Sine and cosine have a ratio of (y/r) and (x/r) respectively. The variable r equals to 1 on the Unit Circle, which is the reason why sine and cosine will never reach undefined since its denominator is 1. Cosecant and secant have a ratio of (r/y) and (r/x) respectively. Moreover, their ratios are reciprocals to sine and cosine. It is possible for cosecant and secant to have an undefined answer since the denominator does not equal to one. Tangent and cotangent have a ratio of (y/x) and (x/y) respectively. None of these two trigs have the variable "r" in their ratio. Thus, there is a likely chance that they will have an undefined answer. Remember, undefined equals asymptotes. All four trigs, except for sine and cosine, can divide by zero. This causes it to be undefined and have asymptotes on the graph.
Reference:
Mrs. Kirch's SSS Packet
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