Optimal sine and sawtooth inequalities.

We determine the optimal inequality of the form $\sum_{k=1}^m a_k\sin kx\leq 1$, in the sense that $\sum_{k=1}^m a_k$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle.