Can someone please explain to me why cotangent graphs look the way they do?

Question

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?

Answer 1

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 $\pi/2.$

Answer 2

$$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{\sin(\frac\pi2-\theta)}{\cos(\frac\pi2-\theta)} = \tan\Big(\frac\pi2-\theta\Big)$$

Answer 3

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.