在三角形ABC中,SINA方=SINB方+SINBSINC+SINC方,求角A
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在三角形ABC中,SINA方=SINB方+SINBSINC+SINC方,求角A
在三角形ABC中,SINA方=SINB方+SINBSINC+SINC方,求角A
在三角形ABC中,SINA方=SINB方+SINBSINC+SINC方,求角A
SINA方=SINB方+SINBSINC+SINC方
根据正弦定理a/sinA=b/sinB=c/sinC
转化a^2=b^2+c^2+bc
bc=-(b^2+c^2-a^2)
余弦定理
cosA=(b^2+c^2-a^2)/(2bc)
=-bc/(2bc)
=-1/2
A=120°
120
sinA=sin(B+C)=sinB cosC+sinC cosB
=>(sinA)^2=(sinB)^2(cosC)^2+(sinC)^2(cosB)^2+2sinBsinCcosBcosC
=(sinB)^2+(sinC)^2+sinBsinC
整理得
(sinB)^2[1-(cosC)^2]+(sinC)^2[1-(cosB)^2]=sin...
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sinA=sin(B+C)=sinB cosC+sinC cosB
=>(sinA)^2=(sinB)^2(cosC)^2+(sinC)^2(cosB)^2+2sinBsinCcosBcosC
=(sinB)^2+(sinC)^2+sinBsinC
整理得
(sinB)^2[1-(cosC)^2]+(sinC)^2[1-(cosB)^2]=sinBsinC(2cosBcosC-1)
2(sinBsinC)^2=sinBsinC(2cosBcosC-1)
sinB,sinC>0
=>2sinBsinC=2cosBcosC-1
=>1=2(cosBcosC-sinBsinC)=2cos(B+C)=-2cosA
=>A=120
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