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Monday, April 21, 2014

BQ#4: Unit T Concept 3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?
ASYMPTOTES. They are the reasons why tangent goes uphill and cotangent goes downhill. As we previously learned, tangent is sine/cosine and the only way to make an asymptote is for it to be undefined. This means cosine must be zero. We know that cosine equals zero at 90º (0,1) and 270º (0,-1). So, on a graph, the asymptote starts at π/2 and ends at 3π/2. Remember, these aren't all the asymptotes, they are just one period. In the picture below, you can see that the asymptotes are the dotted lines. Quadrants two and three consists of one whole period and quadrants one and four are just the halves because it's a snapshot.
Contrasting with the tangent graph, cotangent's asymptotes are placed in a different spot, resulting in a downhill graph. It's ratio is cosine/sine, which means sine must be undefined for it to have an asymptote. Sine is zero at 0º/360º (1,0) and 180º (-1,0), so that is where the asymptotes are placed to make one period (picture below). The cycle is positive, negative and positive,negative. They repeat twice, so there are two periods only in the picture.
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