  Several methods are used to evaluate finite trigonometric sums. In each case,
either the sum had not previously been evaluated, or it had been evaluated, but
only by analytic means, e.g., by complex analysis or modular transformation
formulas. We establish both reciprocity and three sum relations for
trigonometric sums. Motivated by certain sums that we have evaluated, we add
coprime conditions to the summands and thereby define analogues of Ramanujan
sums, which we in turn evaluate. One of these analogues leads to a criterion
for the Riemann Hypothesis, analogous to the Franel-Landau criterion.
