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. 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.
- 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
- Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal. 5: 73–76. JSTOR 3026991.
- 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.
- Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". Amer. Math. Monthly. 52 (9): 507–508. JSTOR 2304540.
- Lehmer, Derik H. (1933). "A note on trigonometric algebraic numbers". Amer. Math. Monthly. 40 (3): 165–166. JSTOR 2301023.