Abstract
We explain how to interpret regular continued fractions related to LLS sequences in terms of integer trigonometric functions. Integer trigonometry has many similarities to Euclidean trigonometry (for instance, integer arctangents coincide with real arctangents; the formulas for adjacent angles are similar). From another point of view they are totally different, since integer sines and cosines are positive integers; there are two right angles in integer trigonometry, etc. In this chapter we discuss basic properties of integer trigonometry. For rational angles we introduce definitions of integer sines, cosines, and tangents. In addition to rational integer angles, there are three types of irrational integer angles: R-irrational, L-irrational, and LR-irrational angles. It is only for R-irrational angles that we have a definition of integer tangents. The trigonometric functions are not defined for L-irrational and LR-irrational angles.
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References
O. Karpenkov, Elementary notions of lattice trigonometry. Math. Scand. 102(2), 161–205 (2008)
O. Karpenkov, On irrational lattice angles. Funct. Anal. Other Math. 2(2–4), 221–239 (2009)
P. Popescu-Pampu, The geometry of continued fractions and the topology of surface singularities, in Singularities in Geometry and Topology 2004 (2007), pp. 119–195
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© 2013 Springer-Verlag Berlin Heidelberg
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Karpenkov, O. (2013). Integer Trigonometry for Integer Angles. In: Geometry of Continued Fractions. Algorithms and Computation in Mathematics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39368-6_5
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DOI: https://doi.org/10.1007/978-3-642-39368-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39367-9
Online ISBN: 978-3-642-39368-6
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