sin(x+π/6)+cos(x+π/6)=(√2)/2
两边平方：1＋2sin(x+π/6)cos(x+π/6)=1/2
∴1+sin﹙2x+π/3﹚＝1/2
∴sin﹙2x+π/3﹚＝-1/2,说明2x+π/3为第三象限或者第四象限的角……①
∵-π/6＜x＜7π/12
∴-π/3＜2x＜7π/6
∴0＜2x+π/3＜3π/2……②
由①②知,2x+π/3为第三象限的角
∴由sin﹙2x+π/3﹚＝-1/2知,2x+π/3＝7π/6
∴2x＝5π/6
∴sin2x＝sin﹙5π/6﹚＝1/2