Cosecant- relates to sine because it is the reciprocal of sine and to graph cosecant you first need to graph for sine and find the asymptotes (which are at undefined). So if we look at cosecant as 1/sinx and we know that sine is 0 at 0 and 180 degrees, we figure out that asymptotes are located there. Remember 1/0 is undefined, hence placing an asymptote there.
Secant- related to cosine because it is the reciprocal: 1/cosx and just like cosecant needs sine, you need to graph cosine and find the asymptotes before graphing secant. From the graphing of the asymptotes you'll notice that asymptotes are located where cosine is 0.
1/cosx= 1/0 at 90 degrees and 270 degrees.
Remember that tangent is sin/cos and cotangent is cos/sin.
So for tangent, whenever cosine is 0 it will be undefined (asymptote) [at 90 & 270 degrees]
As for cotangent, where sin is 0 it is an asymptote. [at 0 & 180 degrees]
Tan/cot graph:
http://etc.usf.edu/clipart/36700/36731/tancotan_36731.htm
Csc/sec graph:
http://www.wmueller.com/precalculus/families/othertrig.html
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