sin (pai/14)sin(3pai/14)sin(5pai/14)求值
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sin (pai/14)sin(3pai/14)sin(5pai/14)求值
sin (pai/14)sin(3pai/14)sin(5pai/14)
求值
sin (pai/14)sin(3pai/14)sin(5pai/14)求值
以上是答案
x=sin (pai/14)sin(3pai/14)sin(5pai/14)>0
x=cos(6π/14)cos(4π/14)cos(2π/14) (sin(π/2-α)=cosα)
=cos(π/7)cos(2π/7)cos(3π/7)
=-cos(π/7)cos(2π/7)cos(4π/7) ...
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x=sin (pai/14)sin(3pai/14)sin(5pai/14)>0
x=cos(6π/14)cos(4π/14)cos(2π/14) (sin(π/2-α)=cosα)
=cos(π/7)cos(2π/7)cos(3π/7)
=-cos(π/7)cos(2π/7)cos(4π/7) (cos(π-α)=-cosα)
-8sin(π/7)*x=8sin(π/7)*cos(π/7)cos(2π/7)cos(4π/7)
=4sin(2π/7)cos(2π/7)cos(4π/7)
=2sin(4π/7)cos(4π/7)
=sin(8π/7)
=-sin(π/7)
x=1/8
收起
分子分母同乘以2cos(π/14)
原式=2sin(π/14)cos(π/14)cos(π/2-5π/14)cos(π/2-3π/14)/2cos(π/14)
=2sin(π/7)cos(π/7)cos(2π/7)/4cos(π/14)
=2sin(2π/7)cos(2π/7)/8sin(π/2-π/14)
=sin(4π/7)/8sin(3π/7)
=1/8
注意：sin(4π/7)=sin(π-4π/7)=sin(3π/7)