用弦化边来做
∵2R(sin^2A-sin^2c)=(根号3*a-b)sinB
asinA-csinC=(根号3*a-b)sinB
a²-c²=根号3*ab-b²
∴cosC=(a²+b²-c²)/2ab=根号3*ab/2ab=根号3/2
C=π/6
S△abc=absinC/2=根号3ab/4≤根号3(a²+b²)/8,当且仅当a=b时等号成立.
此时,A=7π/12
由正弦定理,得
a/sinA=c/sinC=2R
a/sin(7π/12)=2R
a=(根号2+根号6）R
S△abc最大值=根号3a²/4=（2*根号3+3）R²