α∈(0,π/2)
α+π/6∈(π/6,2π/3)
cos(α+π/6)=4/5>0
∴α+π/6∈(π/6,π/2)
∴2α+π/3∈(π/3,π)
cos(a+π/6)=4/5,
sin(a+π/6)=3/5
sin(2a+π/3)=2sin(a+π/6)cos(a+π/6)=2*(4/5)*(3/5)=24/25
cos(2a+π/3)=2cos²(a+π/6)-1=2*(4/5)²-1=7/25
sin(2a+π/12)
=sin[(2a+π/3)+π/4]
=sin(2α+π/3)cosπ/4+cos(2α+π/3)sinπ/4
=(24/25)×(√2/2)+(7/25)×(√2/2)
=31√2/50