Elementos de trigonometria triangular esférica
Autor(a) principal: | |
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Data de Publicação: | 2014 |
Tipo de documento: | Dissertação |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFRR |
Texto Completo: | http://repositorio.ufrr.br:8080/jspui/handle/prefix/728 |
Resumo: | The main objective of this work was to study in triangles constructed on a spherical surface, versions of known results of the plane euclidean geometry and trigonometry in plans triangles. Initially it presents the fundamental concepts of spherical geometry and some elements of spherical triangular trigonometry. For this, begins with a brief review of some of these results and also with some definitions of plane geometry required for the construction of spherical geometry results. That done, are build, in a spherical triangle, versions for the law of sines, law of cosines and other results of the plane triangular trigonometry. Was also seen is the theorem of Girard, where can study the area of a triangle built on the surface of a sphere of radius R and the sum of its internal angles, which is not constant unlike what occurs in triangles plans built on disc of radius r. The Pythagorean theorem is not true in this environment and a counter-example will be presented. Throughout the text will be presented some examples with the use of trigonometric relations, as well as some elementary concepts of geographical coordinates and practical applications of spherical trigonometry in aviation and geography. Finally it is observed that this work strongly uses the mathematics of basic education, facilitating the understanding of the said theory, of students and teachers of basic education, as well as of the professionals who use math. |
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Elementos de trigonometria triangular esféricaElements of the trigonometry of traingle sphericalGeometria esféricaTeorema de GirardTrigonometria nos triângulos esféricosSpherical geometryGirard's theoremTrigonometry in the spherical trianglesCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICAThe main objective of this work was to study in triangles constructed on a spherical surface, versions of known results of the plane euclidean geometry and trigonometry in plans triangles. Initially it presents the fundamental concepts of spherical geometry and some elements of spherical triangular trigonometry. For this, begins with a brief review of some of these results and also with some definitions of plane geometry required for the construction of spherical geometry results. That done, are build, in a spherical triangle, versions for the law of sines, law of cosines and other results of the plane triangular trigonometry. Was also seen is the theorem of Girard, where can study the area of a triangle built on the surface of a sphere of radius R and the sum of its internal angles, which is not constant unlike what occurs in triangles plans built on disc of radius r. The Pythagorean theorem is not true in this environment and a counter-example will be presented. Throughout the text will be presented some examples with the use of trigonometric relations, as well as some elementary concepts of geographical coordinates and practical applications of spherical trigonometry in aviation and geography. Finally it is observed that this work strongly uses the mathematics of basic education, facilitating the understanding of the said theory, of students and teachers of basic education, as well as of the professionals who use math.O principal objetivo deste trabalho foi estudar, em triângulos construídos sobre uma superfície esférica, versões para resultados conhecidos da geometria euclidiana plana e da trigonometria nos triângulos planos. Inicialmente apresentam-se os conceitos fundamentais da geometria esférica e alguns elementos de trigonometria triangular esférica. Para isso, iniciou-se com uma breve revisão de alguns desses resultados e também com algumas definições da geometria plana necessárias para a construção de resultados da geometria esférica. Feito isso, foram construídas, em um triˆangulo esférico, versões para a lei dos senos, a lei dos cossenos e outros resultados da trigonometria triangular plana. Também foi visto o Teorema de Girard, onde pode-se estudar a área de um triângulo construído sobre a superfície de uma esfera de raio R e a soma de seus ângulos internos, que ao contrário do que ocorre nos triângulos planos inscritos em um círculo de raio r, não é constante. Foi apresentado um contraexemplo, neste ambiente, em que o famoso teorema de Pitágoras não vale. Ao longo do texto são apresentados alguns exemplos com a utilização das relações trigonométricas estudadas, bem como alguns conceitos elementares de coordenadas geográficas e aplicações práticas da trigonometria esférica na aviação e na geografia. Finalmente, observa-se que esse trabalho utiliza fortemente a matemática do Ensino Básico, facilitando assim a compreensão e o acesso de alunos e professores do Ensino Médio, bem como profissionais que fazem uso da matemática.Agência 1Universidade Federal de RoraimaBrasilPRPPG - Pró-reitoria de Pesquisa