Hyperbolic cosines and sines theorems for the triangle formed by intersection of three semicircles on Euclidean plane

TL;DR

This paper develops hyperbolic trigonometry on the Euclidean plane by deriving hyperbolic laws from geometric relationships of intersecting semicircles, without referencing hyperbolic geometry.

Contribution

It introduces a novel approach to hyperbolic trigonometry on Euclidean plane using a key formula linking exponential functions and segment ratios.

Findings

Hyperbolic laws derived from Euclidean geometry

Hyperbolic functions naturally arise on Euclidean plane

Relationships between semicircle intersections and hyperbolic angles

Abstract

The purpose the present paper is to construct the hyperbolic trigonometry on Euclidean plane without refereing to hyperbolic plane. In this paper we show that the concept of hyperbolic angle and its functions forming the hyperbolic trigonometry give arise on Euclidean plane in a natural way. The method is based on a key- formula establishing a relationship between exponential function and the ratio of two segments. This formula opens a straightforward pathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic law of cosines I and II and the hyperbolic law of sines are derived by using of the key-formula and the methods of Euclidean Geometry, only. It is shown that these laws are consequences of the interrelations between distances and radii of the intersecting semi-circles.