$\forall n \in\mathbb N$, determine which $n$-th root of unity is closest to $\frac12$.

Question

$\forall n \in\mathbb N$, determine which $n$-th root of unity is closest to $\frac12$.

I'm really struggling with where to even begin with this question. Any help would be appreciated

Answer 1

Well, the answer(s) are given geometrically in the comments.

An analytic approach:

The $n^\textrm{th}$-roots of unity are $\left\{1,\omega, \omega^2, \dots \omega^{n-1} \right\},$ where $\omega = \exp \frac{2\pi i k}{n}.$

We want to minimize $$|z-1/2|^2=\left(e^{2\pi i k/n} -1/2\right)\left(e^{-2\pi i k/n} -1/2\right) = \frac{5}{4} - \cos \frac{2k \pi}{n}.$$

Clearly, $k=0$ gives the minimum at $z=1$. If we disallow $z=1$, then $k=1$ or $k=n-1$ are the next closest along the circle, and also next closest to $1/2$.