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Rational and non-rational values of trigonometric functions

Omer Kurtanović ; Ekonomski fakultet, Univerzitet u Bihaću
Nenad Stojanović ; Poljoprivredni fakultet, Univerzitet u Banja Luci
Fatka Kulenović ; Tehnički fakultet, Univerzitet u Bihaću


Puni tekst: hrvatski pdf 282 Kb

str. 19-32

preuzimanja: 303

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Sažetak

The paper presents an application of Chebyshev polynomials of the first and second kind in proving the non-rationality of certain values of trigonometric functions. More precisely, we determine those rational multiples of the number π for which the values of sine, cosine and tangent are rational numbers, and for which these values are irrational numbers. We show that if \( \cos \alpha \) is a rational number, then so is \( \cos n\alpha \) for every natural number n, and if both \( \sin \alpha \) and \( \cos \alpha \) are rational numbers, then so is \( \sin n\alpha \). Furthermore, it is shown that if m and n are
relatively prime numbers and \(\cos \frac{n}{m}\pi\) is a rational number, then \(\cos \frac{\pi}{m}\) is also a rational number, while for every natural number m > 3, the number \(\cos \frac{\pi}{m}\)
is irrational. We also discuss rationality and irrationality of numbers \(tg\frac{2\pi }{n}\).

Ključne riječi

trigonometric functions, Chebyshev polynomials, rational values of trigonometric functions

Hrčak ID:

258826

URI

https://hrcak.srce.hr/258826

Datum izdavanja:

1.6.2021.

Podaci na drugim jezicima: hrvatski

Posjeta: 936 *