Well, I was thinking that he's writing out the equations for the velocities as a function of x. Then in so doing, he sees the (1-x)/(1+x) thing, and he's all, "Oh hey! That's just a form of cosine under a certain domain. So he replaces that expression with cos(theta), as a simplification.

That's not it, though.

I plotted (1-x)/(1+x) to see what it looks like, and it's nothing like cos(x), aside from the fact that it has a value of 1 when x = 0.
The first thing that made me skeptical was noticing that the (1+x) in the denominator makes it divide by 0 when x = -1, which is not cosiney at all.
It doesn't have a slope of 0 when x = 0, which pretty much makes it a bad approximation of cos, even when |x| << 1.


A vector encodes 2 pieces of information, its direction and magnitude. It cannot be defined with only 1 datum (except for the 0-vector, which has ambiguous direction).