已知sin(α-π/6)=4/5,0≤α≤2π/3,求sin(5π/6-α)+cos(5π/6-α)
问题描述：
已知sin(α-π/6)=4/5,0≤α≤2π/3,求sin(5π/6-α)+cos(5π/6-α)
答
∵0≤α≤2π/3,∴-π/6≤α-π/6≤π/2
∵sin(α-π/6)=4/5,∴cos(α-π/6)=3/5
∴sin(α＋π/6)=sin(α-π/6+π/3)
=sin(α-π/6)cosπ/3+cos(α-π/6)sinπ/3
=4/5×1/2+3/5×√3/2
=（4+3√3）/10
cos(α＋π/6)=cos(α-π/6+π/3)
=cos(α-π/6)cosπ/3-sin(α-π/6)sinπ/3
=3/5×1/2-4/5×√3/2
=（3-4√3）/10
又∵sin(5π/6-α)+cos(5π/6-α)
=sin（π-(α＋π/6)）+cos（π-(α＋π/6)）
=sin(α＋π/6)-cos(α＋π/6)
=（4+3√3）/10-（3-4√3）/10
=（1＋7√3）/10