Various observations on angles proceeding in geometric progression

Abstract

This is a translation of Euler's 1773 "Variae observationes circa angulos in progressione geometrica progredientes", E561 in the Enestr{ö}m index. I translated this paper as a result of my study of Euler's work on the infinite product ${\prod }_{k=1}^{\infty }(1-z^k)$. If one instead considers the finite product ${\prod }_{k=1}^n(1-z^k)$, one can study its behavior on the unit circle. The absolute value of ${\prod }_{k=1}^n(1-e^{ik\theta })$ is $2^n|{\prod }_{k=1}^n\sin k\theta /2|$. My interest in the product ${\prod }_{k=1}^n\sin k\theta /2$ has inspired me to become acquainted with Euler's papers on trigonometric identities, in particular E447, E561, and E562. E561 says nothing about the product ${\prod }_{k=1}^n\sin k\theta /2$, but it has identities which I had not seen before. The identities have a form similar to Viète's infinite product ${\prod }_{k=1}^{\infty }\cos \theta /2^k=\frac{\sin \theta }{\theta }$.

9 pages, E561