Asymptotes are where the graphs will equal undefined. In the case of sine and cosine, they both have "r" as a denominator (Please see chart above). And, since it is a unit circle, r=1. This makes it impossible for sine and cosine to be undefined, which explains why they do not have asymptotes.
Moreover, cosecant and cotangent have similar asymptotes because they both have the denominator of "y". The same appies to secant and tangent, which have the same denominator, "x".
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