已知三角形ABC周长为9,AC=3,4cos2A-cos2C=31求AB的值2求sin（A-派/4）的值派是园周率

Question

已知三角形ABC周长为9,AC=3,4cos2A-cos2C=3
1求AB的值
2求sin（A-派/4）的值
派 是园周率

Answer 1

1、4cos2A-cos2C=3
=4[1-2(sinA)^2]-[1-2(sinC)^2]
3-8(sinA)^2+2(sinC)^2=3
(sinC)^2-4(sinA)^2=0,
sinC=2sinA,(三角形内正弦无负号,故舍去负值）,
sinC/sinA=2,根据下弦定理,sinC/sinA=c/a,
c/a=2,c=2a,2a+a+3=9,a=2,c=4,
AB=c=4,
2、根据余弦定理,
a^2=c^2+b^2-2bccosA,
cosA=7/8,
sinA=√（1-49/64）=√15/8,
sin(A-π/4）=sinAcos(π/4)-cosAsin(π/4)
=(√30-7√2)/16.