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:

In radians, one would require that 0 ≤ x ≤ /2, that x/ be rational, and that sinx be rational. The conclusion is then that the only such values are sin 0 = 0, sin /6 = 1/2, and sin /2 = 1. The theorem appears as Corollary 3.12 in Niven's book on irrational numbers. 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. 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. In his 1933 paper, Lehmer proved the theorem for cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers  and  with , the number  is an algebraic number of degree , where  denotes Euler's totient function. Because rational numbers have degree 1, we must have  and therefore the only possibilities are 1, 2, 3, 4, or 6. Next, he proved a corresponding result for sine using the trigonometric identity . In 1956, Niven extended Lehmer's result to the other trigonometric functions. Other mathematicians have given new proofs in subsequent years. 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

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Further reading

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Rational numbers
Trigonometry
Theorems in geometry
Theorems in algebra