Sin and cos are related to tan by looking at our trig values. Since Tan=Sin/cos or y/x By looking at our value of cosine it never touches our origin which is zero because it would cause the value to become an asymptote and become undefined. However our tan graph is going by certain amounts in which our asymptotes can act like a wall which can guide and lead the tan graph to go to that certain amount. Cosine never intersects with any of the other graphs.
Cotangent has the value of cos/sin or x/y. We can see that sin goes from 0 and touches every one value of pi going to pi 2pi 3pi etc. We can see our cot starts from the positive and transcends into negative within 0 and pi and continues which is our wall that restrains the graph and keeps it contained. Tan and cot are mirrors of the direction of the graph going up or down.
Secant has the value of r/x or r/cos. When cos is 0 it lies on pi/2 3pi/2 and 5pi/2. In these areas sec is split by parabolas and are split by the asymptotes into their own sections. Starting by going up then down and repeats the pattern of the direction of sec. Sin and cos act as a measurement between the graphs of secant.
Cosecant has the value of r/y or r/sin. The asymptotes lie on every pi such as 2pi 3pi etcwhich results in seperate graphs according to csc. Sin is 0 on the values of pi which is 1/0 which is an asymptote because its undefined. The pattern repeats of up and down.