Can someone please explain to me why cotangent graphs look the way they do? I want to know why they basically look like mirror reflected tangent graphs. I get that if $\tan\theta=y/x$, then $\cot\theta=x/y$. But why would this lead to a graph that looks like a tangent graph that was reflected over?
Can you please try to keep the answers at the level of a high school pre-calc student?
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$\begingroup$We have $$ \cot(x) = \frac{\cos(x)}{\sin(x)} = \frac{\sin(\pi/2-x)}{\cos(\pi/2-x)} = \tan(\pi/2-x)$$
so it is reflected and shifted by $\pi/2.$
$\endgroup$ 5 $\begingroup$Just as a visual complement for the answers you already have, notice the following animation.
Green angle: $\theta$, yellow angle: $\pi/2 - \theta$ and $\cot(x)$ is the red curve.
For more details regarding these types of constructions, see this question.
$\endgroup$ $\begingroup$$$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\sin(\frac\pi2-\theta)}{\cos(\frac\pi2-\theta)} = \tan\Big(\frac\pi2-\theta\Big)$$
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