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8,ABCD;..)3.7:3,4,5,6,7,8,9,,.:?.(提示:,:(1)(6,7,8,9),,,9687;(2),964875;(3)0,96408753,0,9648753.)4.8,C1:y=-316x2+3xP,C2B,Q(m,n),xD,QDBPOB.mn.(提示:C2B,B.,,,QDBPOB,.)5.xy=x2+(2k-1)x+k2-1xA(x1,0)B(x2,0),x21+x22=9.:,M,AMB3?,M;,.(提示::(1)0,k(k54);(2)k(k=-1);(3)AB((0,0),(3,0));(4)SAMB=3M(M(2,-2)).)()李成章(,300071):2005-08-26(本讲适合高中).,,,.2090,.,,,,,,.,.,,,.1111,DEFABCBCCAAB,ADBECFM.520061(1)ABCM,sinBAMsinMACsinACMsinMCBsinCBMsinMBA=1;(2)MBCA,sinBMDsinDMCsinMCAsinACBsinCBAsinABM=1;(3)MCAB,sinCMEsinEMAsinMABsinBACsinACBsinBCM=1;(4)MABC,sinAMFsinFMBsinMBCsinCBAsinBACsinCAM=1..,..,.,(,),.,.:222,DBC,EFACAB,ADBECFABCBCM.sinBAMsinMACsinACMsinMCBsinCBMsinMBA=1.1(1),,,,.,MACAB,.,2BFMCME,BMCM,..,ABMC.3ABMC,4:(1)ABCM,sinBAMsinMACsinACMsinMCBsinCBMsinMBA=1;(2)BMAC,sinMBCsinCBAsinBACsinCAMsinAMCsinCMB=1;(3)MCBA,sinCMAsinAMBsinMBAsinABCsinBCAsinACM=1;(4)AMCB,sinACBsinBCMsinCMBsinBMAsinMABsinBAC=1.,(4).43,ABCADBECF.sinBADsinDACsinACFsinFCBsinCBEsinEBA=1,ADBECF.3214,ABC,O,ADBECFH,EDAB6M,FDACN.:(1)OBDF,OCDE;(2)OHMN.4(2001,):(1)OABC,BOC=2BAC.OBC=12(180-BOC)=90-BAC.AFDC,,BDF=BAC.OBFD=P.,BPD=180-PBD-PDB=90.,OBDF.,OCDE.(2)OOGMNG.ADBC,BEAC,CFAB,COG=NME,GOB=FNM,OBE=ANF,EBC=DAN,BCF=MAD,FCO=EMA.AMND1=sinMADsinDANsinANFsinFNMsinNMEsinEMA=sinBCFsinEBCsinOBEsinGOBsinCOGsinFCO=sinCOGsinGOBsinOBEsinEBCsinBCFsinFCO.OGBECF,OGH.OGMN,,OHMN.,OGMN.,,..2ABCDO,ABDCP,ADBCQ,QOQEQF,EF.:PEF.(1997,)5:5,AECEDEDF.QEQFO,,AEF=ADF=180-QDF,FED=QFD.PDA=180-PDQ,DAP=DCQ,EDP=QEC,PAE=ECB=180-QCE,sinAEFsinFED=sinQDFsinQFD=QFQD,sinEDPsinPAE=sinQECsinQCE=QCQE,sinDAPsinPDA=sinDCQsinQDC=QDQC.sinAEFsinFEDsinEDPsinPDAsinDAPsinPAE=sinAEFsinFEDsinEDPsinPAEsinDAPsinPDA=QFQDQCQEQDQC=1.EDAP720061ABCDEF.,PEF.,.,.3ABCO,BCO,AOMN,ADBC.:ADMDN.(1988,)6:6,MDA=,NDA=.=.MNMBNCO,,MAB=MBA=ACB,NAC=NCA=ABC.DABM1=sinADMsinMDBsinDBMsinMBAsinBAMsinMAD=sincossin(B+C)sinMAD=tansin(B+C)sinMAD.,DACN1=tansin(B+C)sinNAD.tan=tan.,=,ADMDN.4ABCDE,AED=ABC=90,BAC=EAD,BDCE=F.:AFBE.(23IMO)7:7,AAHBEH.,AHBDCE.ABCAED,,ABAE=ACAD=BCDE.BAC=EAD,BAD=CAE.,SABD=SACE.sinABDsinCEA=AEECABBD.BCEBDEsinBECsinEBC=BCEC,sinBEDsinDBE=BDED.HAB=EBC,EAH=BED,sinEAHsinHABsinABDsinDBEsinBECsinCEA=sinBEDsinEBCsinABDsinDBEsinBECsinCEA=sinBEDsinDBEsinABDsinCEAsinBECsinEBC=BDAEECBCEDABBDEC=AEBCEDAB=1.ABEAHBDCEF.AHBE,,AFBE..858,ABC,BAC=40,ABC=60,ACABDE,CBD=40,BCE=70,BDCE=F.:AFBC.