Question

已知cos(a-π/6)+sinα=4√3/5,则sin(α-7π/6)的值是

选择题，我算的答案是 -（4+3√3）/10和-（4-3√3）/10，但选项里没有
求详解
谢谢

Answer 1

cos(a-π/6)+sina=4√3/5
cosacosπ/6+sinasinπ/6+sina=4√3/5
√3/2*cosa+3/2*sina=4√3/5
1/2*cosa+√3/2*sina=4/5
∴sin(a+π/6)=4/5
∴sin(a+7π/6)=sin(π+a+π/6)
=-sin(a+π/6)
=-4/5

Answer 2

cos(a-π/6)+sina=4√3/5

=>sin(a+π/6)=4/5

∴cos(a+π/6)=±3/5
sin(a-7π/6+π/6-π/6)
=sin(a+π/6-4π/3)
=sin(a+π/6)cos(4π/3)-cos(a+π/6)sin(4π/3)
=-sin(a+π/6)cos(π/3)+cos(a+π/6)sin(π/3)
=-4/10±3√3/10
=(-4±3√3)/10