Niven's theorem

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In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0 ≤ θ ≤ 90 for which the sine of θ degrees is also a rational number are:[1]


\begin{align}
\sin 0^\circ & = 0, \\[10pt]
\sin 30^\circ & = \frac 12, \\[10pt]
\sin 90^\circ & = 1.
\end{align}

In radians, one would require that 0 ≤ x ≤ π/2, that x/π be rational, and that sin x be rational. The conclusion is then that the only such values are sin 0 = 0, sin π/6 = 1/2, and sin π/2 = 1.

The theorem appears as Corollary 3.12 in Niven's book on irrational numbers.[2][3]

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Notes and references[edit]

  1. ^ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal 5: 73–76. JSTOR 3026991. 
  2. ^ Niven, I. (1956). Irrational Numbers. Wiley. p. 41. MR 0080123. 
  3. ^ Rosenbaum, Robert A. (1958). "Review: Irrational numbers, by Ivan Niven". Bull. Amer. Math. Soc. 64 (2): 68–69. doi:10.1090/S0002-9904-1958-10170-6. 
  • Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". Am. Math. Montly 52 (9): 507–508. JSTOR 2304540. 
  • Lehmer, Derik H. (1933). "A note on trigonometric algebraic numbers". Am. Math. Monthly 40 (3): 165–166. JSTOR 2301023. 

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