化简cosπ/7*cos2π/7*cos4π/7
问题描述:
化简cosπ/7*cos2π/7*cos4π/7
提示:据分析应该用倍角公式.
答
cos(π/7)*cos(2π/7)*cos(4π/7)
=[2sin(π/7)cos(π/7)*cos(2π/7)*cos(4π/7)]/2sin(π/7)
=[sin(2π/7)*cos(2π/7)*cos(4π/7)]/2sin(π/7)
=[2sin(2π/7)*cos(2π/7)*cos(4π/7)]/4sin(π/7)
=[sin(4π/7)*cos(4π/7)]/4sin(π/7)
=[2sin(4π/7)*cos(4π/7)]/8sin(π/7)
=sin(8π/7)/8sin(π/7) (因为sin(8π/7)=-sin(π/7))
=-1/8