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 cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers ${\displaystyle k}$ and ${\displaystyle n}$ with ${\displaystyle n>2}$, the number ${\displaystyle 2\cos(2\pi k/n)}$ is an algebraic number of degree ${\displaystyle \varphi (n)/2}$, where ${\displaystyle \varphi }$ denotes Euler's totient function. Because rational numbers have degree 1, we must have ${\displaystyle \varphi (n)=2}$ and therefore the only possibilities are ${\displaystyle n={}}$1, 2, 3, 4, or 6. Next, he proved a corresponding result for sine using the trigonometric identity ${\displaystyle \sin(\theta )=\cos(\theta -\pi /2)}$.^[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

References

Further reading

• Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". The American Mathematical Monthly. 52 (9): 507–508. JSTOR 2304540.
• Jahnel, Jörg (2010). "When is the (co)sine of a rational angle equal to a rational number?". arXiv:1006.2938 [math.HO].

External links

• Weisstein, Eric W. "Niven's Theorem". MathWorld.
• Niven's theorem at ProofWiki

This page was last edited on 3 October 2023, at 18:19