CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / A trigonometria esfÃrica surgiu das necessidades da Astronomia, na busca de descrever matematicamente o sistema solar. Mentes brilhantes como Euclides, Aristarco de Samos, ApolÃnio de Perga, Hiparco, Menelau de Alexandria, Ptolomeu, entre outros, estudaram sobre os triÃngulos esfÃricos. Neste trabalho, estudaremos os resultados fundamentais a trigonometria esfÃrica buscando uma associaÃÃo com o globo terrestre. Iniciaremos com o estudo dos elementos fundamentais de uma superfÃcie esfÃrica, donde definiremos os triÃngulos esfÃricos e provaremos suas principais propriedades, como soma das medidas dos Ãngulos internos e a fÃrmula de Girard para o cÃlculo de sua Ãrea. Em seguida, apresentamos a classificaÃÃo dos triÃngulos esfÃricos e as principais relaÃÃes entre os lados e os Ãngulos desses triÃngulos, como a lei dos senos e lei dos cossenos, alÃm de um breve estudo dos triÃngulos esfÃricos retÃngulos. Finalmente, consideramos a Terra como uma esfera, denominada globo terrestre, sobre a qual abordamos diversos conceitos geogrÃficos como paralelos, meridianos, latitudes, longitudes, a fim de utilizar da trigonometria esfÃrica para o cÃlculo de distÃncias e Ãngulos sobre a superfÃcie terrestre, criando o forte carÃter interdisciplinar entre MatemÃtica e Geografia. / The spherical trigonometry came from the needs of Astronomy, in the search for mathematically describing the solar system. Brilliant minds like Euclides, Aristarco of Samos, ApolÃnio of Perga, Hiparco, Menelau of Alexandria, Ptolomeu, and others, have studied the spherical triangles. In this work, we study the fundamental results spherical trigonometry seeking an association with the globe. We begin with the study of the fundamental elements of a spherical surface, where we define the spherical triangles and prove their important properties, such as sum of the measures of the internal angles and the Girard formula to calculate its area. Then, we present the classification of spherical triangles and the main relationships between the sides and angles of these triangles, as the law of sines and law of cosines, and a brief study of spherical rectangle triangles. Finally, we consider the Earth as a sphere called earth globe, over which we address various geographical concepts such as parallels, meridians, latitudes, longitudes, in order of use of spherical trigonometry to calculate distances and angles on the Earth's surface, creating strong interdisciplinary character between Mathematics and Geography.

Chapter 1 presents the definitions and basic properties of trigonometric functions including: Sum Identities, Difference Identities, Product-Sum Identities and Sum-Product Identities. These formulas provide effective tools to solve the problems in trigonometry. Chapter 2 handles the most important two theorems in trigonometry: The laws of sines and cosines and show how they can be applied to derive many well known theorems including: Ptolemy¡¦s theorem, Euler Triangle Formula, Ceva¡¦s theorem, Menelaus¡¦s Theorem, Parallelogram Law, Stewart¡¦s theorem and Brahmagupta¡¦s Formula. Moreover, the formulas of computing a triangle area like Heron¡¦s formula and Pick¡¦s theorem are also discussed. Chapter 3 deals with the method of superposition, inverse trigonometric functions, polar forms and De Moivre¡¦s Theorem.