Niven's theorem
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}}](../I/m/778d2cd694809f428b1a92aac1ee3a3dfd52d817.svg)
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]
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
- ↑ Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5: 73–76. JSTOR 3026991.
- 1 2 Niven, Ivan (1956). Irrational Numbers. The Carus Mathematical Monographs. The Mathematical Association of America. p. 41. MR 0080123.
- ↑ 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. MR 2057186.