Cheap algorithm to find measure of angle between vectors

Question

Finding the angle between two vectors is not hard using the cosine rule. However, because I am programming for a platform with very limited resources, I would like to avoid

Answer 1

dot product of two vectors (x1, y1) and (x2, y2) is

x1 * x2 + y1 * y2 

and is equivilent to the product of the lengths of the two vectors times the cosine of the angle between them.

So if you normalize the two vectors first (divide the coordinates by the length)

Where length of V1 L1 = sqrt(x1^2 + y1^2),  
  and length of V2 L2 = sqrt(x2^2 + y2^2),

Then normalized vectors are

(x1/L1, y1/L1),  and (x2/L2, y2/L2),  

And dot product of normalized vectors (which is the same as the cosine of angle between the vectors) would be

 (x1*x2 + y1*y2)
 -----------------
     (L1*L2)

of course this may be just as computationally difficult as calculating the cosine