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Niven's theorem
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<h2 id="history">History</h2> <p>Niven's proof of his theorem appears in his book <i>Irrational Numbers</i>. Earlier, the theorem had been proven by <a href="/facts/D._H._Lehmer/EI61SICa">D. H. Lehmer</a> and J. M. H. Olmstead.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> In his 1933 paper, Lehmer proved the theorem for the cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers k and n with <i>n</i> > 2, the number 2 cos(2<i>πk</i>/<i>n</i>) is an <a href="/facts/Algebraic_number/MkH1PzTK">algebraic number</a> of degree <i>φ</i>(<i>n</i>)/2, where φ denotes <a href="/facts/Euler%2527s_totient_function/mIQwXYzj">Euler's totient function</a>. Because rational numbers have degree 1, we must have <i>n</i> ≤ 2 or <i>φ</i>(<i>n</i>) = 2 and therefore the only possibilities are <i>n</i> = 1,2,3,4,6. Next, he proved a corresponding result for the sine using the trigonometric identity sin(<i>θ</i>) = cos(<i>θ</i> − <i>π</i>/2).<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> In 1956, Niven extended Lehmer's result to the other trigonometric functions.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Other mathematicians have given new proofs in subsequent years.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Pythagorean_triples/I3IJgWhf">Pythagorean triples</a> form right triangles where the trigonometric functions will always take rational values, though the acute angles are not rational.</li> <li><a href="/facts/Trigonometric_functions/czej7ur0">Trigonometric functions</a></li> <li><a href="/facts/Trigonometric_number/zYE1BVLF">Trigonometric number</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". <i>The American Mathematical Monthly</i>. 52 (9): 507–508. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2304540">10.2307/2304540</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2304540">2304540</a>.</li> <li>Jahnel, Jörg (2010). "When is the (co)sine of a rational angle equal to a rational number?". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1006.2938">1006.2938</a> [<a href="https://arxiv.org/archive/math.HO">math.HO</a>].</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/NivensTheorem.html">"Niven's Theorem"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 href="/wiki/Two-Year_College_Mathematics_Journal" target="_blank">/wiki/Two-Year_College_Mathematics_Journal</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123. <a href="/wiki/Ivan_Niven" target="_blank">/wiki/Ivan_Niven</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123. <a href="/wiki/Ivan_Niven" target="_blank">/wiki/Ivan_Niven</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123. <a href="/wiki/Ivan_Niven" target="_blank">/wiki/Ivan_Niven</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Lehmer, Derrick H. (1933). "A note on trigonometric algebraic numbers". The American Mathematical Monthly. 40 (3): 165–166. doi:10.2307/2301023. JSTOR 2301023. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123. <a href="/wiki/Ivan_Niven" target="_blank">/wiki/Ivan_Niven</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>