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]
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 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]
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 and with , the number is an algebraic number of degree , where denotes Euler's totient function. Because rational numbers have degree 1, we must have and therefore the only possibilities are 1, 2, 3, 4, or 6. Next, he proved a corresponding result for sine using the trigonometric identity .[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 alsoedit
Pythagorean triples form right triangles where the trigonometric functions will always take rational values, though the acute angles are not rational
^ abA 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.
Further readingedit
Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". The American Mathematical Monthly. 52 (9): 507–508. JSTOR 2304540.
Jahnel, Jörg (2010). "When is the (co)sine of a rational angle equal to a rational number?". arXiv:1006.2938 [math.HO].
niven, theorem, mathematics, named, after, ivan, niven, states, that, only, rational, values, interval, which, sine, degrees, also, rational, number, displaystyle, begin, aligned, circ, 10pt, circ, frac, 10pt, circ, aligned, radians, would, require, that, that. In mathematics Niven s theorem named after Ivan Niven states that the only rational values of 8 in the interval 0 8 90 for which the sine of 8 degrees is also a rational number are 1 sin 0 0 sin 30 1 2 sin 90 1 displaystyle begin aligned sin 0 circ amp 0 10pt sin 30 circ amp frac 1 2 10pt sin 90 circ amp 1 end aligned In radians one would require that 0 x p 2 that x p be rational and that sin x be rational The conclusion is then that the only such values are sin 0 0 sin p 6 1 2 and sin p 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 8 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 Contents 1 History 2 See also 3 References 4 Further reading 5 External linksHistory editNiven 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 displaystyle k nbsp and n displaystyle n nbsp with n gt 2 displaystyle n gt 2 nbsp the number 2 cos 2 p k n displaystyle 2 cos 2 pi k n nbsp is an algebraic number of degree f n 2 displaystyle varphi n 2 nbsp where f displaystyle varphi nbsp denotes Euler s totient function Because rational numbers have degree 1 we must have f n 2 displaystyle varphi n 2 nbsp and therefore the only possibilities are n displaystyle n nbsp 1 2 3 4 or 6 Next he proved a corresponding result for sine using the trigonometric identity sin 8 cos 8 p 2 displaystyle sin theta cos theta pi 2 nbsp 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 editPythagorean triples form right triangles where the trigonometric functions will always take rational values though the acute angles are not rational Trigonometric functions Trigonometric numberReferences 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 Szekely Gabor 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 Further reading editOlmsted J M H 1945 Rational values of trigonometric functions The American Mathematical Monthly 52 9 507 508 JSTOR 2304540 Jahnel Jorg 2010 When is the co sine of a rational angle equal to a rational number arXiv 1006 2938 math HO External links editWeisstein Eric W Niven s Theorem MathWorld Niven s theorem at ProofWiki Retrieved from https en wikipedia org w index php title Niven 27s theorem amp oldid 1178443903, wikipedia, wiki, book, books, library,
