The main goal of this dissertation is to present a construction of hyperbolic trigonometry, and a study of hyperbolic trigonometric functions so that undergraduate students and high school teachers have better access to that theory. The construction of hyperbolic trigonometry will be prioritizing the application of elementary concepts of math. Will be used the right branch of the hyperbola x2 y2 = 1, making a comparison with the cons- truction of circular trigonometry and taking as starting point a hyperbolic trigonometry sector. Will need calculate the area of this hyperbolic sector, which could be done through of a definite integral, but other basic features will be adopted, answering to main objective of this work. Also present a connection between circular trigonometry and hyperbolic trigonometry, which is extended after for circular trigonometric functions and hyperbolic trigonometric functions. The hyperbolic trigonometric functions and their inverse functions will be studied analytically and graphically. The analytical study will follow so completely elementary, however the graphic study will be done using some elements of the theory of limits of functions. Some application of hyperbolic trigonometry are displayed. Finally, the hyperbolic trigonometry is presented in the set of complex numbers.
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