Unit T Big Question #4

Why does sin and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain?

Asymptote: A line that approaches the given curve but makes no contact with it. 

What is this concept about?

This concept explains why the cosine and sin do not have asymptotes and the other trig ratios do. We will be using ratio reciprocals and ratio identities from Unit Q to explain why trig ratios have asymptotes.

What the viewer needs to pay close attention to in order to understand the concept?

The viewer needs to pay attention to how each distinct place  is undefined on the graph, in other words where the asymptotes are placed when graphing. They also need to pay close attention to whether it is being drawn positive or negative in there period. Pay attention the explanation below on why sine and cosine don't have asymptotes.

Sine/Cosine

Sine does not have an asymptote because the ratio is y/r, R always equals one on the unit circle, therefore it can never be undefined. Same goes with Cosine it's ratio is x/r, the R is always one, so its not undefined. What both of these do have is an amplitude of one which was explained in Big Question #1 part B.

The pictures below explain how ratio identities and reciprocal identities are involved. It as well shows drawings of the asymptotes. 

Tangent

Tangent has a ratio identities of sin/cos, so whenever cosine equals zero the tangent is undefined leading it to have an asymptote. The asymptote on the graph is located wherever cosine equals 0 on the unit circle, in this case pi/2(90deg.) and 3pi/2(270deg.). 

   

Cotangent

Cotangent has the ratio identities of cos/sin, so wherever sine equals 0 the cotangent is undefined leading it to have an asymptote. The asymptote is located where the sin equals 0 , in this case at 0(0deg.) and pi(180deg.). 

Secant

Secant is the reciprocal of cosine, this is known from the reciprocal identities sec=1/cos. So similar to tangent whenever cosine equals 0 it is undefined leading it to an asymptote. The asymptote on the graph is located wherever cosine equals 0 on the unit circle, in this case pi/2(90deg.) and 3pi/2(270deg.). 

Cosecant

Cosecant is the reciprocal of sine and we know this by the reciprocal identities of csc=1/sin. Similar to cotangent whenever sin equals zero is where the asymptote is located.  The asymptote is located where the sin equals 0 , in this case at 0(0deg.) and pi(180deg.).