Niven's theorem
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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}](http://upload.wikimedia.org/math/6/e/c/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][3]
See also [edit]
Notes and references [edit]
- ^ Schaumberger, N. (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal 5: 73–76. JSTOR 3026991.
- ^ Niven, I. (1956). Irrational Numbers. Wiley. p. 41. MR 0080123.
- ^ Rosenbaum, Robert A. (1958). "Review: Irrational numbers, by Ivan Niven". Bull. Amer. Math. Soc. 64 (2): 68–69.
