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
- Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". Am. Math. Monthly. 52 (9): 507–508. JSTOR 2304540.
- Lehmer, Derik H. (1933). "A note on trigonometric algebraic numbers". Am. Math. Monthly. 40 (3): 165–166. JSTOR 2301023.