#### Osječki matematički list, Vol. 21 No. 1, 2021.

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:

str. 19-32

preuzimanja: 303

###### 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 $$tg\frac{2\pi }{n}$$.

258826

###### Datum izdavanja:

1.6.2021.

Podaci na drugim jezicima: hrvatski

Posjeta: 936 *