Sine and cosine do not have asymptotes because their ratios are y/r and x/r, meaning that the radius will ALWAYS be the denominator. You will never have a 0 as your denominator. Because of this, we will never encounter an undefined answer, meaning we will not encounter any asymptotes (since asymptotes = undefined, basically).
However, that is a different case for cosecant, secant, cotangent, and tangent. Cosecant has a ratio of r/y. The "y" value can be 0 in certain cases: if we are at (1,0) or (-1,0). Same for secant; secant has a ratio of r/x and x can be 0 in some cases: (0,1) and (0, -1). Cotangent and tangent have the ratios of x/y and y/x so, again, of course you will have some y values and x values that equal to 0.
Also, a good thing to nice is that cotangent and and cosecant have the same denominator in their ratios: y. Therefore, they will have the same asymptotes. Tangent and secant will have the same asymptotes as well because their denominators in their ratios are x.
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