已知向量a=(cosα,sinα),向量b=(cosβ,sinβ),|a-b|=2根号5/5

解析：∵|a-b|=2√5/5,
∴a^2-2a.b+b^2=4/5
又a^2=│a│^2=(cosα)^2+(sinα)^2=1
b^2=│b│^2=(cosβ)^2+(sinβ)^2=1,
∴a.b=3/5
∴cos（α-β）=cosαcosβ+sinαsinβ=a.b=3/5
∵-π＜β＜0,0＜α＜π/2,
∴0＜α-β＜3π/2,且cos（α-β）=3/5＞0
则0＜α-β＜π/2,-π/2＜β＜0
sinβ=-5/13,cosβ=12/13
∴12cosa-5sina=39/5
联立(cosα)^2+(sinα)^2=1,
解得sinα=（3√46+15）/65