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Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J. Wildberger, currently an associate professor of mathematics at the University of New South Wales. His ideas are set out in his 2005 book (''Divine Proportions: Rational Trigonometry to Universal Geometry'' ). According to New Scientist, part of his motivation for an alternative to traditional trigonometry was to avoid some problems that occur when infinite series are used in mathematics. Rational trigonometry avoids direct use of transcendental functions like sine and cosine by substituting their squared equivalents.〔"(Infinity's end: Time to ditch the never-ending story? )" by Amanda Gefter, New Scientist, 15 August 2013〕 Wildberger draws inspiration from mathematicians predating Georg Cantor's infinite set-theory, like Gauss and Euclid, who he claims were far more wary of using infinite sets than modern mathematicians.〔〔For Wildberger's views on the history of infinity, see the Gefter New Scientist article, but also see Wildberger's History of Mathematics and Math Foundations lectures, University of New South Wales, circa 2009–2014 in more than 120 videos and lectures, available online @youtube〕 To date, rational trigonometry is largely unmentioned in mainstream mathematical literature. ==The approach== Rational trigonometry follows an approach built on the methods of linear algebra to the topics of elementary (high school level) geometry. Distance is replaced with its squared value (quadrance) and 'angle' is replaced with the squared value of the usual sine ratio (spread) associated to either angle between two lines. (Spread also corresponds to a scaled form of the inner product between the lines taken as vectors). The three main laws in trigonometry: Pythagoras's theorem, the sine law and the cosine law, given in rational (squared) form, are augmented by two further laws: the triple quad formula (relating the quadrances of three collinear points) and the triple spread formula (relating the spreads of three concurrent lines), giving the five main laws of the subject. Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with ''a point'' defined as an ordered pair of rational numbers :: and ''a line'' :: as a general linear equation with rational coefficients and . By avoiding calculations that rely on square root operations giving only ''approximate'' distances between points, or standard trigonometric functions (and their inverses), giving only truncated polynomial ''approximations'' of angles (or their projections) geometry becomes entirely algebraic. There is no assumption, in other words, of the existence of real number solutions to problems, with results instead given over the field of rational numbers, their algebraic field extensions, or finite fields. Following this, it is claimed, makes many classical results of Euclidean geometry applicable in ''rational'' form (as quadratic analogs) over any field not of characteristic two. The book ''Divine Proportions'' shows the application of calculus using Rational Trig functions, including 3-d volume calculations. It also deals with rational trig application to situations involving irrationals, such as the proof that Platonic Solids all have rational 'spreads' between their faces.〔See ''Divine Proportions'' for numerous examples of calculus done with Rational Trig functions, as well as problems involving the application of Rational Trig to situations containing irrationals〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rational trigonometry」の詳細全文を読む スポンサード リンク
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