Sine and cosine do not have asymptotes because their ratios are y/r and x/r. Relating to the unit circle, r is a constant that always equals one. For there to be an asymptote, the trig ratio should result in a undefined but since r is always equal to one in the unit circle, that means that sine and cosine can't have asymptotes since its trig ratio never results in undefined. As for the four other trig graphs, there are asymptotes because for cosecant and secant, r is not a denominator so the possibility of having an undefined makes it possible for there to be a asymptote. For tangent and cotangent, there is no r, it's just y/x or x/y so having a denominator of 0 is possible so a undefined solution can result in asymptotes in the graphing. Therefore, only sine and cosine do not have asymptotes while the other four trig graphs do.
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