Niven's theorem

In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0 â‰¤ Î¸ â‰¤ 90 for which the sine of θ degrees is also a rational number are:^[1]

$\begin{align} \sin 0^\circ & = 0, \\[10pt] \sin 30^\circ & = \frac 12, \\[10pt] \sin 90^\circ & = 1. \end{align}$

In radians, one would require that 0 â‰¤ x â‰¤ Ï€/2, that x/Ï€ be rational, and that sin x be rational. The conclusion is then that the only such values are sin 0 = 0, sin Ï€/6 = 1/2, and sin Ï€/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.^{[3]:p. 41} 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.

See also

• Pythagorean triples form right triangles where the trigonometric functions will always take rational values, though the acute angles may not be rational
• Trigonometric functions
• Trigonometric number

References

1. Schaumberger, Norman (1974). "A Classroom Theorem on Trigonometric Irrationalities". Two-Year College Mathematics Journal 5: 73–76. JSTOR 3026991.
2. Niven, I. (1956). Irrational Numbers. Wiley. p. 41. MR 0080123.
3. Niven, Ivan. Irrational Numbers, Carus Mathematical Monographs no. 11, 1956.

Further reading

• Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". Am. Math. Monthly 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.

External links

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