Adam Rifkin stashed this in Math!
Redditor ClimbingWolfBear explains:
I guess you're talking about SOH-CAH-TOA? That gif doesn't explain much about that. The circle on the right has a radius of one, and all it's saying is that from the points on the circle, the sin value is the same as the y value, and the cosine is the same as the x value.
This relates to the ratios by pythagorean's theorem. You know that for a right triangle (and if you sketch a vertical line from the point to the x axis and from the point to the origin, you have one) that if you add the square of both sides you get the square of the hypotenuse (or distance from the origin). In this example, that distance is always 1.
So you can show that sin2(angle)+cos2(angle)=1.
This isn't an understanding you really need until you get into calculus, and most of it you can just gloss over unless you go into some engineering discipline, like me!
The phase shift is off. When sin is zero, cos is always either one or negative one, and vice versa. You could even say that sin(x)=cos(x-pi/2).
Math is hard.