求y=1/sin^2A+4/cos^2A的最小值

求y=1/sin^2A+4/cos^2A的最小值
求y=1/sin^2A+4/cos^2A的最小值

求y=1/sin^2A+4/cos^2A的最小值
y=(sin^2A+cos^2A)/sin^2A+4(sin^2A+cos^2A)/cos^2A
=1+cos^2A/sin^2A+4(sin^2A/cos^2A)+4
=5+cos^2A/sin^2A+4(sin^2A/cos^2A)
≥5+2*2=9
当且仅当cos^2A/sin^2A=4(sin^2A/cos^2A),即tanA=±1/√2时
y取得最小值9