Trigonometric Proofs of the Pythagorean Theorem and Identity

Although I am a computer scientist, my research in geometric modeling and visualization frequently veered into geometry, in particular in Euclidean and algebraic geometry and topology. So, I believe that I know something in the mentioned geometric areas. Throughout the years, some of my friends and collegues in math were in two camps, one believing firmly that Loomis was right while the other was either in doubt or rejecting this "belief." This is very much like the angle trisection situation. Some or actually most or even all people in the math community know that trisecting an arbitrary angle with unmarked ruler and compass is impossible; however, some people still published short articles and books showing they successfully trisected an arbitrary angle. You may find this 2024 edition of Construction of Angle Trisection: An addition in Euclidean Geometry and this 1999 edition of New Theory of Trisection: Solved the Most Difficult Math Problem for Centuries in the Hisotry of Mathematics on Amazon. If you are interested in those interesting and (of course) incorrect accounts of the angle trisectors, please consult Underwoord Dudley's book The Trisectors. If you are interested in a little deeper discussion you may like Anthony Rubino's book The Trisection of Angles or Nicholas D. Kazarinoff's book Ruler and the Round: Classic Problems in Geometric Constructions. In 1897, Felix Kelin wrote a well-known little book Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of an Angle, the Quadrature of the Circle, which can also be found on the web.

Get back to our focus: Is proving the Pythagorean Theorem using trigonometry really impossible? Please note that we are talking about whether it is impossible or not rather than a belief. Some may argue that what Loomis belived is a way of math deveopment or math teaching because it has been the approach in the math education community that the Pythagorean Theorem is used to establish all the fundamental formulae in trigonometry. This argument does not make any sense to me. For example, Euclidean geometry had been taught for centuries; however, many people rejected the non-Euclidean paradigm when it appeared at the beginning of the 19th century. Please refer to a 1906 interesting book by Boberto Bonola Non-Euclidean Geometry for the details. Bonola's book is also available on the web. Consequently, we want to figure out whether a non-standard (or different) approach is possible. More precisely, here is the task: Establishing all (or at least most of) the fundamental formulae in trigonometry without using the Pythagorean Theorem or Identity. This site shows you that it is possible!

The following topics (or pages) are available. Please note that some pages are still under construction. If you wish to read a long expository essay on the same topics, please click here to download a PDF copy. Please also note that this expository essay is being updated from time to time when new information will become available and may not be fully sync-ed with this site.

• What is an "Impossible" Proof?
  1. The Origin of the "Impossible" Proof Claim
• Classical Results
  1. Euclid Could Be the First One Who Proved the Pythagorean Theorem Using Trigonometry
  2. The Angle Difference and Angle Sum Identities for Sine and Cosine and Zimba's 2009 Work
  3. Friedrich Schur' 1899 Analytic Geometry Book
  4. J. Versluys' 1914 Book
  5. Coordinate Axes Rotation
• Other Fundamental Trigonomtric Formulae
  1. Other Trigonometric Fundamental Formulae
  2. The Sum-to-Product and Product-to-Sum Identities (and a Very Simple Proof of the Pythagorean Identity)
  3. The Angle Sum and Angle Difference Identities for the Tangent Function
  4. Lengvárszky's Proof of the Pythagorean Theorem (2018)
• Recent Works
  1. Rachel I. Mason's 1996 Proof (Rachel Was a High School Girl in the Class of 1995)
  2. Paul Maiorano's 1998 Analytic Geometry Proof
  3. Luzia's 2015 First Proof
• Some Calculus Needed!