3t+2=0(t-1)2?(t+2)=0,∵tindicates-2∴t=sinalpha+cosinealpha=1,andthesinealpha?Thecosineofalpha==0.∴sin4alpha+cos4alpha=(sin2alpha+cos2alpha)2,2sin2alpha?Cos2alpha=1or2?0=1Sin6alpha+cos6alpha=(sin2alpha+cos2alpha)(sin4alphasin2alpha?Cos2alpha+cos4alpha)=1Method2:∵sin3alphasin2alpha,orlesscos3alphacos2alphaorless∴sin3alpha+cos3alphasin2alpha+cos2alpha=1orlessEqualsignwhen