Question

0<α<π/2,tanα/2+cotα/2=5/2,求sin(α-π/3)的值

Answer 1

已知:0<α<π/2,tanα/2+cotα/2=5/2,求sin(α-π/3)的值

tanα/2+cotα/2
=[sin(α/2)/cos(α/2)]/[cos(α/2)/sin(α/2)]
=1/[sin(α/2)cos(α/2)]
=2/sinα (sin2α=2sinαcosα)
由tanα/2+cotα/2=5/2,得sinα=4/5
cosα=3/5 (0<α<π/2)
sin(α-π/3)
=sinαcos(π/3)-cosαsin(π/3)
=(4-3√3)/10