Open AccessAlgebraic values of sines and cosines and their arguments
Author(s) -
Edmundas Mazėtis,
Grigorijus Melničenko
Publication year - 2021
Publication title -
lietuvos matematikos rinkinys
Language(s) - English
Resource type - Journals
eISSN - 2335-898X
pISSN - 0132-2818
DOI - 10.15388/lmr.2020.22717
Subject(s) - transcendental number , transcendental function , algebraic number , mathematics , trigonometry , trigonometric functions , rational number , pure mathematics , inverse trigonometric functions , infinity , algebraic extension , transcendental equation , mathematical analysis , geometry , differential equation , differential algebraic equation , ordinary differential equation
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.