*The implications among the three classical trigonometric laws of hyperbolic geometrys*

**David E. Dobbs **

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA

Pages 124-142 | Received 04 January 2023, Accepted 22 February 2023, Published 16 June 2023

**Abstract**

If a,b,c are the hyperbolic lengths of the sides of a hyperbolic triangle \Delta and \alpha, \beta, \gamma are the radian measures of the corresponding interior angles of \Delta (necessarily such that \alpha + \beta + \gamma < \pi), the usual statement of the Hyperbolic Cosine Law (resp., of the Hyperbolic Sine Law; resp., of the trigonometric law in hyperbolic geometry with no Euclidean counterpart, expressing the hyperbolic cosines of a, b and c in terms of \alpha, \beta and \gamma) gives several equations involving a, b, c, \alpha, \beta and \gamma. Now, let the entries in a 6-tuple \mathcal{L}=(a,b,c, \alpha, \beta, \gamma) be positive real numbers such that \alpha + \beta + \gamma < \pi (but do not necessarily assume that these entries arise as above from some hyperbolic triangle). We say that \mathcal{L} *satisfies* HCL (resp., \mathcal{L} *satisfies* HSL; resp., \mathcal{L} *satisfies* HOL) if the entries of \mathcal{L} satisfy all the equations in the usual statement of the Hyperbolic Cosine Law (resp., of the Hyperbolic Sine Law; resp., of the just-mentioned trigonometric law in hyperbolic geometry with no Euclidean counterpart). For any such L (not necessarily induced by some hyperbolic triangle), we prove the following: \mathcal{L} satisfies HOL \Leftrightarrow \mathcal{L} satisfies HCL \Rightarrow \mathcal{L} satisfies HSL; and we give an example showing that the last implication cannot be reversed. This paper could to used to enrich a course on the classical geometries that discusses hyperbolic plane geometry.

**Keywords**: Hyperbolic plane geometry, hyperbolic cosine law, hyperbolic sine law, upper half-plane model, Euclidean geometry, neutral geometry.

**MSC numbers**: Primary 51-02; Secondary 33B10.

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