On an extension of Niven's theorem

Abstract

For a given rational number r, a classical theorem of Niven asserts that if $\cos (\mathrm{r\pi })$ is rational, then $\cos (\mathrm{r\pi })\in \{0,\pm 1,\pm 1/2\}.$ In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that.

Keywords:

• Trigonometric functions
• Niven's theorem
• quadratic irrationality

Acknowledgments

The author would like to thank the anonymous referees for their insightful comments, which greatly help to improve the exposition of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research is funded by National Research Council of Thailand (NRCT) [Research Grant for Mid-Career Scholar number N41A640153].