Why do sine and cosine NOT have asymptotes, but the other four trig graphs do?
To have asymptotes your trig function has to be undefined at some point which is when the denominator equals 0.
Sine and Cosine don't have asymptotes because when you take a right triangle from the unit circle starting at (0,0) sine and cosine have the denominator of r which is the radius of the unit circle. The radius is always 1. Since the radius can never be zero, sine and cosine are never undefined.
All other trig functions have asymptotes because their denominators can equal 0 and thus make the trig function undefined. Also with the same variable denominators like tan and csc, they have the same asymptotes!