BQ#5 – Unit T Concepts 1-3

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 unlike the other four trig graphs. When we refer back to the Unit Circle and find the ratios, we see that sine has a ratio of y/r and cosine has x/r. We know that an asymptote is undefined so that would only mean the denominator has to be 0. However in this case, our denominator will always be r which will never be 0. That is why sine and cosine do not have asymptotes.

For the other trig functions, tangent, cotangent, cosecant, and secant, these have asymptotes. For tangent and cotangent, we know that it deal with x/y or y/x. It is possible that the denominators of the two can be 0 which will make it undefined and result in an asymptote. As for cosecant (r/y) and secant (r/x), we see that the radius is put as the numerator while x and y are on the denominator. Again, they can value 0 which will make them asymptotes. You can notice that when tangent and cosecant both have y as their numerator they land at (-1, 0) or (1, 0). As for cotangent and secant, they both have x for their denominator and the possible points are at (-1, 0) or (1, 0). This is why the other four trig functions are asymptotes.