1+ day, 11+ hour 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…...

1+ day, 22+ hour 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 at the…...

2+ mon, 3+ 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…...

2+ mon, 4+ week ago  (236+ words) In spherical geometry, the interior angles of a triangle add up to more than ". And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds ". On a sphere of radius 1, the area…...

2+ mon, 4+ week ago  (352+ words) Let " 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 " is hyperbolic. What is a circle of radius…...

2+ mon, 4+ week ago  (329+ words) The previous post described a metric for the Poincar" upper half plane. The development is geometrical rather than analytical. There are also analytical formulas for the metric, at least four that I've seen. It's not at all obvious that the…...

3+ mon, 3+ hour ago  (305+ words) One common model of the hyperbolic plane is the Poincar" upper half plane ". This is the set of points in the complex plane with positive imaginary part. Straight lines are either vertical, a set of points with constant imaginary part,…...

3+ mon, 1+ week ago  (62+ words) Five posts on Pythagorean triangles and Pythagorean triples Primitive Pythagorean triangles with the same area Sparse binary Pythagorean triples Matrix Pythagorean triples Approximation by Pythagorean triangles Fibonacci meets Pythagoras The post Pythagorean triples first appeared on John D. Cook. Five posts…...

3+ mon, 1+ week ago  (58+ words) Here are four theorems that generalize the Pythagorean theorem. Follow the links for more details regarding each equation. 1. Theorem by Apollonius for general triangles. 2. Edsgar Dijkstra's extension of the Pythagorean theorem for general triangles. 3. A generalization of the Pythagorean theorem…...

3+ mon, 2+ week ago  (195+ words) The area of a triangle can be computed directly from the lengths of its sides via Heron's formula. Here's is the semiperimeter,'s = (a +'b +'c)/2. Is there an analogous formula for spherical triangles? It's not obvious there should be, but…...