Why do sine and cosine not have asympotes, but the other four trig graphs do?

Sine and cosine will never have asymptotes because of their trig ratios. The ratio for sin= y/r
and the ratio for cos= x/r. According to the unit circle, r will always equal 1. Asymptotes exist when it is undefined, meaning it has a 0 in the denominator. So this means that sine and cosine will always be over 1, making it impossible for them to have an asymptotes. Tangent,cotangent,cosecant, and secant can have asymptotes due to their trig ratios. Their ratios are as follows:
 tan= y/x
cot= x/y
csc= r/y
sec= r/x
Because they have an x or a y value in their denominator this opens up the possibility to be undefined and therefore have an asymptote.  For example, tangent and secant will have an asymptote wherever the x value equals zero, which is at 90 degrees and 270 degrees. Cotangent and cosecant will have asymptotes where the y value is zero so this is at 0 and 180 degrees.The image below displays the trig functions according to the unit circle.

picture:  http://htmartin.myweb.uga.edu/6190/resources/unitcircletrig.gif