Sine and cosine will never have asymptotes. Sine does not have asymptotes because sine is equal to y/r. We know that "r" can never equal zero and in order for us to have asymptotes we need to have zero in the denominator because asymptotes only have when something is undefined. Cosine is equal to y/r and since we know that "r" can't be equal to zero, it can't have any asymptotes.
Tangent and Cotangent have asymptotes because tangent equals y/x and cotangent equals x/y. Tangent and cotangent can have asymptotes because based on the unit circle, there are some angles there where x is zero or y is zero. Remember, asymptotes are formed when we have an undefined value or any number divided by zero. We can have asymptotes in tangent and cotangent because the quadrant angles have zeros in their ordered pairs and those zeroes give us asymptotes.
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