Various observations on angles proceeding in geometric progression

This is a translation of Euler’s 1773 “Variae observationes circa angulos in progressione geometrica progredientes”, E561 in the Enestr{"o}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 Vi`ete’s infinite product $\prod_{k=1}^\infty \cos \theta/2^k=\frac{\sin\theta}{\theta}$.

💡 Research Summary

Euler’s 1773 memoir “Variae observationes circa angulos in progressione geometrica progredientes” (E561) investigates trigonometric identities that arise when angles form a geometric progression. The paper begins with the elementary double‑angle formulas
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