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

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
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

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
|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) = -1/2
cos(x+π/4) = -1/2
2[cos(x/2+π/8)]^2 -1 = -1/2
[cos(x/2+π/8)]^2 = 1
cos(x/2+π/8) =1