Sine and cosine does not have an asymptotes because based on their ratio which is y/r and x/r. The r always have to equal to 1. So both denominator for sine and cosine equal to 1. In order to have an asymptote the denominator have to equal to 0.
However, for cosecant, secant, cotangent, and tangent they have to same ratio. For instance, like cosecant the ratio is r/y. That means the denominator has to be 0 for it to have an asymptote. While for secant which has a ratio of r/x, the value for x have to be 0. That will give us an undefined answer, and when something is undefined = asymptote.