Question

已知向量m=(cosa,sina)和向量n=(（根号2）-sin a,cos a)

已知向量m=(cosa,sina)和向量n=(（根号2）-sin a,cos a)a属于（180，360）且|向量m+向量n|=(8根号2)/5,求cos(a/2+22.5°)

Answer 1

m+n=(cosa-sina+√2,sina+cosa)
∴|m+n|²
=(cosa-sina+√2)²+(sina+cosa)²
=(cosa-sina)²+2√2(cosa-sina)+2+(cosa+sina)²
=2(sin²a+cos²a)+2-2√2(sina-cosa)
=2+2-2√2(sina-cosa)
=4-2√2(sina-cosa)
=2+2-4[(√2/2)sina-(√2/2)cosa]
=2+2-4[sinπ/4sina-cosπ/4cosa]
=4+4[cosπ/4cosa-sinπ/4sina]
=4+4cos(a+π/4)

4+4cos(a+π/4)=(8√2/5)²
4+4cos(a+π/4)=128/25
4cos(a+π/4)=28/25
cos(a+π/4)=7/25

cos(a+π/4)=2cos²(a/2+π/8)-1
2cos²(a/2+π/8)-1=7/25
cos²(a/2+π/8)=16/25
cos(a/2+π/8)=±4/5

a∈（π，2π)
a/2∈（π/2，π)
a/2+π/8∈（5π/8，9π/8)

∴cos(a/2+π/8)=-4/5