In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of in the interval for which the sine of degrees is also a rational number are:[1]
\begin{align} \sin0\circ&=0,\\[10pt] \sin30\circ&=
| 12, | |
| \\[10pt] \sin |
90\circ&=1. \end{align}
In radians, one would require that, that be rational, and that be rational. The conclusion is then that the only such values are,, and .
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, the only rational values of the sine or cosine are,, and ; the only rational values of the secant or cosecant are and ; and the only rational values of the tangent or cotangent are and .[3]
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 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 or and therefore the only possibilities are . Next, he proved a corresponding result for the sine using the trigonometric identity .[4] In 1956, Niven extended Lehmer's result to the other trigonometric functions.[2] Other mathematicians have given new proofs in subsequent years.[3]