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}

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

Notes and references

  1. ^ Schaumberger, N. (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. MR0080123. 

External links