函数Y=sin(x+1/18π)*cos(x+2/9π)的最小值

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函数Y=sin(x+1/18π)*cos(x+2/9π)的最小值
函数Y=sin(x+1/18π)*cos(x+2/9π)的最小值

函数Y=sin(x+1/18π)*cos(x+2/9π)的最小值
y= cos(x+2π/9)sin(x+π/18)
=1/2[(sin(x+x+2π/9+π/18)-sin(2π/9-π/18)]
=1/2[sin(2x+(5π /18))-sinπ/6]
=1/2[sin(2x+(5π /18))-1/2)
=1/2sin(2x+(5π /18))-1/4
当sin(2x+(5π /18))=-1时,即2x+5π/18=(-π/2)+2kπ时,有最小值,
y最小值=-1/2-1/4=-3/4

由积化和差公式,得:
Y=sin(x+1/18π)*cos(x+2/9π)=0.5*(sin(2x+5/18π)-sin(1/6π))
对Y求导,得到最小值点处的x值:
Y'=cos(2x+5/18π)=0,
解得x=1/9π,代入函数Y,得:
Y=sin(1/6π)cos(1/3π)=1/4