Sine and Cosine do not have asymptotes due to that fact that they simply move left and right and get smaller and bigger. The other four trig functions have asymptotes because they rely on the asymptotes to mark where they can start and end with their lines. This is because they move up and down, they also move left and right.
But most important ly is due to that fact that Sine and Cosine become undefined and thus create an asymptote when they are being used with the four other trig graphs.
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