在△ABC中,∠A∠B∠C成等差数列其对边分别为abc且c-a等于AC上的高h 求sin(C-A)/2?

题目

在△ABC中,∠A∠B∠C成等差数列其对边分别为abc且c-a等于AC上的高h 求sin(C-A)/2?

答案

∵∠A∠B∠C成等差数列,设公差为d
∴∠A∠B∠C=3∠B=180,∠B=60
由题意知,d不为0,h=c*sinA=c-a=c-c*sinA/sinC
sinC*sinA=sinC-sinA
sin(60+d)*sin(60-d)=sin(60+d)-sin(60-d)
(sin60*cos d+cos60*sind)(sin60*cos d-cos60*sind)=sin60*cos d+cos60*sind-sin60*cos d+cos60*sind
(√3/2*cos d+sind/2)(√3/2*cos d-sind/2)=sind∴
3*(cos d)^2-(sind)^2=4sind
3-4(sind)^2=4sind
4(sind)^2+4sind-3=0
sind=-3/2(舍去)或sind=1/2
∴d=30,∠A=30,∠C=90
sin(C-A)/2=sin60/2=√3/4
若为求sin((C-A)/2)=sin30=1/2