已知向量m=(sinx,cosx),n=(cosx,根号2-sinx),x属于π到二π,且m+n的模长为倍根号2,求cos(x/2+π/8)的值

题目

已知向量m=(sinx,cosx),n=(cosx,根号2-sinx),x属于π到二π,且m+n的模长为倍根号2,求cos(x/2+π/8)的值

答案

|m+n|^2= (sinx+cosx)^2+(cosx+√2-sinx)^2= 1+2sinxcosx + (cosx-sinx)^2 +2 +2√2(cosx-sinx)= 4+2√2(cosx-sinx) = 2(√2/2)(cosx-sinx) = -1/2cos(x+π/4) = -1/22[cos(x/2+π/8)]^2 -1 = -1/2[cos(x/2+π/8)]^2...