To begin the ratio for sine is y/r and for cosine its x/r. If we recall r is 1 so that means that r can never equal 0 so we can't get an asymptote. We cannot get an asymptote because in order to get an asymptote we have to get an undefined and we can only get an undefined when we divide by zero. Cosecant has a ratios of r/y and cotangent has a ratio of x/y. This means that we will get asymptotes when y=0. Also the asymptotes will be in the same place. The ratios for secant (r/x) and tangent (y/x) both end with x. So we get asymptotes when x=0. All trigs, except for sine and cosine, can be divide by zero this means that we can get an undefined and have asymptotes on the graph.
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