已知α,β∈（π/3,5π/6),若sin（α+π/6）=4/5,cos（β-5π/6）=5/13,则sin（α-β）=?
问题描述：
已知α,β∈（π/3,5π/6),若sin（α+π/6）=4/5,cos（β-5π/6）=5/13,则sin（α-β）=?
答
∵α,β∈（π/3,5π/6)
==>π/2 ∴cos(α+π/6) ∵sin(α+π/6)=4/5,cos(β-5π/6)=5/13
∴cos(α+π/6)=-√[1-(sin(α+π/6))^2]=-3/5
sin(β-5π/6)=-√[1-(cos(β-5π/6))^2]=-12/13
故sin(α-β)=-sin(π+α-β)(应用诱导公式)
=-sin[(α+π/6)-(β-5π/6)]
=cos(α+π/6)sin(β-5π/6)-sin(α+π/6)cos(β-5π/6)(应用和角公式)
=(-3/5)(-12/13)-(4/5)(5/13)
=16/65.