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Sam asked about this problem:

I’m going to show off because (a) I got this an about 2 minutes, and (b) I totally geeked out and spent 40 minutes making an animation for it.

It’s first worth noting that sine is asymmetric, and cosine is symmetric. Switching the sign of one of those ends up being just like spinning your vector around the other way. The numerator, then, represents the y component of two unit vectors added together, while the denominator represents the x component. You can then represent the problem as two vectors end to end, rotating in opposite directions. You will note from the animation below that the end point ends up drawing out a straight line. Since the slope of that line is constant, the relative x and y offsets must be proportional, and the ratio above is constant as well.

{ 3 } Comments

  1. sam shah | May 15, 2008 at 2:44 am | Permalink

    Wow, 2 minutes is way better than me! And thanks for making the animation. I seriously was going to try to learn geometer’s sketch pad to do that. (What did you make it in? I need to learn more things like that.)

    I think there are two conceptual leaps in this problem. The first is to recognize that if you draw a unit circle, you can represent g(x) as the slope between two points. But the points are (cos x, sin x) and (cos(x+a), -sin(x+a)) — the second point is the reflection of (cos(x+a),sin(x+a)) over the x-axis.

    The second conceptual leap for me was to actually draw the diagram. I wanted to prove geometrically that the slope between those points doesn’t depend on x at all. Even after I “saw” what you made so wonderfully clear with the animation above, I needed to prove that it would always be a straight line. So showed that slope isn’t dependent on x.

    Gosh, I can’t stop looking at your animation. Math is so pretty.


  2. Mr. K | May 15, 2008 at 4:37 am | Permalink

    It might have been 3 minutes.

    The big advantage? I already knew (from your post) that the expression was a constant. So I was just looking for something to verify that. Having the same thing inside of sin & cos screams vector to me, after that it was just a matter of pencil & papering some example sketches.

    The individual frames were made using EazyDraw (Mac graphics ware). Each component was rotated 10 degrees in the appropriate direction and then lined up, and saved in 36 different files. The animation was made using GIFfun, which is a Mac graphical front end to a unix command line gif animator.

  3. alphachapmtl | August 23, 2008 at 8:36 am | Permalink

    The path is a hypocycloid, for the case of a circle or radius 1/2 rolling without slipping inside a circle of radius 1.

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  1. […] Mr. K solved the problem in 3 minutes and found a way to show the geometric solution. Head over to his very excellent blog to see it in all it’s glory. Possibly related posts: (automatically generated)getting […]