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| | Wilber Berryhill is the title his mothers and fathers gave him and he completely digs that name. Some time ago he chose to reside in North Carolina and he doesn't strategy on changing it. For years she's been working as a travel agent. To play lacross is some thing I really enjoy performing.<br><br>Feel free to visit my homepage ... love psychic ([http://1.234.36.240/fxac/m001_2/7330 http://1.234.36.240/fxac/m001_2/7330]) |
| {{One source| date=January 2009}}
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| {{third-party| date=May 2013}}
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| {{Refimprove| date=January 2013}}
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| }}
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| '''Rational trigonometry''' is a proposed reformulation of [[metric space|metrical]] [[plane geometry|planar]] and [[solid geometry|solid geometries]] (which includes [[trigonometry]]), by Norman J. Wildberger, an associate professor at [[UNSW]], 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 function]]s like [[sine]] and [[cosine]] by substituting their squared equivalents.<ref name=gefter1>"[http://www.newscientist.com/article/mg21929300.700-infinitys-end-time-to-ditch-the-neverending-story.html Infinity's end: Time to ditch the never-ending story?]" by Amanda Gefter, New Scientist, 15 August 2013</ref> Wildberger draws inspiration from mathematicians predating [[Georg Cantor]]'s [[Georg Cantor#Set theory|infinite set-theory]], like Gauss and Euclid, whom he claims were far more wary of using infinite sets than modern mathematicians.<ref name=gefter1/><ref>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 209/2010, available online @youtube</ref> To date, rational trigonometry is largely unmentioned in mainstream mathematical literature. Early claims by the author that rational trigonometry requires fewer steps to solve typical problems and avoids logical inconsistencies asscociated with classical trigonometry were subject to dispute by at least one other professional mathematician.<ref name="web.maths.unsw.edu.au">http://web.maths.unsw.edu.au/~norman/papers/TrigComparison.pdf</ref>
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| (See [[#Notability and criticism]] below.)
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| ==The approach==
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| 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 [[vector (geometric)|vector]]s). The three main laws in trigonometry: [[Pythagoras' theorem]], the [[sine law]] and the [[cosine law]], given in rational (squared) form, are augmented by two further laws: the [[Rational trigonometry#Triple quad formula|triple quad formula]] (relating the quadrances of three collinear points) and the [[Rational trigonometry#Triple spread formula|triple spread formula]] (relating the spreads of three concurrent lines), giving the [[Rational trigonometry#Laws of rational trigonometry|five main laws]] of the subject.{{citation needed|date=November 2013}}
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| Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with ''a point'' defined as an ordered pair of [[rational number]]s
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| ::<math>(x,y)</math>
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| and ''a line''
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| ::<math>ax + by + c = 0,</math>
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| as a general [[linear equation]] with rational coefficients <math>a, b</math> and <math>c</math>.
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| 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 extension]]s, or [[finite field]]s. Following this, it is claimed, makes many [[Mathematical theorem|classical results]] of [[Euclidean geometry]] applicable in ''rational'' form (as quadratic analogs) over any field not of [[Characteristic (algebra)|characteristic]] two.{{citation needed|date=November 2013}}
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| 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.<ref>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</ref>
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| == Quadrance ==
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| {{unreferenced section|date=November 2013}}
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| Quadrance (and distance as its square root) both measure separation of points in Euclidean space.<ref name="horizons"/> Following Pythagoras' theorem, the quadrance of two points <math>A_1=(x_1,y_1)</math> and <math>A_2=(x_2,y_2)</math> in a plane is therefore defined as the sum of squares of differences in the <math>x</math> and <math>y</math> coordinates:
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| : <math>Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2.\,</math>
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| Unlike vector addition of distances with line segments, adding the quadrances of two vectors to obtain their combined or resultant magnitude always entails finding the third leg of the associated triangle they form, even in case of collinear segments (a degenerate triangle) where the same calculation made with distance-vectors simplifies to addition. In effect, the [[triangle inequality]] is modified under the conditions of rational trigonometry to the equivalent of Pythagoras' theorem.
