The Exercise

A rotation $R_{\hat n}(\theta)$ of the Bloch sphere about an axis $\hat n$, applied over and over, comes back to the identity after finitely many steps exactly when $\theta$ is a rational multiple of $2\pi$. The exercise takes an angle $\theta$ with $\cos\theta = 3/5$, for instance the angle of a right triangle with sides $3$, $4$ and $5$ that lies opposite the side of length $4$, and asks to show that it is not such a multiple: that $\theta/2\pi$ is irrational, not a quotient of two integers.^✦

The Claim

What Is Checked

Its angle $\theta$ is any real solution of $\cos\theta = 3/5$, not only the acute one.