求值:Cos2/7π+Cos4/7π+Cos6/7π

2个回答

  • 是Cos(π2/7)+Cos(π4/7)+Cos(π6/7)吧?

    Cos(π2/7)+Cos(π4/7)+Cos(π6/7)

    =2cos(3π/7)cos(π/7)+2(cos3π/7)^2-1……(一)

    =2(cosπ/7+cos3π/7)cos3π/7-1

    =4cos(2π/7)cos(π/7)cos(3π/7)-1……(二)

    =4sin(π/7)cos(π/7)cos(2π/7)cos(3π/7)/sin(π/7)-1

    =2sin(2π/7)cos(2π/7)cos(3π/7)/sin(π/7)-1(分子分母同乘sin(π/7))

    =sin(4π/7)cos(3π/7)/sin(π/7)-1

    =(sinπ+sinπ/7)/sin(π/7)-1……(三)

    =1-1

    =0

    (一)和(三)用了和差化积公式:cosA+cosB=2cos[(A+B)/2]cos[(A-B)/2]);(二)用了积化和差公式:sinAcosB=[sin(A+B)+sin(A-B)]/2