Question

已知向量m=(cosθ,sinθ)和n=(-sinθ,cosθ)，θ∈［π，2π］．

(1)求|m+n|的最大值；

(2)当|m+n|=时，求cos()的值．

Answer 1

解：(1)m+n=(cosθ-sinθ+,cosθ+sinθ),                                 

|m+n|=

=

=.                                                     

∵θ∈［π，2π］，∴≤θ+≤.

∴cos(θ+)≤1,|m+n|_max=2．                                        

(2)由已知|m+n|=,得cos(θ+)=.                                  

又cos(θ+)=2cos²(+)-1,

∴cos²(+)=.                                                     

∵θ∈［π，2π］,∴≤+≤.

∴cos(+)=-．