# 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]

{\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}}

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]

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.

## 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.