题目

若f(n)=sin(nπ)/6,n∈N试求:f(1)*f(3)*f(5)*f(7)*…*f(101)的值
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提问时间：2021-02-25

答案

解析：∵sin(π/6)sin(3π/6)sin(5π/6)
sin(7π/6)sin(9π/6)sin(11π/6)
=sin(π/6)sin(π/2)sin(π/6)sin（-π/6）
sin(3π/2)sin(-π/6)=1/2*1*1/2*(-1/2)*(-1)*(-1/2)=-1/2^4
而sin(π/6)sin(3π/6)sin(5π/6)
sin(7π/6)sin(9π/6)sin(11π/6)
=sin(13π/6)sin(15π/6)sin(17π/6)
sin(19π/6)sin(21π/6)sin(23π/6)
=……=sin(85π/6)sin(87π/6)sin(89π/6)
sin(91π/6)sin(93π/6)sin(95π/6)
(sin(97π/6)=sin(π/6)=1/2
sin(99π/6)=sin(π/2)=1,
sin(101π/6)=sin(5π/6)=1/2
∴f(1)*f(3)*f(5)*f(7)*…*f(101)
=sin(π/6)sin(3π/6)sin(5π/6)
sin(7π/6)sin(9π/6)sin(11π/6)……sin(95π/6)sin(97π/6)sin(99π/6)sin(101π/6)
=(-1/2^4)^8*1/2*1*1/2
=1/2^32*1/4
1/2^34