### Carsten Elsner, Takao Komatsu and Iekata Shiokawa

FHDW, Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, D-30173 Hannover, Germany
e-mail: carsten.elsner@fhdw.de

Graduate School of Science and Technology, Hirosaki University, Hirosaki, 036-8561 Japan
e-mail: komatsu@cc.hirosaki-u.ac.jp

Department of Mathematics, Keio University, Yokohama, 223-8522 Japan
e-mail: shiokawa@math.keio.ac.jp

Abstract.   We compute upper and lower bounds for the approximation of certain values ξ of hyperbolic and trigonometric functions by rationals x/y such that x, y satisfy Diophantine equations. We show that there are infinitely many coprime integers x, y such that

|y ξ - x| (log log y)/(log y)

and a Diophantine equation holds simultaneously relating x, y and some integer z. Conversely, all positive integers x, y with yc0 solving the Diophantine equation satisfy

|y ξ - x| (log log y)/(log y)

Moreover, we approximate sin(πα) and cos(πα) by rationals in connection with solutions of a quadratic Diophantine equation when tan(πα/2) is a Liouville number.

2000 Mathematics Subject Classification.   11D09, 11D25, 11J04, 11J70.

Key words and phrases.   Diophantine approximation, Diophantine equations, trigonometric and hyperbolic functions.

Full text (PDF) (free access)

DOI: 10.3336/gm.44.2.02

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