(1998,)81:ABC=60,CBD=40,BAC=40,BCE=70,ABD=20,ACB=80,ACE=10.BAF=x,,FAC=40-x.ABCF1=sinBAFsinFACsinACFsinFCBsinCBFsinFBA=sinxsin(40-x)sin10sin70sin40sin20.sin(40-x)sinx=sin10sin40sin70sin20=2sin10=sin10sin30=sin(40-30)sin30.sin(40-x)sinx=sin40cotx-cos40x(0,),,BAF=x=30.ABC+BAF=90.,AFBC.2:AAHBCH.BAH=30,HAC=10.ABCsinBAHsinHACsinACEsinECBsinCBDsinDBA=sin30sin10sin10sin70sin40sin20=sin402cos20sin20=1.,AHBDCE.,FAH,AFBC.969,PABC,PAB=10,PBA=20,PCA=30,PAC=40.:ABC.(1996,):ACB=x,BCP=x-30.APCB1=sinAPBsinBPC#sinNPCBsinNBCA#sinNCABsinNBAP=sin150bsin100b#sin(x-30b)sinx#sin50bsin10b.sin(x-30b)sinx=2cos10b#sin10bsin50b=sin(50b-30b)sin50b.cos30b-cotx#sin30b=cos30b-cot50b#sin30b.,cotx=cot50b.,x=50b.BAC=10b+40b=50b=x=NACB,,vABC.10710,ABC,AB=AC,NA=20b,ABACDE,EBC=60b,NDCB=50b.BED.1:BED=x,,NCED=40b+x.BCED1=sinNBCDsinNDCE#sinNCEDsinNDEB#sinNEBDsinNDBC=sin50bsin30b#sin(x+40b)sinx#sin20bsin80b.sin(x+40b)sinx=sin80b2cos40b#sin20b=2cos20b=sin70bsin30b=sin(30b+40b)sin30b.sin(x+40b)sinx=cos40b+cotx#sin40bx(0,P),,NBED=x=30b.2:BED=x,,NEDB=160b-x.EDCB1=sinNDEBsinNBEC#sinNECBsinNBCD#sinNCDBsinNBDE=sinxsin40b#sin80bsin50b#sin50bsin(160b-x).920061sin(160b-x)sinx=2cos40b=sin50bsin30b=sin130bsin30b=sin(160b-30b)sin30b.sin160b#cotx-cos160b=sin160b#cot30b-cos160b.,NBED=x=30b.,DBECDBCE,?,.,,.引理2sin20b#sin40b#sin80b=34.引理的证明:2sin20b#sin40b#sin80b=2sin20b#cos10b#sin40b=(sin30b+sin10b)sin40b=12sin40b+sin10b#sin40b=12sin40b+12(cos30b-cos50b)=12sin60b=34..11811,ABCD,NCAB=30b,NABD=26b,NACD=13b,NDBC=51b.ADB.(1989,)1:ADB=x,,NADC=x+43b.CAB=30b,NABD=26b,NDBC=51b,NACD=13b,ACB=73b,NBDC=43b.BCDA1=sinNDCAsinNACB#sinNCBAsinNABD#sinNBDAsinNADC=sin13bsin73b#sin77bsin26b#sinxsin(x+43b).sin(x+43b)sinx=sin13b#sin77bsin73b#sin26b=sin13b#cos13bsin73b#sin26b=12sin73b=sin30bsin73b=sin150bsin107b=sin(107b+43b)sin107b.cos43b+cotx#sin43b=cos43b+cot107b#sin43b.,NADB=x=107b.2:ADB=x,,NDAB=154b-x.DACB1=sinNADBsinNBDC#sinNDCBsinNBCA#sinNCABsinNBAD=sinxsin43b#sin86bsin73b#sin30bsin(154b-x).sin(154b-x)sinx=sin86b2sin43b#sin73b=cos43bsin73b=sin(154b-107b)sin107b.sin(154b-x)sinx=sin154b#cotx-cos154bx(0,P),,NADB=x=107b.12912,ABC,AC=BC,NACB=80b,ABCM,MAB=10b,NMBA=30b.AMC.(1983,)1:AC=BC,NACB=80b,CAB=NCBA=50b.MAB=10b,NMBA=30b,,10CAM=40b,NCBM=20b.BCM=x,,NACM=80b-x.ABCM1=sinNCAMsinNMAB#sinNABMsinNMBC#sinNBCMsinNMCA=sin40bsin10b#sin30bsin20b#sinxsin(80b-x).sin(80b-x)sinx=sin40b#sin30bsin10b#sin20b=cos20bsin10b=sin(80b-10b)sin10b.sin(80b-x)sinx=sin80b#cotx-cos80bx(0,P),,NBCM=x=10b.AMC=NMAB+NABC+NBCM=70b.2:AC=BC,NACB=80b,CAB=NCBA=50b.MAB=10b,NMBA=30b,,NCAM=40b,NCBM=20b,NAMB=140b.AMC=x,,NCMB=220b-x.MABC1=sinNAMCsinNCMB#sinNMBCsinNCBA#sinNBACsinNCAM=sinxsin(220b-x)#sin20bsin50b#sin50bsin40b.sin(220b-x)sinx=12cos20b=sin(220b-70b)sin70b.sin(220b-x)sinx=sin220b