Math Analysis: BQ#5:Unit T- Concepts 1-3

1.) Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.

Sine and cosine do NOT have asymptotes because that is how they are, they are in a wavy line when they are fully done in the graph and they run infinite on the x-axis. It refers to the unit circle because the ratio for cosine is x/r and since r is always 1 cosine can never be undefined since r can never be zero which is what makes a function undefined when the denominator is zero . The same goes for sine, in the unit circle the ratio for sine is y/r and since r is always 1 sine can never be undefined since r can never be zero, which is what is needed if you want to result in an asymptote to get that you need the denominator to be zero.

http://www.kirupa.com/developer/animation/images/cosinegraph.jpg

Cosecant, Secant, Tangent, and Cotangent all have asymptotes because of how x and y are zero in this case. Going back to the identities we know that secant is 1/cos which is, using the ratio of secant which is r/x. X can be any number and when it is zero secant automatically becomes undefined and when it is undefined we know that its going to have a asymptote. The same goes for cosecant, which is r/y, except that when y is cosecant it becomes undefined and has an asymptote. Tangent is sin/cos so when cosine is 0 it becomes undefined and makes an asymptote. The same goes for cotangent except that since cotangent is cos/sin, when sine is 0 cotangent becomes undefined and makes an asymptote as well.