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{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac 12},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}](/2014-wikipedia_en_all_02_2014/I/media/6/e/c/a/6ecaeade9f47e5c68fb2bd36d7d056c3.png)
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]
See also
- Pythagorean triples form right triangles where sin will always have a rational result
- Trigonometric functions
- Trigonometric number
Notes and references
- ↑ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal 5: 73–76. JSTOR 3026991.
- ↑ Niven, I. (1956). Irrational Numbers. Wiley. p. 41. MR 0080123.
- ↑ 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.
External links
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