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]

${\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}}$

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 the cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers k and n with n > 2, the number 2 cos(2πk/n) is an algebraic number of degree φ(n)/2, where φ denotes Euler's totient function. Because rational numbers have degree 1, we must have n ≤ 2 or φ(n) = 2 and therefore the only possibilities are n = 1,2,3,4,6. Next, he proved a corresponding result for the sine using the trigonometric identity sin(θ) = cos(θ − π/2).^[4] In 1956, Niven extended Lehmer's result to the other trigonometric functions.^[2] Other mathematicians have given new proofs in subsequent years.^[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