BA与AC夹角的余弦为:1/3
即:cos(π-A)=1/3,即:cosA=-1/3
1
sin((B+C)/2)^2+cos(2A)+1
=(1-cos(B+C))/2+2cosA^2
=1/2+cosA/2+2cosA^2
=1/2-1/6+2/9=5/9
2
a^2=b^2+c^2-2bccosA=b^2+c^2+2bc/3=3
b^2+c^2≥2bc,故:2bc+2bc/3≤3
即:bc≤9/8,而:sinA=2√2/3
S△ABC=(1/2)bcsinA=(√2/3)bc≤3√2/8
即S△ABC的最大值是:3√2/8