Question

求f(x)=[1/(sinα)^2]+[9/(cosα)^2]的值域

Answer 1

1=sin^2(α)+cos^2(α)

f(x)=[sin^2(α)+cos^2(α)]/sin^2(a)=[9sin^2(α)+9cos^2(α)]/cos^2(α)

=1+[cos^2(α)/sin^2(α)]+9+9sin^2(α)/cos^2(α)

=10+[cos^2(α)/sin^2(α)]+[9sin^2(α)/cos^2(α)]≥10+2√[cos^2(α)/sin^2(α)]·[9sin^2(α)/cos^2(α)]

=10+2√9

=16

值域为：[16,+∞)