We know that the ratio for tangents are y/x which is sine/cosine. We also know that sine and cosine have all values so if cosine were to be zero than that would make the ration undefined! Hmm, what was that thing about undefined again? OH, right! Undefined means there is an asymptote! Here is where we look at the graph and decide where tangent will be negative or positive. According to the unit circle, we already know that tangent is positive in the first quadrant which but negative in the second! SO when looking at a graph, we see the tangent line high above the x-axis in range of 0 to pi/2 but as soon as we go pi/2 to pi, what happens? it becomes negative and goes downhill but because it starts off as negative (by the asymptote) and then reverses to positive, a tangent graph is normally uphill!. This is because it is just another presentation of the Unit circle!
Cotangent is much like its reciprocal but has its own qualifications. Its ratio is x/y- or cosine/sine. To find our asymptotes now we want SINE to be equal to zero. According to the Unit Circle sine has asymptotes at zero and pi! Then we continue with the same steps as for tangent and decide in which quadrants is cotangent positive and negative in: positive in one and three but negative in two and four! The Asymptote really gives us where the graph really begins. And because we can see that the cotangent graph starts out as a positive above the x-axis but then transcends into the negative, it is called a downhill!
(Mrs. K's unit T: Exploration Packet)
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