Well, sine and cosine don't have asymptotes but the rest of the trig functions do have asymptotes because at certain marks on the graph, sine can be 0 and cosine can be 0. These two functions relate to the others because all of the others have sine and/or cosine within them in one way or another.
a) tangent. tangent is sine/cosine. When cosine is 0, that means that the trig function is undefined so there is an asymptote. Tangent's asymptotes are at pi/2 and 3pi/2.
b) cotangent: cotangent is the inverse of tangent which means because tangent is sine/cosine, cotangent is cosine/sine. When sine is 0, there is an asymptote because of how the whole thing is made undefined at that particular moment. Tangent's asymptotes are at pi/2 and 3pi/2 so, to contrast, this means that cotangent's asymptotes are at 0 and pi. Using the graphs, notice how the asymptotes of the graph of y = tan(x) are the x-intercepts of the graph of y = cot(x). There are vertical asymptotes at each end of the cycle. The asymptote that occurs at
repeats every
units
c) secant: secant is the inverse of cosine which means that it is 1/cosine and when cosine is 0 in this case, secant becomes undefined as a whole since it is the reciprocal of cosine. Anytime we have an undefined as our answer, we automatically know that there is going to be an asymptote. There are vertical asymptotes. The asymptote that occurs at
repeats every
units
the pink is the parent graph of cosine and the brown is secant.
d) cosecant: cosecant is the reciprocal of sine making cosecant 1/sine. So, when sine is zero in this case, cosecant becomes undefined anytime sine is zero, thus resulting in an asymptote. The x-intercepts of y = sin x are the asymptotes for y = csc x. There are vertical asymptotes. The asymptote that occurs at
repeats every
units
the brown is the cosecant graph and the blue is the parent graph of sine.
Excellent Source: http://www.regentsprep.org/Regents/math/algtrig/ATT7/othergraphs.htm
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