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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: 462

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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 cosα is a rational number, then so is cosnα for every natural number n, and if both sinα and cosα are rational numbers, then so is sinnα. Furthermore, it is shown that if m and n are
relatively prime numbers and cosnmπ is a rational number, then cosπm is also a rational number, while for every natural number m > 3, the number cosπm
is irrational. We also discuss rationality and irrationality of numbers tg2πn.

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: 1.627 *

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