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Niven's theorem

The only rational angles in first quadrant whose sine is rational are 0, 30 and 90 degrees

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

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

^ 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.