Niven's_theorem Knowpia

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 {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}

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 edit

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 $k$ and $n$ with $n>2$ , the number $2\cos(2\pi k/n)$ is an algebraic number of degree $\varphi (n)/2$ , where $\varphi$ denotes Euler's totient function. Because rational numbers have degree 1, we must have $\varphi (n)=2$ and therefore the only possibilities are $n={}$ 1, 2, 3, 4, or 6. Next, he proved a corresponding result for sine using the trigonometric identity $\sin(\theta )=\cos(\theta -\pi /2)$ .^[4] In 1956, Niven extended Lehmer's result to the other trigonometric functions.^[2] Other mathematicians have given new proofs in subsequent years.^[3]

References edit

^ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5 (1): 73–76. doi:10.2307/3026991. JSTOR 3026991.
^ ^a ^b ^c ^d Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123.
^ ^a ^b 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. JSTOR 4145241. MR 2057186.
^ Lehmer, Derrick H. (1933). "A note on trigonometric algebraic numbers". The American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023.

External links edit

Weisstein, Eric W. "Niven's Theorem". MathWorld.
Niven's theorem at ProofWiki

[1] Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5 (1): 73–76. doi:10.2307/3026991. JSTOR 3026991.

[niven-2] Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123.

[bgs-3] 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. JSTOR 4145241. MR 2057186.

[Lehmer-4] Lehmer, Derrick H. (1933). "A note on trigonometric algebraic numbers". The American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023.

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