  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 $\prod_{k=1}^\infty \cos \theta/2^k=\frac{\sin\theta}{\theta}$.
