求sin(π/14)*sin(3π/14)*sin(5π/14)的值
题目
求sin(π/14)*sin(3π/14)*sin(5π/14)的值
答案
sinπ/14 sin3π/14 sin5π/14
= cos6π/14 cos4π/14 cos2π/14
=cosπ/7 cos2π/7 cos3π/7……(1)
因为2sinθcosθ=sin2θ,
又sin3π/7=sin4π/7,sin6π/7=sinπ/7
所以 (1)*sinπ/7= sinπ/7 cosπ/7 cos2π/7 cos3π/7
=1/2 sin2π/7 cos2π/7 cos3π/7
=1/4 sin4π/7 cos3π/7
=1/4 sin3π/7 cos3π/7
=1/8 sin6π/7
=1/8 sinπ/7
而(1)*sinπ/7=1/8 sinπ/7
所以(1)式=1/8
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