求证1/sin^2a+3/cos^2a>=4+2根号下31/sin²a +3/cos²a=(sin²a+cos²a)/sin²a +3(sin²a+cos²a)/cos²a=1+cos²a/sin²a+3+3sin²a/cos²a=4+cos²a/sin²a+3sin²a/cos²a≥4+2√[(cos
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求证1/sin^2a+3/cos^2a>=4+2根号下3
1/sin²a +3/cos²a
=(sin²a+cos²a)/sin²a +3(sin²a+cos²a)/cos²a
=1+cos²a/sin²a+3+3sin²a/cos²a
=4+cos²a/sin²a+3sin²a/cos²a
≥4+2√[(cos²a/sin²a)(3sin²a/cos²a)]=4+2√3
1 /sin²a +3 /cos²a
=csc²a +3sec²a
=cot²a+1 +3 (tan²a+1)
=4+cot²a+3tan²a≥4 +2√(cot²a*3tan²a)=4 +2√3.
左边=csc^2a+3sec^2a
=1+ctg^2a+3+3tan^2a
=4+(ctg^2a+3tan^2a)
>=4+2根号3
当且仅当ctg^2a=3tan^2a,即tana=±3^(-1/4)时,等号成立csc,ctg 我们还没有学,能不能用其他方法sin^2a+cos^2a=1 这个学了吗?学了左边=(sin^2a+cos^2a)/sin^2a+3(s...
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左边=csc^2a+3sec^2a
=1+ctg^2a+3+3tan^2a
=4+(ctg^2a+3tan^2a)
>=4+2根号3
当且仅当ctg^2a=3tan^2a,即tana=±3^(-1/4)时,等号成立
收起
用柯西不等式,简单。
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