e Pós-GraduaçãoPROFMAT - Programa de Mestrado Nacional Profissional em MatemáticaUFRROliveira, Joselito dehttp://lattes.cnpq.br/7059770609022356Santos, Rodson da Silva2022-04-19T14:36:26Z20222022-04-19T14:36:26Z2014info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesishttp://repositorio.ufrr.br:8080/jspui/handle/prefix/728porAttribution-NonCommercial-ShareAlike 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-sa/3.0/br/info:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFRRinstname:Universidade Federal de Roraima (UFRR)instacron:UFRR2022-04-23T05:00:16Zoai:repositorio.ufrr.br:prefix/728Repositório InstitucionalPUBhttp://repositorio.ufrr.br:8080/oai/requestangelsenhora@gmail.comopendoar:2022-04-23T05:00:16Repositório Institucional da UFRR - Universidade Federal de Roraima (UFRR)false |
dc.title.none.fl_str_mv |
Elementos de trigonometria triangular esférica Elements of the trigonometry of traingle spherical |
title |
Elementos de trigonometria triangular esférica |
spellingShingle |
Elementos de trigonometria triangular esférica Santos, Rodson da Silva Geometria esférica Teorema de Girard Trigonometria nos triângulos esféricos Spherical geometry Girard's theorem Trigonometry in the spherical triangles CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
title_short |
Elementos de trigonometria triangular esférica |
title_full |
Elementos de trigonometria triangular esférica |
title_fullStr |
Elementos de trigonometria triangular esférica |
title_full_unstemmed |
Elementos de trigonometria triangular esférica |
title_sort |
Elementos de trigonometria triangular esférica |
author |
Santos, Rodson da Silva |
author_facet |
Santos, Rodson da Silva |
author_role |
author |
dc.contributor.none.fl_str_mv |
Oliveira, Joselito de http://lattes.cnpq.br/7059770609022356 |
dc.contributor.author.fl_str_mv |
Santos, Rodson da Silva |
dc.subject.por.fl_str_mv |
Geometria esférica Teorema de Girard Trigonometria nos triângulos esféricos Spherical geometry Girard's theorem Trigonometry in the spherical triangles CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
topic |
Geometria esférica Teorema de Girard Trigonometria nos triângulos esféricos Spherical geometry Girard's theorem Trigonometry in the spherical triangles CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA |
description |
The main objective of this work was to study in triangles constructed on a spherical surface, versions of known results of the plane euclidean geometry and trigonometry in plans triangles. Initially it presents the fundamental concepts of spherical geometry and some elements of spherical triangular trigonometry. For this, begins with a brief review of some of these results and also with some definitions of plane geometry required for the construction of spherical geometry results. That done, are build, in a spherical triangle, versions for the law of sines, law of cosines and other results of the plane triangular trigonometry. Was also seen is the theorem of Girard, where can study the area of a triangle built on the surface of a sphere of radius R and the sum of its internal angles, which is not constant unlike what occurs in triangles plans built on disc of radius r. The Pythagorean theorem is not true in this environment and a counter-example will be presented. Throughout the text will be presented some examples with the use of trigonometric relations, as well as some elementary concepts of geographical coordinates and practical applications of spherical trigonometry in aviation and geography. Finally it is observed that this work strongly uses the mathematics of basic education, facilitating the understanding of the said theory, of students and teachers of basic education, as well as of the professionals who use math. |
publishDate |
2014 |
dc.date.none.fl_str_mv |
2014 2022-04-19T14:36:26Z 2022 2022-04-19T14:36:26Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://repositorio.ufrr.br:8080/jspui/handle/prefix/728 |
url |
http://repositorio.ufrr.br:8080/jspui/handle/prefix/728 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-ShareAlike 3.0 Brazil http://creativecommons.org/licenses/by-nc-sa/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NonCommercial-ShareAlike 3.0 Brazil http://creativecommons.org/licenses/by-nc-sa/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Roraima Brasil PRPPG - Pró-reitoria de Pesquisa e Pós-Graduação PROFMAT - Programa de Mestrado Nacional Profissional em Matemática UFRR |
publisher.none.fl_str_mv |
Universidade Federal de Roraima Brasil PRPPG - Pró-reitoria de Pesquisa e Pós-Graduação PROFMAT - Programa de Mestrado Nacional Profissional em Matemática UFRR |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFRR instname:Universidade Federal de Roraima (UFRR) instacron:UFRR |
instname_str |
Universidade Federal de Roraima (UFRR) |
instacron_str |
UFRR |
institution |
UFRR |
reponame_str |
Repositório Institucional da UFRR |
collection |
Repositório Institucional da UFRR |
repository.name.fl_str_mv |
Repositório Institucional da UFRR - Universidade Federal de Roraima (UFRR) |
repository.mail.fl_str_mv |
angelsenhora@gmail.com |
_version_ |
1802112040502820864 |