The fact that the dot product and the cosine of the angle between two vectors are mutually computable is easy to show (see the two sides in the two answers at Dot product in coordinates).
But looking at the dot product, I would never have thought that it somehow captures something about the angle (and vice versa).
How did the connection get discovered? Who were the major players? Did it just fall out of the development of matrix operations for linear algebra (or did the dot product come first) or are these only related by hindsight or what?