已知三角形ABC,角A,B,C所对的边分别为a,b,c且满足a(3a-2b)=3(c+b)(c-b),求sin2C+cos（2A+2B)的值

题目

已知三角形ABC,角A,B,C所对的边分别为a,b,c且满足a(3a-2b)=3(c+b)(c-b),求sin2C+cos（2A+2B)的值

答案

a(3a-2b)=3(c+b)(c-b),
变形得（a²+b²-c²）/2ab=1/3
就是cosC=1/3
则sinC=2√2/3
sin2C+cos（2A+2B)
=sin2C+cos2C
=2sinCcosC+cos²C-sin²C
=（4√2-7）/9