I mean, I get that if I have, say:

  [0 1 1 1 0 1 0 0]
  [1 0 0 0 1 0 1 1]

that these are perpendicular to each other, which I will easily call ninety degrees, and that two such collinear vectors are at zero degrees.

But I somehow wouldn't go from that intuition into specific cosines. Like "Oh, look, if I divide out the lengths from the dot product, I'm getting 0.5! Why that's the cosine of 60 degrees!"