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| == Spread ==
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| [[Image:Spread as ratio.svg|thumb|right|Suppose ''ℓ''<sub> 1</sub> and ''ℓ''<sub> 2</sub> intersect at the point <var>A</var>. Let <var>C</var> be the foot of the perpendicular from <var>B</var> to ''ℓ''<sub> 2</sub>. Then the spread is ''s'' = ''Q''/''R''.]]
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| Spread gives one measure to the separation of two lines as a single [[dimensionless number]] in the range <math>[0,1]</math> (from ''parallel'' to ''perpendicular'') for Euclidean geometry. It replaces the concept of angle but has several differences from angle, discussed in the section below. Spread can have several interpretations.
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|
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| *''Trigonometric'' (most elementary): it is the sine-ratio for the quadrances in a right triangle and therefore equivalent to the square of the sine of the angle.<ref name="horizons"/>
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| *''Vector:'' as a rational function of the directions (practically, the slopes) of a pair of lines where they meet.
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| *''Cartesian:'' as a rational function of the three co-ordinates used to ascribe the two vectors.
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| *''Linear algebra'' (coming from the dot product): a normalized rational function of the square of the [[determinant]] the vectors (or pair of intersecting lines) divided by the product of their quadrances.
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| === Calculating spread ===
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| * '''Trigonometric'''
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| Suppose two lines, ''ℓ''<sub> 1</sub> and ''ℓ''<sub> 2</sub>, intersect at the point <var>A</var> as shown at right. Choose a point ''B'' ≠ ''A'' on ''ℓ''<sub> 1</sub> and let <var>C</var> be the foot of the perpendicular from <var>B</var> to ''ℓ''<sub> 2</sub>. Then the spread <var>s</var> is
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| : <math>s(\ell_1, \ell_2) = \frac{Q(B, C)}{Q(A, B)} = \frac{Q}{R}.</math><ref name="horizons"/>
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| * '''Vector/slope''' (two-variable)
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| Like angle, spread depends only on the relative slopes of two lines (constant terms being eliminated) and spread with parallel lines is preserved. So given two lines whose equations are
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| :<math>a_1x + b_1y= \mathrm{constant}</math> and <math>a_2x + b_2y= \mathrm{constant}</math>
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| we may rewrite them as two lines which meet at the origin <math>(0,0)</math> with equations
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| :<math>a_1x + b_1y= 0</math> and <math>a_2x + b_2y= 0</math>
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| In this position the point <math>(-b_1,a_1)</math> satisfies the first equation and <math>(-b_2,a_2)</math> satisfies the second and the three points <math>(0,0),(-b_1,a_1)</math> and <math>(-b_2,a_2)</math> forming the spread will give three quadrances:
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| :<math>Q_1=(b_1^2+a_1^2),</math>
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| :<math>Q_2=(b_2^2+a_2^2),</math>
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| :<math>Q_1=(b_1- b_2)^2+(a_1-a_2)^2</math>
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| The ''cross law'' – see below – in terms of spread is:
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| :<math>1-s = \frac{(Q_1+Q_2-Q_3)^2}{4Q_1Q_2}.\,</math>
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| which becomes:
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| :<math>1-s=\frac{(a_1^2+a_2^2+b_1^2+b_2^2-(b_1-b_2)^2-(a_1-a_2)^2)^2}{4(a_1^2+b_1^2)(a_2^2+b_2^2)}\,</math>
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| This simplifies, in the numerator, to: <math>(2a_1a_2+2b_1b_2)^2,</math> giving:
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| :<math>1-s=\frac{(a_1a_2+b_1b_2)^2}{(a_1^2+b_1^2)(a_2^2+b_2^2)}\,</math>
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| Then, using the important identity due to [[Brahmagupta–Fibonacci identity|Fibonacci:]] <math>(a_2b_1-a_1b_2)^2+(a_1a_2+b_1b_2)^2=(a_1^2+b_1^2)(a_2^2+b_2^2),</math>
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| the standard expression for spread in terms of slopes (or directions) of two lines becomes:
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| : <math>s = \frac{(a_1 b_2 - a_2 b_1)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}.\,</math>
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| * '''Cartesian''' (three-variable)
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| This replaces <math>(-b_1,a_1)</math> with <math>(x_1,y_1), (-b_2,a_2)</math> with <math>(x_2,y_2)</math> and the origin <math>(0,0)</math> (as the point of intersection of two lines) with <math>(x_3,y_3)</math> in the previous result:
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| : <math>s = \frac{((y_1 - y_3) (x_2 - x_3) - (y_2 - y_3) (x_1 - x_3))^2}{((y_1 - y_3)^2 + (x_1 - x_3)^2)((y_2 - y_3)^2 + (x_2 - x_3)^2)}.\,</math>
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| === Spread compared to angle ===
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| [[Image:Spread between two lines.svg|thumb|right|The spread of two lines can be measured in four equivalent positions.]]
