Abstract
This chapter discusses the derivation of the formulas of hyperbolic geometry, which express the numerical relations among the sides and angles of a triangle. Each of these formulas is analogous to some familiar Euclidean formula. Corresponding to the right triangle formulas a2 + b2 = c2 and sin A = a/c. The chapter discusses properties of associated right triangles. It presents the theorem that states that the sine of an acute angle in a right triangle is equal to the hyperbolic sine of the opposite side divided by the hyperbolic sine of the hypotenuse. The hyperbolic cosine of the hypotenuse of a right triangle is equal to the product of the hyperbolic cosines of the other sides. The measures of very small figures in hyperbolic geometry fit the formulas of Euclidean geometry very closely and any desired precision of it can be obtained by taking the figures sufficiently small. Euclidean geometry applies very well to the physical world of experience and is, therefore, extensively used in science, engineering, and many other fields dealing with geometrical concepts.
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