三角函数…

来源：百度知道 编辑：UC知道 时间：2024/05/15 04:30:44
y=3sin(x+20)+5sin(x+80)的最大值？ 曾经有位高人给解答过此题，但我自己做来只能到y=11/2sin(x+20+a),之后便做不下去了

方法一
tga=5√3/11,0<a<90°
cosa=11/14
5.5/cosa=5.5/(11/14)=7

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
方法二
y=3sin(x+10)+5sin(x+70) =3sin(x+10)+5sin(x+10+60) =3sin(x+10)+5sin(x+10)*cos60+5cos(x+10)sin60 =11/2*sin(x+10)+5√3/2*cos(x+10) 所以最大值=√((11/2)^2+(5√3/2)^2) =√(121/4+75/4) =√(196/4) =√49 =7