How do the graphs of sine and cosine relate to each of the others?

Tangent and cotangent have asymptotes where their respective ratios are equal to zero. We know that Tangent=sine/cosine. So when cosine is 0, tangent will have asymptotes. So this means that tangents asymptotes will be at pi/2 (90 degrees) and 3pi/2 (270 degrees). These are the points where cosine, or the x value, is equal to zero. This can also be thought of as cosecant is the reciprocal of sine.  Cosecant is 1/sinx and we know based on the unit circle that sine is 0 at 0 and 180 degrees, so its asymptotes lie there. Whenever we have a zero in the denominator, it is undefined, and undefined means asymptote!  Secant is also the reciprocal of cosine  and its ratio is 1/cosx. Cosine equals 0 at 90 degrees and 270 degrees, so its asymptotes are at pi/2 and and 3pi/2. We know that cotangent=cosine/sine so when sine is 0, cotangent will have asymptotes. So cotangents asymptotes are at 0 and pi(180 degrees), the points where sine is equal to zero. The images below show that cotangents asymptote is in the same place at pi for cosecant.
photos:  http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg94.gif
http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg2.gif