loading3. 证明：∵sinA?sinB?sinC?2sin
A?BA?Bcos?sin(A?B) 22A?BA?BA?BA?B ?2sincos?2sincos2222A?BA?BA?B?2sin(cos?cos)
222CAB?2cos?2coscos
222ABC?4coscoscos
222ABC∴sinA?sinB?sinC?4coscoscos
222a2?ac?b2?bcab?1， 4．证明：要证??1，只要证2ab?bc?ac?cb?ca?c即a2?b2?c2?ab
而∵A?B?120,∴C?600
0a2?b2?c22cosC?,a?b2?c2?2abcos600?ab
2ab∴原式成立。
CA3b ?ccos2?2221?cosC1?cosA3sinB ∴sinA? ?sinC??222 即sinA?sinAcosC?sinC?sinCcosA?3sinB
5．证明：∵acos2 ∴sinA?sinC?sin(A?C)?3sinB
即sinA?sinC?2sinB，∴a?c?2b
参考答案（数学5必修）第一章 [提高训练C组] 一、选择题
1.C sinA?cosA?2sin(A?),
4?而0?A??,2.B
?4?A??4?5?2????sin(A?)?1 424a?bsinA?sinB??sinA?sinB csinCA?BA?BA?B ?2sin cos?2cos2223.D cosA?11,A?600,SVABC?bcsinA?63 22004.D A?B?900则sinA?cosB,sinB?cosA，0?A?45, sinA?cosA，45?B?90,sinB?cosB
5.C a2?c2?b2?bc,b2?c2?a2??bc,cosA??,A?1200
0012sinAcosBsin2AcosBsinA??,?,sinAcosA?sinBcosB 6.B
cosAsinBsin2BcosAsinB sin2A?sin2B,2A?2B或2A?2B??
二、填空题
1. 对 sinA?sinB,则2. 直角三角形
ab??a?b?A?B 2R2R1(1?cos2A?1?cos2B)?cos2(A?B)?1, 21(cos2A?cos2B)?cos2(A?B)?0, 2cos(A?B)cos(A?B)?cos2(A?B)?0
cosAcosBcosC?0
??3. x?y?z A?B?,A??B,sinA?cosB,sinB?cosA,y?z
22 c?a?b,sinC?sinA?sinB,x?y,x?y?z
A?CA?CA?CA?C cos?4sincos2222A?CA?CACACcos?2cos,coscos?3sinsin
2222221AC则sinAsinC?4sin2sin2 3221cosA?cosC?cosAcosC?sinAsinC
3AC??(1?cosA)(1?cosC)?1?4sin2sin2
22ACAC??2sin2?2sin2?4sin2sin2?1?1
2222??tanA?tanC5. [,) tan2B?tanAtanC,tanB??tan(A?C)?
32tanAtanC?1tanA?tanC tanB??tan(A?C)? 2tanB?14．1 sinA?sinC?2sinB,2sintan3B?tanB?tanA?tanC?2tanAtanC?2tanB
tan3B?3tanB,tanB?0?tanB?3?B?
?3
6．1 b?ac,sinB?sinAsinC,cos(A?C)?cosB?cos2B
22?cosAcosC?sinAsinC?cosB?1?2sin2B
?cosAcosC?sinAsinC?cosB?1?2sinAsinC ?cosAcosC?sinAsinC?cosB?1
?cos(A?C)?cosB?1?1
三、解答题
a2?b2sin(A?B)a2sinAcosBsin2A?,2??1. 解：2 22a?bsin(A?B)bcosAsinBsinB
cosBsinA?,sin2A?sin2B,2A?2B或2A?2B?? cosAsinB∴等腰或直角三角形
2. 解：2RsinA?sinA?2RsinC?sinC?(2a?b)sinB,
asinA?csinC?(2a?b)sinB,a2?c2?2ab?b2,
a2?b2?c22a?b?c?2ab,cosC??,C?4502ab2222
c?2R,c?2RsinC?2R,a2?b2?2R2?2ab, sinC2R22R?2ab?a?b?2ab,ab? 2?22221222R2S?absinC?ab??,Smax?2442?2另法：S?2?12R 2122absinC?ab??2RsinA?2RsinB 244?2?2RsinA?2RsinB?2R2sinAsinB 41?2R2??[cos(A?B)?cos(A?B)]
212?2R2??[cos(A?B)?]22 2R22??(1?)22?Smax?2?12R 此时A?B取得等号 23. 解：sinA?sinC?2sinB,2sinA?CA?CA?CA?C cos?4sincos2222sinB1A?C2B14BB7?cos?,cos?,sinB?2sincos? 222424224A?C??2,A?C???B,A?3?B?B?,C?? 4242sinA?sin(3?3?3?7?1?B)?sincosB?cossinB? 4444sinC?sin(?4?B)?sin?4cosB?cos?4sinB?7?1 4a:b:c?sinA:sinB:sinC?(7?7):7:(7?7)
4. 解：(a?b?c)(a?b?c)?3ac,a2?c2?b2?ac,cosB?1,B?600 2 tan(A?C)?tanA?tanC3?3,?3?,
1?tanAtanC1?tanAtanC tanAtanC?2?3，联合tanA?tanC?3?3 00????tanA?2?3??tanA?1?A?75?A?45 得?，即? 或?或?00tanC?1tanC?2?3???????C?45?C?75 当A?75,C?45时，b?0043?4(32?6),c?8(3?1),a?8 sinA43?46,c?4(3?1),a?8 sinA当A?45,C?75时，b?00000∴当A?75,B?60,C?45时，a?8,b?4(32?6),c?8(3?1), 当A?45,B?60,C?75时，a?8,b?46,c?4(3?1)。
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