若cos(π/6+α)乘以cos(π/3-α)=-1/4,α∈[π/3,π/2],则α=？？

Question

若cos（π/6+α）乘以cos（π/3-α）=-1/4,α∈[π/3,π/2]，则α=？?

Answer 1

解：因为 (π/6+α)+(π/3-α)=π/2，
所以 cos(π/6+α)=sin(π/3-α)，
又 cos(π/6+α)xcos(π/3-α)=-1/4,
所以 sin(π/3-α)cos(π/3-α)=-1/4
2sin(π/3-α)cos(π/3-α)=-1/2
sin(2π/3-2α)=-1/2
因为 α∈[π/3,π/2]，
所以 2π/3-2α∈[-π/3，0]，
所以 2π/3-2α=--π/3，
α=π/2。