已知3(sinα)^2-2sinα+2(sinβ)^2=0,试求(sinα)^2+(sinβ)^2的取值范围

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已知3(sinα)^2-2sinα+2(sinβ)^2=0,试求(sinα)^2+(sinβ)^2的取值范围
已知3(sinα)^2-2sinα+2(sinβ)^2=0,试求(sinα)^2+(sinβ)^2的取值范围

已知3(sinα)^2-2sinα+2(sinβ)^2=0,试求(sinα)^2+(sinβ)^2的取值范围
sina2+sinb2=sina-sina2/2
只需求该区间就可以了
令x=sina
可得f(x)=x-x2/2 -1=

由已知移项得3(sinα)^2-2sinα+1=cos2β
所以-1≤3(sinα)^2-2sinα+1≤1，由此得0≤sinα≤2/3
那么(sinα)^2+(sinβ)^2=(sinα)^2+sinα-3/2(sinα)^2
=-1/2(sinα)^2+sinα
...

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由已知移项得3(sinα)^2-2sinα+1=cos2β
所以-1≤3(sinα)^2-2sinα+1≤1，由此得0≤sinα≤2/3
那么(sinα)^2+(sinβ)^2=(sinα)^2+sinα-3/2(sinα)^2
=-1/2(sinα)^2+sinα
=-1/2(sinα-1)^2+1/2
∵ 0≤sinα≤2/3 -1≤sinα-1≤-1/3 ∴1/9≤(sinα)^2+(sinβ)^2≤1

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