1). How do the graphs of sine and cosine relate to the others? (With an emphasis on asymptotes)

Well then, the very first thing you need to answer this is, what do ya know, the graphs. Who would've thought that, right? Anyhow, here goes nothing...

a). Tangent in relation to sine and cosine

• Here we see a graph of sine, cosine, and tangent (black) as well as some asymptotes (various shaded regions). Focus on the light green and orange regions. It is in these two regions that tangent is a full period, therefore we can observe the relationship it has according to its asymptotes through these regions. We can see that asymptotes occur wherever cosine (green) has a point on the x-axis (in other words a value of zero.)  This corresponding relationship can be explained in the trig ratio: tan = sin/cos. Since we know that we get asymptotes by dividing by zero, using this trig ratio we know that there will be an asymptote whenever cosine, the denominator, is zero.

b). Cotangent in relation to sine and cosine

• Here is the same graph with cotangent instead of tangent. We can apply the same logic to this one as we did above. Since cot = cos/sin, there will be an asymptote wherever the sine value is zero.

c). Secant in relation to cosine

• Shown here is cosine (middle and green) and secant (top/bottom and blue). Since secant is the reciprocal of cosine, it's relationship to the cosine graph is affected in various ways. The most obvious is the x-axis reflection of the peaks and valleys of the cosine graph. However, what we are looking for is how this relationship causes the asymptotes (black dotted lines). The reason asymptotes are there is once again because of a trig ratio, this time being that sec = 1/cos, and like before, since cosine is the denominator, any place where the cosine value is zero we will have an asymptote.

d). Cosecant in relation to sine

• This one is just like the secant and cosine graphs. The only difference is that the ratio applied is csc  = 1/sin and therefore wherever the sine value is equal to zero on the x- axis then there will be an asymptote.