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:

:$\begin \sin 0^\circ & = 0, \\$0pt\sin 30^\circ & = \frac 12, \\0pt\sin 90^\circ & = 1.
\end

In radians, one would require that 0 ≤ ''x'' ≤ /2, that ''x''/ be rational, and that sin''x'' be rational. The conclusion is then that the only such values are sin 0 = 0, sin /6 = 1/2, and sin /2 = 1.

The theorem appears as Corollary 3.12 in Niven's book on irrational numbers.

The theorem extends to the other trigonometric functions as well. 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.A proof for the cosine case appears as Lemma 12 in

See also

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

References

Further reading

*
*
*

External links

*
* {{ProofWiki , id=Niven's_Theorem , title=Niven's Theorem

Rational numbers
Trigonometry
Theorems in geometry
Theorems in algebra