Triangles are a terrible way to teach trig. Let's look at circles instead.

Trig is just a set of functions that describe how to turn Angle and Distance into Rise and Run (y-axis distance and x-axis distance. If you think about it, an angle is the same kind of thing as a slope, it's just calculated differently. And the distance just says how far that slope goes before you reach the point you're looking at.

Here, the "radius" of the circle is the Distance between its center and its edge, and the "angle" tells is where on the edge our point is.

We want to turn those into the X and Y that tells where the point is, instead. The problem is, "sine" and "cosine" only take one variable, and output one variable, don't they? The reason it works is, we can assume that the Radius equals 1, and then we can just multiply our X or Y by it, and everything works fine!

It just so happens that the "...do something to..." function for X is Cosine, and the "...do something to..." function for Y is Sine.

The reason trig works on triangles is that the line from the center of the circle makes two right triangles, and the Radius is the Hypotenuse, and X and Y are either Adjecant or Opposite, depending on where you're looking.