BQ#4 – Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” Cotangent graph downhill? Use unit circle ratios to explain.
Asymptotes play such a large part in the reason why "normal" tangent graphs go up and why "normal" cotangent graphs go down.
Tangent, as discussed before, has the ratio of y/x or sine/cosine. With this knowledge, we know that whenever our x-value/cosine is 0, we will have an asymptote. Well, if we think about it, at pi/2 (0,1) and 3pi/2 (0,-1), our x-value is 0. These are where are asymptotes lie. Now, thinking back to our unit circle, we know that from pi/2 to pi, we go into Quadrant II, where only sine is positive--in other words, tangent is negative. So, in our first half of the space between our two asymptotes, our values are negative. Moving on, we know that pi to 3pi/2 in the unit circle is Quadrant III where tangent is positive. So, our other half of the space between our two asymptotes is going to be positive. Looking at the entire picture, we would see that our tangent graph begins at the bottom and goes up, making it uphill!
Cotangent has the same idea. Its ratio is x/y or cosine/sine. We want sine/our y-value to be 0 in order to find our asymptotes. So, the only places in the unit circle where we have 0 as our y-value are at 0 radians (1,0) and pi (-1,0). Therefore, our asymptotes are at 0 and pi. Again--same idea. From 0 to pi/2, we have Quadrant I where all trig functions are positive; cotangent is positive. However, from pi/2 to pi, we have only sine being positive so cotangent is negative. Guess what? That is what we are going to be seeing on our cotangent graph. Our first half of the space between the asymptotes is basically Quadrant I, so our graph will have positive values. However, our second half of the period is in Quadrant II where we established the fact that cotangent is negative. Our graph will go down. So, looking at the entire graph as a whole, we see that our graph starts from top to bottom, going downhill.