Tangent lines to the sine function with rational slope

Today I was wondering the following thing (I won’t bore you with how I ended up with this question): Are there any rational values of for which the line is tangent to the graph of Clearly the answer is yes: But my gut feeling was that this was the only such After some head scratching,…

An amazing paragraph from Euler’s Introductio

Today I’d like to share an amazing paragraph from Euler’s 1748 textbook Introductio in analysin infinitorum (Introduction to analysis of the infinite). This two–volume book is what Carl Boyer calls “The foremost textbook of modern times,” edging out, for example, Descartes’s Géométrie, Gauss’ Disquisitiones, and Newton’s Principia. Boyer writes that “Euler accomplished for analysis what Euclid…

Trigonometric functions and rational multiples of pi

Recall that a real number is algebraic if it is the root of a polynomial with integer coefficients and that it is transcendental otherwise. For example is algebraic because it is a root of the polynomial , but is transcendental because it is not the root of any such equation. (On a recent blog post…

More about the neat calculator trick

Yesterday I wrote about a neat calculator trick that I had just learned. We saw that if the calculator was set to degree mode, then times a high enough power of 10 is approximately . A commenter named Robert suggested looking at the difference between this approximation for and itself. He remarked that the error…

The math behind a neat calculator trick

[Update: after you read this post, read my follow-up post.] I received an interesting comment on yesterday’s blog post from Nemo. It was a cool calculator trick that I’d never seen before. Nemo wrote: Reminds me of my favorite calculator trick. Set your calculator to degree mode (NOT radians). Type in a bunch of 5’s:…

Interesting approximations using trigonometry

Today on Twitter John D. Cook, writing as @AlgebraFact, posted the following tweet: In radians, sin(11) is very nearly -1. (It happens to be -0.9999902…) I thought that was awesome! So, I (@divbyzero) replied that cos(333) is approximately 1. (It is 0.999961…) Then @michiexile chimed in, pointing out that cos(355) is closer to -1 than…