The only rational values of an angle in degrees for which both the angle and its sine are rational are 0, 30, 90, 150, 180 degrees and their sine values. Proved by Ivan Niven, it is a compact result connecting trigonometry and Diophantine approximation.
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StatementNiven's theorem identifies exactly which angles theta with 0 degrees <= theta <= 90 degrees have both theta and the sine of theta degrees rational; only a small finite set of such angles exists. 1 Classification
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Source Ivan M. Niven (Wikipedia)
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1. Niven's Theorem (Wikipedia)
Wikimedia FoundationLead section, opening theorem statement
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:
Proved By: Ivan Niven, Lead paragraph
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
View the Source Ivan M. Niven (Wikipedia)
Principal Result Of: Ivan Niven, Article textQuote, Principal Result Of: Ivan Niven, Article text
Niven numbers, Niven's constant, and Niven's theorem are named for Niven.
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