作业帮sin(2x+y)=3siny,x≠kπ+π/2,x+y≠kπ+π/2(k∈Z)求证：tan(x+y)=2tanx
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sin(2x+y)=3siny,x≠kπ+π/2,x+y≠kπ+π/2(k∈Z)求证：tan(x+y)=2tanx
sin(2x+y)=3siny,x≠kπ+π/2,x+y≠kπ+π/2(k∈Z)求证：tan(x+y)=2tanx
sin(2x+y)=3siny,x≠kπ+π/2,x+y≠kπ+π/2(k∈Z)求证：tan(x+y)=2tanx
观察 结论tan(x+y)=2tanx 和条件sin(2x+y)=3siny中
角度的关系,可知
2x+y=(x+y)+(x)
y=(x+y)-(x)
这样可由条件推出结论
条件变为
3sin[(x+y)+x]=sin[(x+y)-x]————根据三角函数两角和公式sin(α+β)=sinαcosβ+cosαsinβ
sin(α-β)=sinαcosβ -cosαsinβ 展开得
3sin(x+y)cosx-3cos(x+y)sinx=sin(x+y)cosx+cos(x+y)sinx,化简
2sin(x+y)cosx=4cos(x+y)sinx
因为x≠kπ+π/2 ,所以cosx≠0
因为x+y≠kπ+π/2,所以cos(x+y)≠0
于是,两边同除以cos(x+y)cosx得
tan(x+y)=2tanx
令a=x+y，则条件变为
3sin(a-x)=sin(a+x)，展开得
3sinacosx-3cosasinx=sinacosx+cosasinx，移项
2sinacosx=4cosasinx
tana=2tanx，于是
tan(x+y)=2tanx
tan(x+y)=2tanx(x,x+y≠kπ+π/2,k∈Z),证3siny=sin(2x+y)答对有奖!
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