在三角形ABC中,abc是三角形ABC内角A、B、C的对边(a-b)/c=(sinB+sinC)/（sinA+sinB） 求角A的值 已知a=2... 三角形ABC的内角A,B,C的对边分别为a,b,c.已知si...
\u4e09\u89d2\u5f62ABC\u7684\u5185\u89d2ABC\u7684\u5bf9\u8fb9\u5206\u522b\u4e3aabc\uff0c\u4e14asin(A+B-C)=csin(B+C)\u6c42\u89d2C\u7684\u503c\u7531\u6b63\u5f26\u5b9a\u7406\u5f97a=2RsinA,b=2RsinB,c=2RsinC,
\u6545\u6709 asin(B-C)+bsin(C-A)+csin(A-B) =2R(sinAsin(B-C)+sinBsin(C-A)+sinCsin(A-B)) =2R(sinA(sinBcosC-cosBsinC)+sinB(sinCcosA-cosCsinA)+sinC(sinAcosB-cosAsinB)) =2R(sinAsinBcosC-sinAcosBsinC+sinBsinCcosA-sinBcosCsinA+sinCsinAcosB-sinCcosAsinB)=0
\u7b54\u9898\u4e0d\u6613\uff0c\u6ee1\u610f\u7684\u8bdd\u7ed9\u4e2a\u8d5e\u3002
\u25b3ABC\uff0csinB=sin(A+C)=sinAcosC+cosAsinC
\u6240\u4ee5\u6709sinAsinC+cosAsinC=0=(sinA+cosA)sinC
=\u221a2sin(A+\u220f/4)sinC\uff0c\u25b3ABC\uff0c\u220f>C>0\uff0c\u220f>A>0\u6240\u4ee5A+\u220f/4=\u220f\uff0cA=3\u220f/4\u3002
\u6240\u4ee5sinC/c=sinA/a=sinC/\u221a2=(\u221a2/2)/2\uff0c
sinC=1/2\uff0c\u25b3ABC\uff0cA=3\u220f/4\uff0c\u6240\u4ee5C=\u220f/6\u3002
\u540c\u89d2\u4e09\u89d2\u51fd\u6570
\uff081\uff09\u5e73\u65b9\u5173\u7cfb\uff1a
sin^2(\u03b1)+cos^2(\u03b1)=1
tan^2(\u03b1)+1=sec^2(\u03b1)
cot^2(\u03b1)+1=csc^2(\u03b1)
\uff082\uff09\u79ef\u7684\u5173\u7cfb\uff1a
sin\u03b1=tan\u03b1*cos\u03b1 cos\u03b1=cot\u03b1*sin\u03b1
tan\u03b1=sin\u03b1*sec\u03b1 cot\u03b1=cos\u03b1*csc\u03b1
sec\u03b1=tan\u03b1*csc\u03b1 csc\u03b1=sec\u03b1*cot\u03b1
（1）
根据正弦定理：a/sinA=b/sinB=c/sinC=2R
(a-b)/c=(sinB+sinC)/(sinA+sinB)=(b+c)/(a+b)
a^2-b^2=c^2+bc
b^2+c^2-a^2=-bc
根据余弦定理：cosA=(b^2+c^2-a^2)/(2bc)=-bc/(2bc)=-1/2
A=120°
（2）a=2，则：b^2+c^2-a^2=-bc=b^2+c^2-4>=2bc-4，bc<=4/3
当且仅当b=c时取得最小值b=c=2√3/3
原式=3(b^2+c^2)min=3(4/3+4/3)=8
由正弦定理：a/sinA=b/sinB=c/sinC可得
(a-b)/c=(sinB+sinC)/（sinA+sinB）
(a-b)/c=(b+c)/(a+b)
a²-b²=bc+c²
a²=b²+c²+bc=a² =b²+c²-2bccosA
cosA=-1/2
A=120
已知a=2...]
a²=b²+c²+bc=4≥2bc+bc=3bc
bc≤4/3
当且仅当b=c时取得最大值4/3
.b=c=2√3/3
3(b²+c²)≥6bc=8
3(b²+c²)的最小值8
(a-b)/c=(sinB+sinC)/（sinA+sinB），
由正弦定理,(a-b)/c=(b+c)/(a+b),
∴a^-b^=bc+c^,
∴b^+c^-a^=-bc,
∴cosA=-1/2,
∴A=120°。
b^+c^+bc=a^=4,2bc<=b^+c^,
∴(3/2)(b^+c^)>=4,
当b=c时取等号，
∴3（b^+c^)的最小值=8.
(a-b)/c=(sinB+sinC)/（sinA+sinB）
(sinA-sinB)/sinC=(sinB+sinC)/（sinA+sinB）
(sinA-sinB)（sinA+sinB）=(sinB+sinC)sinC
sin^A-sin^B=sinBsinC+sin^C
a^2-b^2=bc+c^2
b^2+c^2-a^2=-bc
(b^2+c^2-a^2)/2bc=-1/2
cosA=-1/2
A=120
a=2
b^2+c^2+bc=4
4=b^2+c^2+bc<=3/2(b^2+c^2)
b^2+c^2>=8/3
3(b^2+c^2)>=8
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