Sin and cos don’t have asymptotes but other trig functions do because cos is x/r which is x/1 on the unit circle. If r was 0 then it would be undefined but in this case r can't be 0, it is always 1. This is the same thing for sin which is y/r or y/1 and why they don’t have asymptotes. Sec and csc in the other hand do have asymptotes because sec is 1/cos and whenever cosine is 0 then it is undefined which makes it an asymptote. Csc is 1/sin and whenever sin is 0 the answer would also have an asymptote. For tan it equals sin/cos and whenever cos is 0 then the answer would be undefined which will have an asymptote. Cot it is cos/sin and whenever sin is 0 it will also be an asymptote. This is why sin and cos aren’t asymptotes but the other trig functions are.
https://wiki.eee.uci.edu/images/6/6e/Triangle_on_unit_circle.jpg
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