数学三角证明求证:cos(派/2n+1)*cos(2派/2n+1)*cos(3派/2n+1)*…*cos(n派/2n+1

1个回答

  • 是在网上找到了关于sin的

    sin(π/(2n+1))sin(2π/(2n+1))sin(3π/(2n+1)).sin(nπ/(2n+1))=√(2n +1)/2^n

    设Z=cos2π/(2n+1)+ isin2π/(2n+1)

    则x^(2n+1)=1的根为1,z,...z^2n

    得x^2n+...+x+1=(x-z)(x-z^2)...(x-z^2n)

    2n+1=|(1-z)||(1-z^2)|...|(1-z^2n)|...(1)

    又|(1-z^k)|=2sinkπ/(2n+1)...(2)

    |1-z^k| = |1-(cos(2kπ/(2n+1)) +sin(2kπ/(2n+1)) )|

    =|1-cos(2kπ/(2n+1))) -sin(2kπ/(2n+1)) )|

    =√((1-2cos(2kπ/(2n+1)) +cos^2 (2kπ/(2n+1))) + sin^2 (2kπ/(2n+1)))

    =√(2-2cos(2kπ/(2n+1)) )

    =√(4sin^2(kπ/(2n+1))

    =2sin(kπ/(2n+1)