It's a subset of the plane, so you can visualize it or draw a picture. What does it look like?
If f : [0,1] -> R2 defined by f(t) = (cost, sint) then the map is continuous and the closed interval [0,1] is connected. So the image of f should be connected.and i think the image of f is exactly the unit circle centerd at origin. is it right?
but is there any general method for proving or disproving any subset of R2 connected?