已知向量m=(cosθ,sinθ)和向量n=√2-sinθ,cosθ),θ∈(∏,2∏),且丨m+n丨=8√2/5,求cos(θ/2+π/8)

题目

已知向量m=(cosθ,sinθ)和向量n=√2-sinθ,cosθ),θ∈(∏,2∏),且丨m+n丨=8√2/5,求cos(θ/2+π/8)

答案

cos(θ/2+π/8)=-4/5
m+n=(cosθ+√2-sinθ,sinθ+cosθ)
丨m+n丨=√[(cosθ+√2-sinθ)^2+(sinθ+cosθ)^2]
=√[4+2√2cosθ-2√2sinθ]
=√[4+4(cosπ/4cosθ-sinπ/4sinθ)]
=√[4+4cos(θ+π/4)]
=√{4+4[2cos(θ/2+π/8)cos(θ/2+π/8)-1]}
=2√2│cos(θ/2+π/8)│
=8√2/5
│cos(θ/2+π/8)│=4/5
θ∈(π,2π),θ/2+π/8∈(5π/8,9π/8)
cos(θ/2+π/8)=-4/5