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Why we kept the trig in golf: Mathematical simplicity is not always the same as conceptual simplicity

Someone read the golf example and asked:

You define the threshold angle as arcsin((R – r)/x), but shouldn’t it be arctan((R – r)/x) instead?

Is it just that it does not matter with these small angles, where
sine and tangent are about the same, or am I missing something?

My reply:

This sin vs tan thing comes up from time to time.

As you note, given the dimensions involved, the two functions are for all practical purposes equivalent. If you look at the picture I drew of the little ball and the big ball and the solid and dotted lines, the dotted lines are supposed to just touch the edge of the inner circle. In that case, if you drop a perpendicular from where the dotted line hits the circle, that perpendicular goes through the center of the circle, hence the angle of interest is arcsin((r-R)/x): the solid line of length x is the hypoteneuse of the triangle.

As many people have pointed out, the whole model is approximate in that it assumes the ball enters the whole only if the entire ball is over the hole. But actually the ball could fall in, if only half of the ball is over the whole. So, arguably, it should be asin(R/x). But that’s also only an approximation!

So we could replace all that trig with a simple R/x, and I’m guessing it would fit the data just as well. So why didn’t I do that? Doesn’t the trig just complicate things? I kept the trig because, to me, it’s cleaner to derive the trig solution and just use it, then have mathematical conversations about simplifying it. Mathematically, arcsin((R-r)/x) is more complicated than R/x, but conceptually I think it’s simpler to go with arcsin((R-r)/x) as it has a direct geometrical derivation. And, for teaching purposes, I like having a model that’s very clearly tailored to this particular problem.

Mathematical simplicity is not always the same as conceptual simplicity. A (somewhat) complicated mathematical expression can give some clarity, as the reader can see how each part of the formula corresponds to a different aspect of the problem being modeled.


  1. it depends on what you mean by distance to the hole. If you mean the distance from the center of the ball to the center of the hole, then the relevant angle is arctan… if you mean the center of the ball to the offset location where the line meets the circle of radius R-r then the angle is arcsin

    I’d say most people measure distance to the hole as center of ball to center of hole, so I agree with the correspondent that arctan makes more sense. you should change it because of the reasons you mention, namely that it’s confusing to have arcsin

  2. To me it seems clear that asin(R/x) is best (i.e. the center of the ball just glances the hole). But more importantly, it would be great to tell the reader something like, “you might argue for different forms for this equation, based on whatever geometry seems relevant. You can try different equations, and see how the variation in model outcomes compares to the differences with other models on this page.” (I.e. the reader will find that it doesn’t matter practically, that the geometric approach is pretty insensitive to these subtle points.)

    This is a wonderful example, by the way.

    • I think the diagram needs a “right angle” symbol. Given the diagram as is, it’s easy to mistake what’s being drawn. I think I see Andrew’s point that the right angle is actually at the point of tangency with the circle of radius R-r… ok fine. But as drawn it looks like the right angle is at the center of the hole.

  3. Dale Lehman says:

    Clearly, you are not a golfer (I am – a bad one, is there any other kind?). You said “But actually the ball could fall in, if only half of the ball is over the whole.” In actuality, even if the ball is completely over the hole, it may pop out. And, if it is half over the whole, the probability it will not go in is 96.78942%.

  4. Josh M says:

    I will skip over the myriad whole/hole puns I’ve got, and instead just point out that there are some incorrect word choices in the blog post. It’s a very small problem in the whole of the work on a hole, though. (Sorry, couldn’t resist at least one of them.)

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