I also picked up sine and cosine at around that age; it was thanks to programming.
Various example BASIC programs (e.g. Lissajous curve and circle plotters and such) used those functions and from their behavior I kind of picked up the relationship to both undulation and the circle. I found out about the arctangent function, and from that I realized that it could be used for 3D projection. I made a proof-of-concept program that mapped some connected points in 3D (boxes and whatnot) to 2D by calculating the visual angles using arctangents: i.e. I plotted the angle between the line of sight and each point, rather than a simple planar projection. It resulted in the objects appearing in a very nice "fish eye" perspective on the screen, which was very pleasing. From all that I learned about radians also and how there are "two pi" of them in a circle, since the functions accepted radians and not degrees.
You know how they say that necessity is the mother of invention; it's also the mother of learning trigonometric functions and whatever else.
Similar age for sin/cos. I wanted to move a game object in a circle that I used to develop amateur content for. I remember being wowed by light displays some scripters put together at the time (there was no particle system at the time).
By the time it came around, I was disappointed to learn about all the "triangle stuff". I want quite there enough to make the link between right angle triangles and a circle but it made that topic so much easier to connect.
Thanks to programming and the thought patterns it drums into you, I was able to do my maths course work in a few pages (far less than anyone else) but still achieve high marks. I remember one top performer in my class complained to my teacher because of how little I'd produced! The teacher said "doesn't matter - he used an approach that let him do that".
I learned how to use sine and cosine for drawing circles when I was probably 9 or 10. It was years before I found out what they actually mean and how they work. I didn't even know what the curves looked like.
