Exact evaluations and reciprocity theorems for finite trigonometric sums

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first method uses contour integration and extends a previous method used by two of the authors. In the second, we work in two cyclotomic fields to evaluate new sums involving roots of unity, which lead to the evaluations of several sums involving trigonometric functions. Reciprocity theorems for certain trigonometric sums are also established.