BQ#4: Unit T Concept 3: Upstair Downstairs...Graphs

Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill? Use unit circle ratios to explain.

The parent tangent and cotangent graphs are in different directions because their asymptotes are in different places. According to the Ratio Identities, tan=sin/cos and cot=cos/sin. To get an asymptote, the ratio must be undefined (in other words, the denominator has to equal 0.)

How does all that fit in? Tangent has a denominator of cosine, so the places where cosine equals 0 on the Unit Circle and the tangent graph are 90º (pi/2) and 270º (3pi/2). However, cotangent has a denominator of sin, so the places where there would be an asymptote are 0º (0), 180º (pi), and 360º (2pi). Even though the pattern of positive and negatives are the same (+ - + -), because the asymptotes are in different places, the graphs have to be draw differently in order to not touch the asymptotes and follow the rules.