Fundamental group of the circle S^1

The question is to prove that the fundamental group of the circle S^1 is isomorphic to the group of integers under addition.

So I think I should show that the following map Phi is an isomorphism.

Phi: F(S^1, (1,0)) --> Z defined by Phi([f])= f*(1) where f* is the lifting path of f ( pof*=f) and f*(1) is the degree of f and p is the map

p:Reals--> S^1 defined by p(t)=(cos 2 pi t, sin 2 pi t).

I am able to show that Phi is onto, but I am having trouble showing that it is 1-1 and that it is well defined .