If p/q is a close approximation to π, then cos(p), taken in radians, will be close to +1 or -1, because it implies that p is close to an integer multiple of π, and therefore cos(p) is close to the cosine of an integer multiple of π:

   p/q =~ π 
   p =~ qπ
   cos(p) =~ cos(qπ)

In turn, the cosine of an integer multiple of π is +1 or -1 because the period of cosine (as a trigonometric function) is 2π radians, with maxima at 0, 2π, 4π, ... and minima at π, 3π, 5π, ...