△abc中,sinA+sinC=2sinB ,a^2-c^2=ac-bc,求sin2B:sinC

题目

△abc中,sinA+sinC=2sinB ,a^2-c^2=ac-bc,求sin2B:sinC

答案

sinA+sinC=2sinB
a+c=2b
b=(a+c)/2,代人a^2-c^2=ac-bc
a^2-c^2=ac-((a+c)/2)c
2a^2-ac-3c^2=0
(2a-3c)(a+c)=0
2a-3c=0
a=(3/2)c,代人a+c=2b
(3/2)c+c=2b
b=(5/4)c
而：b^2=a^2+c^2-2ac*cosB
(25/16)c^2=(9/4)c^2+c^2-2(3/2)c^2*cosB
cosB=9/16
另外：sinB/sinC=b/c
所以：sin2B:sinC=2sinBcosB/sinC
=2(sinB/sinC)cosB
=2(b/c)*(9/16)
=2(5/4)(9/16)
=45/32