recursively moving around the unit circle with constant angle to the origin

Starting with solving this general formula for two vectors [x,y] and [a,b]:

The above says the dot product between them equals to c and the magnitude of the cross product equals to d (counterclockwise). And then it is solved for a,b.
So we begin by thinking about if [x,y] starts as the vector [1,0]... then we can find possible values for c and d like so, picking any value as close or far away from 1 as we'd like for x and solving...

So plug those in for d and c respectively...Or alternatively use a rational value from the rational parameterization of the circle. These are the values for the dot and cross product from the vector to the generated point to the vector [1, 0] because the dot product will just be the x coordinate of the point we generated and the cross product will just be the y.   Let's call this Formula 1.

Now c and d is also our first vector after [1,0] so we can plug that in for x and y as well and get...

This vector [a,b] is the same angle to [.9801,.141] as [.9801,.141] was to [1,0], because we made the dot and cross products stay the same.

Now we can plug this [a,b] in for x and y in formula 1 and get...

And in this way work our way around the unit circle by that small angle (which we actually don't know) without ever having to figure a cosine or sine! An approximation of the angle can be found by seeing how long it takes for this process to get close to a known value like [0,1]