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| Unlike angle, which can define a relationship between ''rays'' emanating from a point, by a [[Angle#Measuring angles|circular measure parametrization]], and where a pair of lines can be considered four pairs of rays, forming four angles, 'spread' is a fundamental concept in rational trigonometry, describing ''two lines'' by a single measure of a rational function (see above).<ref name="horizons"/> Being equivalent to the ''square'' of a sine, the spread of both an angle and its [[supplementary angle]] are equal.
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| {| style="border-spacing: 1em 0; text-align: center"
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| |-
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| ! [[Degree (angle)|Degree]] !! [[Radian]] !! Spread
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| |-
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| | 0 || 0 || 0
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| |-
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| | 30 || (1/6)π || 1/4
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| |-
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| | 45 || (1/4)π || 1/2
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| |-
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| | 60 || (1/3)π || 3/4
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| |-
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| | 90 || (1/2)π || 1
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| |-
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| | 120 || (2/3)π || 3/4
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| |-
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| | 135 || (3/4)π || 1/2
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| |-
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| | 150 || (5/6)π || 1/4
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| |-
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| | 180 || π || 0
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| |}
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| Spread is not proportional, however, to the separation between lines as angle would be; with spreads of 0, 1/4, 1/2, 3/4, and 1 corresponding to unevenly spaced angles 0, 30, 45, 60 and 90 degrees.
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| Instead, (recalling the supplementary property) two equal, co-terminal spreads determine a third spread, whose value will be a solution to the triple spread formula for a triangle (or three concurrent lines) with spreads of <math>s, s</math> and <math>r</math>:
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| ::<math>(2s + r)^2 = 2(2s^2 + r^2) + 4s^2r</math>
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| ::<math>4s^2 + 4sr + r^2 = 4s^2 + 2r^2 + 4s^2r</math>
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| giving the quadratic polynomial (in <math>s</math>):
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| ::<math>r^2 + 4s^2r - 4sr = 0</math>
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| ::<math>r^2 - 4s(1-s)r = 0</math>
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| and solutions
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| ::<math>r = 0</math> (trivial) or
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| ::<math>r = 4s(1-s)</math>
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| This is equivalent to the trigonometric identity :
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| ::<math>\sin^2 2\theta=4\sin^2\theta(1-\sin^2\theta)</math>
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| of the angles <math>\theta,\theta</math> and <math>2\theta</math> of a triangle, using
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| ::<math>S_2(s)=S_2(\sin^2\theta)=\sin^2(2\theta)=r(s)</math>
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| to denote a ''second'' [[spread polynomial]] in <math>s</math>.
