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Journal of Mathematics
Volume 2013, Article ID 920528, 10 pages
Research Article

Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane

1Joint Institute for Nuclear Research, Laboratory of Information Technology, Street Joliot Curie 6, CP 141980, P.O. Box 141980, Dubna, Russia
2Universidad Nacional Autonoma de Mexico, Facultad de Estudios Superiores, Avenida 1 Mayo, 54740 Cuautitlán Izcalli, MEX, Mexico

Received 17 December 2012; Accepted 21 February 2013

Academic Editor: Sujit Samanta

Copyright © 2013 Robert M. Yamaleev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The hyperbolic cosines and sines theorems for the curvilinear triangle bounded by circular arcs of three intersecting circles are formulated and proved by using the general complex calculus. The method is based on a key formula establishing a relationship between exponential function and the cross-ratio. The proofs are carried out on Euclidean plane.