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.

Both tangent and cotangent's graph direction is due to their asymptotes. Asymptotes lie on the x-axis where x=0. Finding the direction of the graph can be determined by the trig ratio and unit circle.

*color stands for each quadrant in order from 1-4*
(template taken from Unit T Intro Desmo Activity & drawn on by self) 

For tangent, the ratio is y/x. To find the asymptote, x would have to be 0 to be undefined. So when x is 0, we find that the asymptotes lie at (0,1) and (0,-1) which in radians would be pi/2 and 3pi/2. Knowing the asymptotes, we can start on the direction of the graph. Back to the Unit Circle, we know that tangent is only positive in the first and third quadrant and negative in the second and last quadrant. In between the asympotes that we found are the second and third quadrant. Since we know that it's negative in the second, then positive in the third, it goes uphill. The graph will eventually start to repeat itself as the period goes on.

However, for a cotangent graph, it goes down hill. The ratio of cotangent is x/y. Again, asymptotes have a denominator of 0 to make it undefined so we have (1,0) and (-1,0). In radians, it's 0pi and pi. So on our x-axis, these points would be where the asymptote lines are. We know that cotangent is just like tangent in it's positive and negative location. Quandrant I and 3 are positive while quadrant 2 and 4 are negative. We see that the asymptote boundaries are in the first and second quadrant. So it will go from a positive position, down to a negative in the second quadrant. This is how we get our downhill shape.