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]
![\begin{align}
\sin 0^\circ & = 0, \\[10pt]
\sin 30^\circ & = \frac 12, \\[10pt]
\sin 90^\circ & = 1.
\end{align}](../I/m/6ecaeade9f47e5c68fb2bd36d7d056c3.png)
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]
See also
- Pythagorean triples form right triangles where sin will always have a rational result
- Trigonometric functions
- Trigonometric number
References
Further reading
- Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". Am. Math. Montly 52 (9): 507–508. JSTOR 2304540.
- Lehmer, Derik H. (1933). "A note on trigonometric algebraic numbers". Am. Math. Monthly 40 (3): 165–166. JSTOR 2301023.