已知α,β∈(0,π),且cosα=4/5,sin(α+β)=-5/13,则sinβ=

Question

已知α,β∈(0,π),且cosα=4/5,sin(α+β)=-5/13,则sinβ=

Answer 1

∵α∈(0,π),且cosα=4/5>0
∴α∈(0,π/2)
sinα=√[1-(cosα)^2]=3/5
∵β∈(0,π)
∴α+β∈(0,3π/2)
∵sin(α+β)=-5/13cos(α+β)=-√[1-(sin(α+β))^2]=-12/13
故sinβ=sin[(α+β)-α]
=sin(α+β)cosα-cos(α+β)sinα
=(-5/13)(4/5)-(-12/13)(3/5)
=16/65.