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According to the unit circle, tangent is positive in the first and third quadrants which is why on the graph, tangent is above the x axis (positive) from 0 to pi/2 and pi to 3pi/2. Also, because the asymptotes for tangent are pi/2 and 3pi/2, it can explain why tangent is going towards those asymptotes meaning that because tangent stops being positive at the end of the first quadrant, it goes toward the asymptote to finish the cycle, and then starts going negative to start the second quadrant and so on. Tangent is in red and since red is going uphill, along with this explanation, that is why a normal tangent graph is uphill.
According to the unit circle, cotangent is positive in the first and third quadrants which is why on the graph, cotangent is above the x axis (positive) from 0 to pi/2 and pi to 3pi/2. Also, because the asymptotes for cotangent are 0 and pi, it can explain why cotangent is going towards those asymptotes, meaning that because cotangent stops being positive at the end of the first quadrant, it goes toward the asymptote to finish the cycle, and then starts going negative to start the second quadrant and so on. Cotangent is in blue and since blue is going downhill, along with this explanation, that is why a normal cotangent graph is downhill. The reason why it is downhill is also because it is the inverse of tangent so the direction is almost like a mirror image going in a different direction since they are reciprocals of each other.
Source:http://www.cartage.org.lb/en/themes/Sciences/Mathematics/Trigonometry/differentfund/different%20Functions%20.html
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