cos(π/6-α)+sinα=4√3/5.
cosπ/6cosα+sinπ/6sinα+sinα=4√3/5.
(√3/2)cosα+(1/2)sinα+sinα=4√3/5.
(√3/2)cosα+(3/2)sinα=4√3/5.
√3[(1/2)cosα+(√3/2)sinα=4√3/5.
√3(sinπ/6cosα+cosπ/6sinα)=4√3/5.
√3sin(π/6+α)=4√3/5.
sin(π/6+α)=4/5.
∵sin(α+7π/6)=sin(π+π/6+α).
=sin[π+(π/6+α)].
=-sin(π/6+α).
∴sin(α+7π/6)=--4/5.