高中数学竞赛数列讲义.pdf

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1、2014»bÜÜ9•p¥êÆJpùÂùÇ<ÜÜ2014c1888¹¹¹1n¼ê22êØ1143êØ21611.n¼ê~~~KKK1.1.¦Šµsin2osin18osin22osin38osin42osin58osin62osin78osin82oy².

2、^ðª4sinxsin(60+x)sin(60x)=sin3x:(sin2osin58osin62o)(sin18osin42osin78o)(sin22osin38osin82o)sin6osin54osin64o=(1)64sin18o=(2)256p51=(3)1024

3、~~~KKK1.2.¦Šµtan25o+tan210o+:::+tan285oy².

4、^ðªtan2x+tan2(60+x)+tan2(60x)=9tan3x+6µ(tan25+tan255+tan265)+(tan215+tan245+tan275)+:::+(tan225+tan235+tan265)+tan230+tan260=9(tan215+tan230+:::+tan275)+30+tan230+tan260(4)2221=9(tan15+tan45+tan75)+63+(5)321=9(9tan45+6)+63+(6)35

5、95=(7)3ps:˜„·‚k
nX12n2k3tan=2n2nk=1pp~~~KKK1.3.z{µ1+4sin2acos2a+4sinacosa+cos4a5sin4a4sinacosa+6sin2ay².pp=(1+2sinacosa)2+cos4asin4a4sin4a4sinacosa+6sin2a(8)p=1+2sinacosa+cos2asin2a4sin4a4sinacosa+6sin2a(9)2p=1+2sinacosa+1+4sin2acos2a4sinacosa(10)p=1+2sinacosa+(

6、12sinacosa)2(11)=2(12)~~~KKK1.4.e2sinxsiny3cosxcosy=0¦Š11+2sin2x+3cos2x2sin2y+3cos2yy².^‡$2tanxtany=311sin2x+cos2xsin2y+cos2y+=+(13)2sin2x+3cos2y2sin2y+3cos2y2sin2x+3cos2y2sin2y+3cos2ytan2x+1tan2y+1=+(14)2tan2x+32tan2y+35=(15)6abc~~~KKK1.5.ex;y;z;a;b;c÷v==¦yµtan(x+y)tan(x+

7、z)tan(y+z)a+b2b+c2c+a2sin(yz)+sin(zx)+sin(xy)=0abbccay².d^‡:a+btan(x+y)+tan(x+z)sin(2x+y+z)==abtan(x+y)tan(x+z)sin(yz)·‚k:a+b2cos(2x+2z)cos(2x+2y)sin(yz)=sin(2x+y+z)sin(yz)=ab2¦ÚTª•0~~~KKK1.6.z{µsin8x+cos8y+6sin4xcos4x+4sin2xcos2x(sin4x+cos4x)y².3(a+b)4¥a=si

8、n2x;b=sin2yTª•1~~~KKK1.7.3n/ABC¥,¦yµ1+cosA1+cosB1+cosCp2+2+2>33ABCx2Ay².3تcosx>1¥x=221+cosA22A>AA8¦Ú:22218pLHS>++>>33ABC883~~~KKK1.8.3bn/ABC¥,¦yµpp3sinA+sinB+sinC3tanA+tanB+tanC1+4+2>22y².dþŠØª:q2(sinA+sinB+sinC)+tanA+tanB+tanCLHS>2233

9、^ت2sinx+tanx>3x;

10、>x>0:22(sinA+sinB+sinC)+(tanA+tanB+tanC)>3‘yps:a;b;c>0;x2(0;)y²2ra2+b2+c2sinxa+b+ctanx2+>a+b+c3x3x~~~KKK1.9.3n/ABC¥,e2A+3B=y²4(a+b)<5c3BBy².^‡$A=;C=+du½n••Iy²:222sin3B+sinBsinA+sinB525<$

11、^úª,·‚Psin=t5¿:0<=<•0

12、+2t+1<$(4t1)>0cosB42~~~KKK1.10.3n/ABC¥,•Øu45o,¦y:pcotA+cotB+cotC+3cot