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Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficientcondition is established for the three numbers $1$, $\cos (\pi r_1)$, and $\cos(\pi r_2)$ to be rationally independent. Extending a classical fact sometimesattributed to I. Niven, the result even yields linear independence over largernumber fields. The tools employed in the proof are applicable also in the caseof more than two trigonometric numbers. As an application, a completeclassification is given of all planar triangles with rational angles and sidelengths each containing at most one square root. Such a classification washitherto known only in the special case of right triangles.

Linear independence of trigonometric numbers