BQ #4: Unit T Concept 3: Why is a “normal” tangent graph uphill, but a “normal” COtangent graph downhill?

Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill?

The pattern of tangent on the Unit Circle is positive, negative, positive, and negative. From this patter, we know that the period repeats itself within the Unit Circle, or within 2pi. The ratio of tangent is sine/cosine, which means that asymptotes can exist. An asymptote exists when the denominator is 0; and in this case, cosine would need to equal 0. We know that on the Unit Circle, cosine is 0 at 90 degrees (or pi/2) and 270 degrees (or 3pi/2). So within the asymototes, a period would start from the negative and move onto the positive, creating an uphill graph. (Look at the picture below for a visual description)

Cotangent is somewhat similar to tangent. It also has the same patter of positive and negative, however its ratio is cosine/sine. As sine as its denominator, the asymptotes would be located differently. On the Unit Circle, sine is 0 at 0 degrees (0pi) and at 180 degrees (pi). Between the asymptotes, the graph would start with positive and move onto negative, creating a downhill graph.

Overall, the tangent and cotangent graphs depend on where the asymptotes are located at.