已知y=3sin(x+20)+5sin(x+80)求最小值

题目

已知y=3sin(x+20)+5sin(x+80)求最小值
如题

答案

y=3sin(x+20°)+5sin(x+80°）
=3sin(x+20°)+5sin(x+20°+60°）
=3sin(x+20°)+5*[sin(x+20°)*cos60°+cos(x+20°)*sin60°]
=3sin(x+20°)+2.5sin(x+20°)+2.5√3*cos(x+20°)
=5.5sin(x+20°)+2.5√3*cos(x+20°)
=5.5*[sin(x+20°)+(5√3/11)*cos(x+20°)]
=5.5*[sin(x+20°)+tga*cos(x+20°)]
=5.5*[sin(x+20°)*cosa+cos(x+20°)*sina]/cosa
=7*sin(x+a+20°)
y=3sin(x+20°)+5sin(x+80°）最小值=-7,此时：sin(x+a+20°)=-1