My current understanding of some aspects of trigonometry:

Similar triangles will have the same theta and same sine/cosine. Since similar triangles have sides that are the same ratio and the same angles, theta ought to be the same. I'm not exactly sure what's relevant about this fact, but I imagine it is useful.

In a unit circle (in which the origin is (0, 0) and radius = 1, draw a line anywhere from the center to the circumference. By definition, this line = 1. Place a point anywhere on the x-axis. > 0 and < 1 (just because I want to stick with positive values, I know it holds for negatives). We will call this point E.

The corresponding y value will be the point above point x that intersects the circumference of the circle. We will call this point F. Draw a line whose points are EF. This is a right angle to the x-axis.

Then, from the center of the circle (at the origin) draw a line to F. We will call this line OF. By definition, OF has length 1. We now have a right triangle with sides: hypotenuse OF, opposite EF, and adjacent OE.

We get all kinds of cool stuff with this. We realize that we can do this with any line drawn from the radius to the circumference of the circle. When you start messing around with the x axis, you notice that cosine increases as x increases/sine decreases. You learn that as y increases, sine increases and cosine decreases. You notice that as theta increases, sine increases/cosine decreases. As theta decreases, sine decreases/cosine decreases.

Onto Intuitions:

First, I want to make it clear that I'm interested in this from a pure-mathematics perspective. Meaning, I don't care if these facts make it easy to make bridges, describe ladders, or calculate angles and distances of physical objects. I'm curious what these facts tell us about triangles, sine, cosine, similar triangles, circles, etc. I'm curious about the "beauty" of what these things are representing and telling us.

What do these trigonometric functions do? What innate mathematical concepts do they reveal? What exactly are the trigonometric functions? What math utility do they have for other math fields? What exactly is this overall relationship between the unit circle, the angle of theta, the increasing and decreasing of cosine/sine relative to the size of x and y?

It feels like I'm missing an intuitive understanding of the relationship between all of these things and a pattern I'm not seeing. Trigonometry is beautiful, and I would like to understand it much more deeply.

Note: if I made any mistakes with variable names or just minor errors, I wouldn't be surprised. Trying to type this from my phone. I'm concerned with the conceptual stuff though. Any corrections would be appreciated.