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]
![{\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/778d2cd694809f428b1a92aac1ee3a3dfd52d817)
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 
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
Further reading
External links
- Weisstein, Eric W. "Niven's Theorem". MathWorld.
- Niven's theorem at ProofWiki