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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: : 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.〔Niven, Ivan. ''Irrational Numbers'', Carus Mathematical Monographs no. 11, 1956.〕 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 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Niven's theorem」の詳細全文を読む スポンサード リンク
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