已知y=3sin(x+20°)+5sin(x+80°）,求该函数最大值

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已知y=3sin(x+20°)+5sin(x+80°）,求该函数最大值
已知y=3sin(x+20°)+5sin(x+80°）,求该函数最大值

已知y=3sin(x+20°)+5sin(x+80°）,求该函数最大值
tga=5√3/11,0

y=3sin(x+20°)+5sin(x+80°），
=3sin(x+50°-30°)+5sin(x+50°+30°）
化简：
Y=4√3sin(x+50°）+cos(x+50°)
令sin(x+50°）=t,cos(x+50°)=√(1-x∧2），-1≤t≤1
y=4√3t+√(1-x∧2）≥2*√√〔（4√3t）∧2*(1-x∧2）〕
y≥2*√√48〔(1/4-(x-1/2)∧2〕
ymax=2*√2√3

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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°）最大值=7

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