三角形中的有关计算与证明41.已知三角形ABC的三个内角A,B,C成等差数列,且(1/cosA)+(1/cosC)=-(根号2/cosB),求cos[(A-C)/2]的值2.已知三角形ABC中sinA(sinB+cosB)-sinC=0,sinB+sin2C=0,求角A,B,C的大小
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![三角形中的有关计算与证明41.已知三角形ABC的三个内角A,B,C成等差数列,且(1/cosA)+(1/cosC)=-(根号2/cosB),求cos[(A-C)/2]的值2.已知三角形ABC中sinA(sinB+cosB)-sinC=0,sinB+sin2C=0,求角A,B,C的大小](/uploads/image/z/12567884-68-4.jpg?t=%E4%B8%89%E8%A7%92%E5%BD%A2%E4%B8%AD%E7%9A%84%E6%9C%89%E5%85%B3%E8%AE%A1%E7%AE%97%E4%B8%8E%E8%AF%81%E6%98%8E41.%E5%B7%B2%E7%9F%A5%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E7%9A%84%E4%B8%89%E4%B8%AA%E5%86%85%E8%A7%92A%2CB%2CC%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E4%B8%94%281%2FcosA%29%2B%281%2FcosC%29%3D-%28%E6%A0%B9%E5%8F%B72%2FcosB%29%2C%E6%B1%82cos%5B%28A-C%29%2F2%5D%E7%9A%84%E5%80%BC2.%E5%B7%B2%E7%9F%A5%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E4%B8%ADsinA%28sinB%2BcosB%29-sinC%3D0%2CsinB%2Bsin2C%3D0%2C%E6%B1%82%E8%A7%92A%2CB%2CC%E7%9A%84%E5%A4%A7%E5%B0%8F)
三角形中的有关计算与证明41.已知三角形ABC的三个内角A,B,C成等差数列,且(1/cosA)+(1/cosC)=-(根号2/cosB),求cos[(A-C)/2]的值2.已知三角形ABC中sinA(sinB+cosB)-sinC=0,sinB+sin2C=0,求角A,B,C的大小
三角形中的有关计算与证明4
1.已知三角形ABC的三个内角A,B,C成等差数列,且
(1/cosA)+(1/cosC)=-(根号2/cosB),求cos[(A-C)/2]的值
2.已知三角形ABC中sinA(sinB+cosB)-sinC=0,sinB+sin2C=0,求角A,B,C的大小
三角形中的有关计算与证明41.已知三角形ABC的三个内角A,B,C成等差数列,且(1/cosA)+(1/cosC)=-(根号2/cosB),求cos[(A-C)/2]的值2.已知三角形ABC中sinA(sinB+cosB)-sinC=0,sinB+sin2C=0,求角A,B,C的大小
1.A,B,C成等差数列,所以B=60度 设A=60+x,B=60-x
根号2/cosB=1/根号2
cos[(A-C)/2]=cosx=t>0
1/cos(60-x)+1/(60+x)=-1/2^0.5
(cos(60+x)+cos(60-x))/(cos(60-x)cos(60+x))=-1/2^0.5
4t/(4t^2-3)=-1/2^0.5
解方程得t=(根号5-根号2)/2(负值舍去)
cos[(A-C)/2]=(5^0.5-2^0.5)/2
2.由sinA(sinB+cosB)-sinC=0得
sinA(sinB+cosB)-sin(A+B)=0
sinA(sinB+cosB)-sinAcosB-cosAsinB=0
sinAsinB-cosAsinB=0
sinB0
sinA=cosA
A=45度
由sinB+sin2C=0得
sinB+2sinCcosC=0
sinB/2=sin(A+B)cos(A+B)
sinB/2=(sinAcosB+cosAsinB)(cosAcosB-sinAsinB)
sinA=1/2^0.5
sinB/2=1/2*(cosB+sinB)(cosB-sinB)
sinB=1-2sinB^2
sinB=0.5或-1
因为B