已知三角形ABC三边成等差数列，
a:b:c =sinA:sinB:sinC
2b =a+c ====>2sinB =sinA+sinC
===>2sin(A+C) =sinA+sinC
sinA=sin(C+90)=cos[90-(C+90)] =cosC
cosA=cos(C+90) =sin[90-(C+90)]=-sinC
2sin(A+C) =2sinAcosC+2cosAsinC =2[(cosC)^2-(sinC)^2]
===>2[(cosC)^2-(sinC)^2] =cosC +sinC
2(cosC +sinC)(cosC -sinC)=(cosC +sinC)
消去(cosC +sinC)
==>2...全部
a:b:c =sinA:sinB:sinC
2b =a+c ====>2sinB =sinA+sinC
===>2sin(A+C) =sinA+sinC
sinA=sin(C+90)=cos[90-(C+90)] =cosC
cosA=cos(C+90) =sin[90-(C+90)]=-sinC
2sin(A+C) =2sinAcosC+2cosAsinC =2[(cosC)^2-(sinC)^2]
===>2[(cosC)^2-(sinC)^2] =cosC +sinC
2(cosC +sinC)(cosC -sinC)=(cosC +sinC)
消去(cosC +sinC)
==>2(cosC -sinC) =1/2
sinC =t cosC =根号(1-t^)解方程
===>t= sinC =(根号7 -1)/4
===>sinA =(根号7 +1)/4
sinB =(sinA+sinC)/2 =1/4
==>a:b:c =sinA:sinB:sinC =(根号7+1):(根号7):(根号7-1)
该回答在8月9日 20:53由回答者修改过
修改回答 ┆ 采纳答案 ┆ 评论 ┆ 举报
回答：曼丽
级别：智者
8月9日 21:18 2b=a+c, 2sinB=sinA+sinC, 4sin(B/2)cos(B/2)=2sin[(A+C)/2]cos[(A-C)/2] , 4sin(B/2)cos(B/2)=2cos(B/2)cos[(A-C)/2], 2sin(B/2)=cos45°， sin(B/2)=√2/4, cos(B/2)=√[1-(√2/4)^]=√7/2√2,
sinB=2×(√2/4)×(√7/2√2)=√7/4, 2C=90°-B, cos2C=sinB, 1-2(sinC)^=√7/4, sinC=(√7-1)/4, sinA=2sinB-sinC)=(√7+1)/4,
∴ a∶b∶c=)(√7+1)∶√7∶(√7-1)
。
收起
