BQ#3 Unit T Concepts 1-3

2) How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.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 or cosine within them in one way or another.  

a) Tangent 
tangent is sine/cosine so when cosine is 0 that means that the trig function is undefined so there is an asymptote. The Tangent's asymptotes are at pi/2 and 3pi/2.

http://mythi-trig.fateback.com/images/basicall.gif

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 the whole thing is going to be undefined at that 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 you 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 pi repeats every pi units.

http://image.tutorvista.com/content/feed/u1593/cotangent.GIF

c) Secant 
secant is the inverse of cosine which means that it is 1/cosine and when cosine is 0, 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 pi/2 repeats every pi units.

http://www.analyzemath.com/trigonometry/graph_cosecant.gif

d) Cosecant 
cosecant is the reciprocal of sine making cosecant 1/sine so when sine is zero, cosecant becomes undefined anytime sine is zero 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 pi repeats every pi units.

http://www.scs.sk.ca/cyber/elem/learningcommunity/math/math30c/curr_content/mathc30/GRAPHICS/RECIPROCAL_GRAPHS/MOVIE8.GIF                                                                                                  WORKS CITED PAGE
TANGENT
COTANGENT
SECANT
COSECANT