Sine and cosine graphs do not have asymptotes. The reason being is that when looking back at Unit Circle ratios, sine is y/r and cosine is x/r. "r" was always one no matter what and for an asymptote to occur, the answer must be undefined. However because "r" is always one and never zero, sine and cosine graphs do not have asymptotes.
In cosecant and cotangent graphs, the respective ratios are 1/sine and cosine/sine. Their graphs have asymptotes because sine can equal to zero at 0 and pi. Because sine can be zero, when using the ratios, we can never divide by zero because it leads to undefined answers. This is the reason why cosecant and tangent graphs have asymptotes.
In secant and tangent graphs, the respective ratios are 1/cosine and sine/cosine. Their graphs have asymptotes because cosine can equal to zero at 90 (pi/2) and 270 (3pi/2). Because cosine can be zero, when using the ratios, we can never divide by a zero because that is undefined. Because it is undefined, that means there is an asymptote, which is why those graphs have asymptotes.
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