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| Tripling spreads likewise involves a triangle (or three concurrent lines) with one spread of <math>r</math> (the previous solution), one spread of <math>s</math> and obtaining a ''third'' spread polynomial, <math>t</math> in <math>s</math>. This turns out to be:
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| ::<math>S_3(s)=s(3-4s)^2=t(s)</math>
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| Further multiples of any basic spread of lines can be generated by continuing this process using the triple spread formula.
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| Every ''multiple'' of a spread which is rational will thus be rational, but the converse does not apply. For example, by the [[half-angle formula]], two lines meeting at a 15° (or 165°) angle have spread of:
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| ::<math>\sin^2 (30^\circ/2)=(1-\cos 30^\circ)/2=(1 - \sqrt{3}/2)/2=(2-\sqrt{3})/4 \approx 0.0667.</math>
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| and thus exists by algebraic extension of the rational numbers.
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| ==Spread polynomials==
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| As seen for double and triple spreads, the ''n''th multiple of any spread, <math>s</math> gives a polynomial in that spread, denoted <math>S_n(s)</math>, as one solution to the triple spread formula.
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| In the conventional language of [[circular functions]], these ''n''th-degree ''spread polynomials'', for ''n'' = 0, 1, 2, ..., can be characterized by the identity:{{Citation needed|date=November 2013}}
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| :<math>\sin^2(n\theta) = S_n(\sin^2\theta).\,</math>
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| ===Identities===
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| ====Explicit formulas====
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| :<math>S_n(s) = s\sum_{k=0}^{n-1} {n \over n - k} {2n-1-k \choose k} (-4s)^{n-1-k}.</math> (S. Goh){{Citation needed|date=March 2009}}
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| :<math>S_n(s) = \frac{1}{2} - \frac{1}{4} \left ( 1-2s+2 \sqrt {s^2 -s} \right )^{n} - \frac{1}{4} \left ( 1-2s-2 \sqrt {s^2 -s} \right )^{2}.</math> (M. Hovdan)
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| :<math>S_n(s) = - \frac{1}{4} \left ( \left ( \sqrt {1 -s} +i\sqrt {s} \right )^{2n}-1 \right )^{2}\left ( \sqrt {1 -s} -i\sqrt {s} \right )^{2n}.</math> (M. Hovdan)
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| From the definition it immediately follows that
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| :<math>S_n(s) = \sin^2\left(n\arcsin\left(\sqrt{s}\right)\right).</math> {{Citation needed|date=November 2013}}
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| ====Recursion formula====
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| :<math>S_{n+1}(s) = 2(1-2s) S_n(s) - S_{n-1}(s) + 2s.\,</math> {{Citation needed|date=November 2013}}
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| ====Relation to Chebyshev polynomials====
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| The spread polynomials are related to the [[Chebyshev polynomials]] of the first kind, ''T''<sub>''n''</sub> by the identity
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| :<math>1 - 2S_n(s) = T_n(1 - 2s).\,</math>
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| This implies
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| :<math>S_n(s) = {1 - T_n(1 - 2s) \over 2} = 1 - T_n\left(\sqrt{1-s}\right)^2.</math>
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| The second equality above follows from the identity
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| :<math>2T_n(x)^2 - 1 = T_{2n}(x) \, </math>
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| on Chebyshev polynomials. {{Citation needed|date=November 2013}} | |
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| ====Composition====
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| The spread polynomials satisfy the composition identity
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| :<math>S_n(S_m(s)) = S_{nm}(s).\,</math> {{Citation needed|date=November 2013}}
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| ====Coefficients in finite fields====
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| When the coefficients are taken to be members of the [[finite field]] ''F''<sub>''p''</sub>, then the sequence { ''S''<sub>''n''</sub> }<sub>''n'' = 0, 1, 2, ...</sub> of spread polynomials is periodic with period (''p''<sup>2</sup> − 1)/2. In other words, if ''k'' = (''p''<sup>2</sup> − 1)/2, then ''S''<sub>''n'' + ''k''</sub> = ''S''<sub>''n''</sub>, for all ''n''. {{Citation needed|date=November 2013}}
