已知向量m=（cosx,sinx）和向量n=（根号2-sinx,cosx）,x属于（π,2π）,且|m+n|=（8/5）乘以根号2,求cos（x/2+π/8)的值

Question

已知向量m=（cosx,sinx）和向量n=（根号2-sinx,cosx）,x属于（π,2π）,且|m+n|=（8/5）乘以根号2,求cos（x/2+π/8)的值

Answer 1

由此可知：(cosx+√2-sinx)^2+(sinx+cosx)^2=(8√2/5)^2
即为：1+2(√2- sinx)cosx+2-2√2sinx+1+2sinxcosx=128/25
即4+2√2(cosx-sinx)=128/25.即 cos(x-π/4)=28/100.
即2(cos(x/2-π/8))^2-1=28/100.即cos(x/2-π/8)^2=16 /25.
则cos(x/2-π/8)=4/5或-4/5
而cos(x/2+π/8)=cos(x/2-π/8+π/4)=cos(x-π/8)√2/2+sin(x-π/8)√2/2
=cos(x-π/8)√2/2+√(1-(cos(x/2-π/8))^2)))√2/2.
当 cos(x/2-π/8)=4/5时,cos(x/2+π/8)=4√2/10±3√2/10,即为7√2/10,或√2/10.
当 cos(x/2-π/8)=-4/5时,cos(x/2+π/8)=-4√2/10±3√2/10,即为-7√2/10,或-√2/10.