Niven's theorem

Template:Short description In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of Template:Mvar in the interval Template:Math for which the sine of Template:Mvar degrees is also a rational number are:

$\begin{array}{cc}\sin 0^{\circ }& =0,\\ \sin 30^{\circ }& =\frac{1}{2},\\ \sin 90^{\circ }& =1.\end{array}$

In radians, one would require that Template:Math, that Template:Math be rational, and that Template:Math be rational. The conclusion is then that the only such values are Template:Math, Template:Math, and Template:Math.

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 Template:Mvar, the only rational values of the sine or cosine are Template:Math, Template:Math, and Template:Math; the only rational values of the secant or cosecant are Template:Math and Template:Math; and the only rational values of the tangent or cotangent are Template:Math and Template:Math.

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 the cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers Template:Mvar and Template:Mvar with Template:Math, the number Template:Math is an algebraic number of degree Template:Math, where Template:Mvar denotes Euler's totient function. Because rational numbers have degree 1, we must have Template:Math or Template:Math and therefore the only possibilities are Template:Math. Next, he proved a corresponding result for the sine using the trigonometric identity Template:Math. 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

Further reading

• Template:Cite journal
• Template:Cite arXiv

External links

• Template:MathWorld