1+ week, 2+ day ago  (251+ words) I ran across a geometry theorem with the following diagram. The theorem corresponding to the diagram is interesting, but I found reproducing the diagram more interesting. The segment AB is a diameter and the line CD is perpendicular to the…...

1+ mon, 1+ week ago  (180+ words) Several posts lately have looked at the function f(x) = cos(sin(x) + x). This post will look at the function from a different angle. It’s a smooth function with period 2π, and it’s very flat at odd multiples of π, i.e. the…...

2+ mon, 1+ week ago  (165+ words) TimF left a comment on my guitar pick post saying the image was a “squircle-ish analog for an isosceles triangle.” That made me wonder what a more direct analog of the squircle might be for a triangle. A squircle is…...

2+ mon, 2+ week ago  (221+ words) A couple days ago I wrote a post about turning a trick into a technique, finding another use for a clever way to construct simple, accurate approximations. I used as my example approximating the Bessel function J(x) with (1 + cos(x))/2. I…...

4+ mon, 3+ day ago  (294+ words) In the previous two posts, we looked at why Mathematica and SymPy did not simplify sinh(arccosh(x)) to √(x² − 1) as one might expect. After understanding why sinh(arccosh(x)) doesn’t simplify nicely, it’s natural to ask why sin(arccos(x)) does…...

4+ mon, 2+ week ago  (258+ words) Here is an identity that doesn’t look correct but it is. For real x and y, I found the identity in [1]. The author’s proof is short. First of all, Taking square roots completes the proof. Now note that the statement…...

4+ mon, 2+ week ago  (291+ words) I ran across an old article [1] that gave a sort of multiplication table for trig functions and inverse trig functions. Here’s my version of the table. I made a few changes from the original. First, I used LaTeX, which didn’t…...

7+ mon, 1+ week ago  (376+ words) This post is a side quest in the series on navigating by the stars. It expands on a footnote in the previous post. There are six pieces of information associated with a spherical triangle: three sides and three angles. I…...

7+ mon, 2+ week ago  (236+ words) In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius…...

7+ mon, 2+ week ago  (352+ words) Let ℍ be the upper half plane, the set of complex real numbers with positive imaginary part. When we measure distances the way we’ve discussed in the last couple posts, the geometry of ℍ is hyperbolic. What is a circle…...