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| ====Orthogonality====
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| When the coefficients are taken to be [[real number|real]], then for ''n'' ≠ ''m'', we have
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| :<math>\int_0^1 \left(S_n(s) - {1 \over 2}\right) \left(S_m(s) - {1 \over 2}\right){ds \over \sqrt{s(1-s)}}=0.</math> {{Citation needed|date=November 2013}}
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| For ''n'' = ''m'', the integral is ''π''/8 unless ''n'' = ''m'' = 0, in which case it is ''π''/4. {{Citation needed|date=November 2013}}
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| ====Generating functions====
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| he ordinary [[generating function]] is
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| :<math>\sum_{n=1}^\infty S_n(s)x^n = {sx(1+x) \over (1-x)^3 + 4sx(1-x)}.</math> {{Citation needed|date=November 2013}}
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| The exponential generating function is
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| :<math>\sum_{n=1}^\infty {S_n(s)\over n!} x^n = {1 \over 2} e^x \left [ 1-e^{-2sx} \cos\left (2x \sqrt{s(1-s)}\right )\right ] .</math> {{Citation needed|date=November 2013}}
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| ====Differential equation====
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| ''S''<sub>''n''</sub>(''s'') satisfies the second order linear non-homogenous differential equation
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| :<math>s(1-s)y'' + (1/2-s)y' + n^2(y-1/2) = 0.\,</math> {{Citation needed|date=November 2013}}
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| ==Spread Periodicity Theorem==
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| For any [[integer]] ''s'' and any [[prime]] ''p'', there is a [[natural number]] ''m'' such that ''S''<sub>''n''</sub>''(s)'' is divisible by ''p'' precisely when ''m'' divides ''n''. This number ''m'' is a divisor of either ''p-1'' or ''p+1''. The proof of this number theoretical property was first given in a paper by Shuxiang Goh and N. J. Wildberger.<ref>{{Cite journal | arxiv = 0911.1025 | postscript = | title = Spread polynomials, rotations and the butterfly effect | author = Shuxiang Goh, N. J. Wildberger | date = November 5, 2009}}</ref> It involves considering the projective analogue to [[Rational trigonometry#Quadrance|quadrance]] in the [[Projective line#For a finite field|finite projective line]] '''P'''<sup>1</sup>(''F''<sub>''p''</sub>). | |
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| ===Table of spread polynomials, with factorizations===
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| The first several spread polynomials are as follows:
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| :<math>
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| \begin{align}
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| S_0(s) & = 0 \\[10pt]
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| S_1(s) & = s \\[10pt]
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| S_2(s) & = 4s-4s^2 \\
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| & = 4s(1-s) \\[10pt]
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| S_3(s) & = 9s-24s^2+16s^3 \\
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| & = s(3-4s)^2 \\[10pt]
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| S_4(s) & = 16s-80s^2+128s^3-64s^4 \\
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| & = 16s(1-s)(1-2s)^2 \\[10pt]
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| S_5(s) & = 25s-200s^2+560s^3-640s^4+256s^5 \\
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| & = s(5-20s+16s^2)^2 \\[10pt]
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| S_6(s) & = 36s-420s^2+1792s^3-3456s^4+3072s^5-1024s^6 \\
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| & = 4s(1-s)(1-4s)^2(3-4s)^2 \\[10pt]
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| S_7(s) & = 49s-784s^2+4704s^3-13440s^4+19712s^5-14336s^6+4096s^7 \\
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| & = s(7-56s+112s^2-64s^3)^2 \\[10pt]
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| S_8(s) & = 64s-1344s^2+10752s^3-42240s^4+90112s^5-106496s^6 \\
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| & {} \qquad + 65536s^7-16384s^8 \\
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| & = 64s(s-1)(1-2s)^2(1-8s+8s^2)^2 \\[10pt]
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| S_9(s) & = 81s - 2160s^2 + 22176s^3 - 114048s^4 + 329472s^5 - 559104s^6 \\
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| & {} \qquad + 552960s^7 - 294912s^8 + 65536s^9 \\
