设α为锐角,若cos（α+π/6）=4/5,则sin（2α+π/12）=?

Question

设α为锐角,若cos（α+π/6）=4/5,则sin（2α+π/12）=?

Answer 1

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(2α+π/3)
=2cos²(α+π/6)-1
=2*(4/5)²-1
=32/25-1
=7/25>0
∴2α+π/3∈(π/3,π/2)
sin(2α+π/3)=24/25
sin(2α+π/12)
=sin(2α+π/3-π/4)
=sin(2α+π/3)cosπ/4-cos(2α+π/3)sinπ/4
=24/25*√2/2-7/25*√2/2
=24√2/50-7√2/50
=17√2/50