Algebraic values of sines and cosines and their arguments

• Main Authors: Edmundas Mazėtis, Grigorijus Melničenko
• Format: Article
• Language: English
• Published: Vilnius University Press 2021-03-01
• Series: Lietuvos Matematikos Rinkinys
• Subjects: trigonometric functions sin \alpha, cos \alpha, tg \alpha ir ctg \alpha; rational numbers; algebraic numbers; transcendental numbers; Lindemann–Weierstrass theorem
• Online Access: https://www.zurnalai.vu.lt/LMR/article/view/22717

The article introduces the reader to some amazing properties of trigonometric functions. It turns out that if the values of the arguments of the functions sin x, cos x, tg x and ctg x, expressed in radians, are algebraic numbers, then the values of these functions are transcendental numbers. Hence, it follows that the values of all angles of the pseudo-Heronian triangle, including the values of all angles of the Pythagoras or Heron triangle, expressed in radians, are transcendental numbers. If the arguments of functions sin x and cos x, expressed in radians, are equal to x = r 2 \pi, where r are rational numbers, then the values of the functions are algebraic numbers. It should be noted that in this case the argument x = r 2\pi  is transcendental and, if expressed in degrees, becomes a rational.