f(x)=cos2x/sin(x+π/4) 若f(x)=4/3,求sin2x的值

题目

f(x)=cos2x/sin(x+π/4) 若f(x)=4/3,求sin2x的值

答案

cos2x/sin(x+π/4) =4/3
cos2x/sin(x+π/4) =4/3
(cos² x-sin² x)/sin(x+π/4) =4/3
(cosx-sinx)(cosx+sinx)/sin(x+π/4) =4/3
(cosx-sinx)[√2(√2/2cosx+√2/2sinx)]/sin(x+π/4) =4/3
(cosx-sinx)[√2(sinπ/4cosx+cosπ/4sinx)]/sin(x+π/4) =4/3
(cosx-sinx)[√2sin(x+π/4)]/sin(x+π/4) =4/3
√2(cosx-sinx)=4/3(平方）
2(cosx-sinx)²=16/9
(cosx-sinx)²=8/9
cos²x+sin²x-2sinxcosx=8/9
1-2sinxcosx=8/9
2sinxcosx=1/9
sin2x=1/9