Trigonometry in elliptic geometry II | Universal Hyperbolic Geometry 42 | NJ Wildberger

We explain how to prove the main laws of rational elliptic/spherical trigonometry, using basic linear algebra involving dot products, 3x3 matrices and their inverses. The main object of interest is an elliptic triangle, and in the previous video, UHG41, we stated the basic laws: the Spread law, the Cross law and the Dual cross law. Here we prove them. These are among the most important formulas in geometry, and hence in mathematics. As easy consequences of these laws, we prove Thales' theorem, Pythagoras' theorem, the Dual Pythagoras' theorem, and the Triple Quad Formula in this elliptic or projective setting. These are complementary, and closely related to, corresponding results in (planar) affine trigonometry. Remarkably, all these formulas have exactly the same form as they do in Universal Hyperbolic Geometry. Video Content: 00:00 Introduction 5:48 Hyperbolic rational trigonometry laws 9:33 Polarity of elliptic triangles 18:40 Rewriting quadrances and spreads 23:30 What about the cross law? 29:11 Dual cross law 34:49 Dual Pythagoras' theorem ************ Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF... Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos)    • Elementary Mathematics Explained (K-6)   (40 videos)    • Ancient Mathematics and insights of Howard...   (7 videos)    • Wild West Banking: A mathematician goes We...   (7 videos)    • Playing Go: the ancient oriental board game   (19 videos)    • Maths and Music   (21 videos)    • Year 9 Mathematics (review fractions, deci...   (10 videos)    • Wild Trig: An introduction to Rational Tri...   (94 videos) Wild Egg Maths Playlists:    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)