题目

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）=?

提问时间：2020-11-03

答案

∵α,β∈（π/3,5π/6)
==>π/2<α+π/6<π,-π/2<β-5π/6<0
∴cos(α+π/6)<0,sin(β-5π/6)<0
∵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.