Alright, so a big reason for this would be our asymptotes. As we know, the trigonometric functions are tan(x) = sin(x) / cos(x), and cot(x) = cos(x) / sin(x). As we know, tangent has asymptotes whenever cos(x) = 0, and cotangent has asymptotes when sin(x) = 0. To make this simpler, we know that cos(x) = x / r, and sin(x) = y / r. This means that both x and y have to be 0. For tangent, we know the asymptotes are located at 0 pi, 1 pi, and 2 pi, while cotangent has asymptotes at pi / 2 and 3 pi /2. The reason that tangent goes uphill, however, is because in quadrant two, where the graph begins, is because cosine is negative. This places our first 1/2 of the graph in the negative territory. Tangent goes into the positive section in quadrant three, because both the x and y values are negative in the third quadrant, meaning that you divide two negatives, which gives you a positive. The opposite can be said for cotangent. Since our first asymptote for cotangent is on 0 pi, our graph starts in quadrant one. Since both cosine and sine are positive in quadrant one, we start out in the positive section. As we go into the second quadrant, only sine is positive while cosine is negative, which means that cotangent is negative. Then, we run into our asymptote. As we restart our graph in quadrant three, both cosine and sine are negative, giving us a positive graph. In quadrant four, only cosine is positive while sine is negative, making cotangent negative.
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