As we can see from the above picture, sine and cosine have unique ratios when compared with all the other trig values. They both have a denominator of r. In the Unit Circle, r is always equal to 1. Now, the question is why don't sine and cosine have asymptotes whereas all the other ratios do?
We get an asymptote whenever we divide by zero (in other words, whenever we have a denominator of zero.) This results in undefined answers, in which an asymptote accounts for.
Sine and cosine have denominators of r which is always equal to one in this case. Therefore when using sine and cosine we will never be dividing by zero and never be getting asymptotes.
The rest of the trig functions, however, have denominators of x and y. These denominators can be any number, and when that number is zero, then you will get undefined and therefore an asymptote.
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