Image found here.
Image found here. (thank you Google!)
Sine and cosine do not have asymptotes because they are always divided by 1 (by "r")
Yet once we move away from being divided by "r" we start reaching asymptote area.
If you reference to the photos you will notice undefined under tangent. Do not forget about their reciprocal ratios of cosecant (r/y), secant (r/x) and tangent (x/y).
Remember that we get asymptotes when our circle ratios equal undefined which results when you divide by 0.
For sine and cosine you will never be dividing by 0 as you may for all other four circle ratios. But because you'll never be dividing by 0, you will never reach undefined, so you'll never have asymptotes!