Here is a graph comparing sine and cosecant.

The wave that is in red is y=sin(x), while the one in purple is y=csc(x). The relation between sine and cosecant is that cosecant is 1/sine. There reason why the graph of cosecant is not connected to each other is because there are times, where sin(x)=0, thus turning the csc(x)=1/0. But that is considered to be undefined. So, we do not plot those certain points. When sin(x)=1, the csc(x) is also 1. that is why they touch at those points. As the value of sin(x) gets closer to 0, the y-coordinate of the graph y=csc(x) increases very rapidly, since csc(x)=1/sin(x).

Here is a graph comparing Cosine and Secant.

The graph in blue is represented by y=cos(x), while the one in green is represented by y=sec(x). Pretty much the explanation is just the same except that the graph shifted. We know that cos((π /2)-x)=sin(x), thus the reason why the graph seems to be shifted. We also know that sin((π /2)-x)=cos(x). Therefore, we can say that csc((π /2)-x)=sec(x) and sec((π /2)-x)=csc(x). That’s also why the graph of cosecant seems to be a shifted secant and vice versa.

Lastly, Here is a graph comparing tangent and cotangent.

From the past lessons, we learned that cotangent is equal to 1/tangent. We also know that tan(0)=0, and since cotangent is equal to 1/tangent it turns to 1/0, again that is undefined making it not possible to graph that point. the graph of cotangent seems to be a mirror image of the graph of tangent but it also seems to be shifted.