sin(π+π/6)sin(2π+π/6)sin(3π+π/6).sin(2010π+π/6)

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sin(π+π/6)sin(2π+π/6)sin(3π+π/6).sin(2010π+π/6)

sin(π+π/6)sin(2π+π/6)sin(3π+π/6).sin(2010π+π/6)
sin(π+π/6)sin(2π+π/6)sin(3π+π/6).sin(2010π+π/6)

sin(π+π/6)sin(2π+π/6)sin(3π+π/6).sin(2010π+π/6)
sin(π+π/6)= -sinπ/6 = -1/2;
sin(2π+π/6)=sinπ/6= 1/2;.
所以原式等价于 = (-1/2)(1/2)(-1/2).(1/2)
含负号的项共有 2010/2=1005个,是奇数,故原式= -(1/2)^2010.

sin(π+π/6)sin(2π+π/6)sin(3π+π/6)......sin(2010π+π/6)
=(-1/2)(1/2)(-1/2)(1/2)......(-1/2)(1/2)
=-1/2^2010