a为锐角 cos(a+π/6)=4/5 求sin(2a+π/12)

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a为锐角 cos(a+π/6)=4/5 求sin(2a+π/12)

a为锐角 cos(a+π/6)=4/5 求sin(2a+π/12)
a为锐角 cos(a+π/6)=4/5 求sin(2a+π/12)

a为锐角 cos(a+π/6)=4/5 求sin(2a+π/12)
∵0<a<π/2,∴π/6<a+π/6<π/2+π/6<π,又cos(a+π/6)=4/5>0,∴a+π/6为锐角,
∴sin(a+π/6)=√{1-[cos(a+π/6)]^2}=√(1-16/25)=3/5.
∴sin(2a+π/12)
=sin[(2a+π/3)-π/3+π/12]=sin[(2a+π/3)-π/4]
=sin(2a+π/3)cos(π/4)-cos(2a+π/3)sin(π/4)
=(√2/2)×2sin(a+/6)cos(a+π/6)-(√2/2)×{2[cos(a+π/6)]^2-1}
=√2×(3/5)×(4/5)-(√2/2)[2×(4/5)^2-1]
=12√2/25-(√2/2)×(7/25)=24√2/50-7√2/50=17√2/50.