Niven's theorem
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]
- [math]\displaystyle{ \begin{align} \sin 0^\circ & = 0, \\[10pt] \sin 30^\circ & = \frac 12, \\[10pt] \sin 90^\circ & = 1. \end{align} }[/math]
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]
The theorem extends to the other trigonometric functions as well.[2] For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1.[3]
History
Niven's proof of his theorem appears in his book Irrational Numbers. Earlier, the theorem had been proven by D. H. Lehmer and J. M. H. Olmstead.[2] In his 1933 paper, Lehmer proved the theorem for cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers [math]\displaystyle{ k }[/math] and [math]\displaystyle{ n }[/math] with [math]\displaystyle{ n\gt 2 }[/math], the number [math]\displaystyle{ 2\cos(2\pi k/n) }[/math] is an algebraic number of degree [math]\displaystyle{ \varphi(n)/2 }[/math], where [math]\displaystyle{ \varphi }[/math] denotes Euler's totient function. Because rational numbers have degree 1, we must have [math]\displaystyle{ \varphi(n)=2 }[/math] and therefore the only possibilities are [math]\displaystyle{ n={} }[/math]1, 2, 3, 4, or 6. Next, he proved a corresponding result for sine using the trigonometric identity [math]\displaystyle{ \sin(\theta)=\cos(\theta-\pi/2) }[/math].[4] In 1956, Niven extended Lehmer's result to the other trigonometric functions.[2] Other mathematicians have given new proofs in subsequent years.[3]
See also
- Pythagorean triples form right triangles where the trigonometric functions will always take rational values, though the acute angles are not rational
- Trigonometric functions
- Trigonometric number
References
- ↑ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal 5 (1): 73–76. doi:10.2307/3026991.
- ↑ 2.0 2.1 2.2 2.3 Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. https://archive.org/details/irrationalnumber00nive.
- ↑ 3.0 3.1 A proof for the cosine case appears as Lemma 12 in Bennett, Curtis D.; Glass, A. M. W.; Székely, Gábor J. (2004). "Fermat's last theorem for rational exponents". American Mathematical Monthly 111 (4): 322–329. doi:10.2307/4145241.
- ↑ Lehmer, Derrick H. (1933). "A note on trigonometric algebraic numbers". The American Mathematical Monthly 40 (3): 165–166. doi:10.2307/2301023.
Further reading
- Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". The American Mathematical Monthly 52 (9): 507–508.
- Jahnel, Jörg (2010). "When is the (co)sine of a rational angle equal to a rational number?". arXiv:1006.2938 [math.HO].
External links
- Weisstein, Eric W.. "Niven's Theorem". http://mathworld.wolfram.com/NivensTheorem.html.
- Niven's theorem at ProofWiki
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