Euklidska, hiperbolička i sferna trigonometrija

master's thesis

• Author: Iva Kavčić
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-03-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog diplomskog rada je dokazati osnovne relacije između mjera kutova i duljina stranica trokuta u euklidskoj i hiperboličkoj ravnini te na površini sfere. Rad započinje povijesnim pregledom razvoja trigonometrije, od samih početaka u 18. st. pr. Kr. kad se razvila iz praktičnih potreba vezanih za graditeljstvo, astronomiju i navigaciju do danas kad je to zasebna matematička disciplina. Slijede definicije i svojstva trigonometrijskih funkcija sinus, kosinus, tangens i kotangens te njihovih inverznih funkcija. Potom proučavamo sličnosti i razlike između trigonometrijskih relacija u tri različita modela: modelu euklidske ravnine, modelu hiperboličke ravnine te modelu sfere. Svaki od modela najprije definiramo i iskazujemo svojstva istog, zatim promatramo trigonometriju u modelu. Naposljetku dolazi neutralni sinusov teorem koji vrijedi u svakom od promatranih modela.

Abstract (english)

The aim of this graduate thesis is to prove basic relations between angle measures and edge lengths of a triangle in Euclidean plane, hyperbolic plane and on the surface of the sphere. The thesis begins with a historical overview of trigonometry, from the very beginnings in the 18th century BC when it developed from practical needs and was used in engineering distinct, astronomy and navigation, until today when it is a mathematical discipline. Next are definitions and properties of trigonometric functions sine, cosine, tangent, cotangent and their inverse functions. Thereafter, we analyzed similarities and differences between trigonometric relationships in three different models: the Euclidean plane model, the hyperbolic plane model and the sphere model. Firstly we defined each model, declared properties and then we analyzed trigonometry of it. Finally, we state the neutral law of sines which is valid in each of the models.

• Keywords: trokut; euklidska ravnina; hiperbolička ravnina; površina sfere; trigonometrija; trigonometrijske funkcije
• Keywords (english): triangle; Euclidean plane; hyperbolic plane; surface of the sphere; trigonometry; trigonometric functions
• Language: croatian
• URN:NBN: urn:nbn:hr:217:572876
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-08-31 12:27:16