The first time I was able to properly visualize sine and cosine was from an old 8mm film reel produced in the 1950s by IBM which showed essentially this same animation:
I was introduced to sine in elementary school. Then we had 6 months of trigonometry in high school. We even touched infinite series at the end of high school.
Then at univ we had some trigonometry again but not much, some infinite series and stuff.
Nowhere along the way I was shown that sine is actually how high above the ground you are when traveling around the circle. It was always some length of side in right triangle divided by length of another side. That was technically the same thing but nowhere close as intuitive and natural.
Even when at some point I got interested in math and start thinking about sine as y coordinate it was always in relation to angle of a triangle (rooted in the center of the circle, one point at radius, the other at x axis below it) and not in relation of actual distance traveled along the circle. At some point radians appeared and we were just taught that's other way to measure angles, so you know: 0 degrees is 0, 90 degrees is 1/2 of PI and 360 degrees is 2 PI. I thought it was kinda weird to this that way but w/e probably people have their reasons...
The concept is so beautiful and natural yet it was made as artificial as possible during math education I had.
No wonder so many kids don't get or like math :(
My pet theory is that people usually are shown all these things, they just happen to lack a piece or a few of the underlying concepts, so it floats on by.
(a decent set of textbooks should have lots of these didactic gizmos in it, but that doesn't solve the problem of getting the student ready to look at it or making sure they look at it after they are ready)
> sine is actually how high above the ground you are when traveling around the circle.
That makes total sense, and I wish it would have been explained like that to me, too.
My math teacher in HS spent an hour and a half explaining how derivatives work by just working through stuff on the board after he insisted no one take notes, and it made everything so clear. And reading through the Tau Manifesto finally made me understand what was going on in trig at a basic level.
I totally agree with you about how artificial math education can feel, and I think if larger concepts were introduced more like that many more people would be able to get much better at it.
I read something a couple years ago about how some school was experimenting in a math program starting in 8th grade where they would teach algebra, geometry, and calculus concepts all at the same time. I guess the students were really picking things up fast, and understanding how everything fit together better.
>> sine is actually how high above the ground you are when traveling around the circle.
I learned my trigonometry when trying to figure out how to rotate and move objects around in 2D space for a computer game. I learned then, that sin is responsible for y axis, and cos for x axis.
But I never ever phrased this as "how high above the ground you are when traveling around the circle". I like the elegance of that; things would probably have clicked for me immediately back then, had I heard a sentence like this (and ditto for cosine being "how far to the right are you").
Bottomline, I'm in agreement with both OP and GP here.
His other works are also worth checking: http://1ucasvb.tumblr.com/ (for example, he has a super-nice explanation of Fourier series).