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| & = s(-3+4s)^2(-3+36s-96s^2+64s^3)^2 \\[10pt]
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| S_{10}(s) & = 100s - 3300s^2 + 42240s^3 - 274560s^4 + 1025024s^5 \\
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| {} & \qquad - 2329600s^6 + 3276800s^7 - 2785280s^8 + 1310720s^9 - 262144s^{10} \\
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| & = 4s(1-s)(5 - 20s+16s^2)^2(1-12s+16s^2)^2\\[10pt]
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| S_{11}(s) & = 121s - 4840s^2 + 75504s^3 - 604032s^4 + 2818816s^5 \\
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| {} & \qquad -8200192s^6 + 15319040s^7 - 18382848s^8 + 13697024s^9 -5767168s^{10} + 1048576s^{11}\\
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| & = s(11 -220s + 1232s^2 -2816s^3 +2816s^4 -1024s^5)^2
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| \end{align}
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| </math>
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| == Laws of rational trigonometry ==
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| Wildberger states that there are five basic laws in rational trigonometry. He also states, correctly, that these laws can be verified using high-school level mathematics. Some are equivalent to standard trigonometrical formulae with the variables expressed as quadrance and spread.<ref name="horizons"/>
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| In the following five formulas, we have a triangle made of three points ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, . The spreads of the angles at those points are ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, , and ''Q''<sub>1</sub>, ''Q''<sub>2</sub>, ''Q''<sub>3</sub>, are the quadrances of the triangle sides opposite ''A''<sub>1</sub>, ''A''<sub>2</sub>, and ''A''<sub>3</sub>, respectively. As in classical trigonometry, if we know three of the six elements ''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, , ''Q''<sub>1</sub>, ''Q''<sub>2</sub>, ''Q''<sub>3</sub>, and these three are not the three ''s'', then we can compute the other three.
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| === Triple quad formula ===
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| The three points ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, are [[collinear]] if and only if:
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| : <math>(Q_1 + Q_2 + Q_3)^2 = 2(Q_1^2 + Q_2^2 + Q_3^2).\,</math>
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| It can either be proved by [[analytic geometry]] (the preferred means within rational trigonometry) or derived from [[Heron's formula]], using the condition for collinearity that the triangle formed by the three points has zero area.
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| {{Hidden begin|title=Proof (click at right to show/hide)}}
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| [[Image:TripleQuadProofIllustration.png|thumbnail|300px|right|Illustration of nomenclature used in the proof.]]
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| The line <math>AB\,</math> has the general form:
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| :<math>ax + by + c = 0\,</math>
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| where the (non-unique) parameters ''a'', ''b'' and ''c'', can be expressed in terms of the coordinates of points ''A'' and ''B'' as:
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| :<math>a = A_y - B_y\,</math>
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| :<math>b = B_x - A_x\,</math>
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| :<math>c = A_xB_y - A_yB_x\,</math>
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| so that, everywhere on the line:
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| :<math>(A_y - B_y)x + (B_x - A_x)y + (A_xB_y - A_yB_x) = 0.\,</math>
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| But the line can also be specified by two simultaneous equations in a parameter ''t'', where ''t'' = 0 at point ''A'' and ''t'' = 1 at point ''B'':
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| :<math>x = (B_x - A_x)t + A_x\text{ and }y = (B_y - A_y)t + A_y\,</math>
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| or, in terms of the original parameters:
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| :<math>x = bt + A_x\,</math> and <math>y = -at + A_y.\,</math>
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| If the point ''C'' is collinear with points ''A'' and ''B'', there exists some value of ''t'' (for distinct points, not equal to 0 or 1), call it ''λ'', for which these two equations are simultaneously satisfied at the coordinates of the point ''C'', such that:
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| :<math>C_x = b\lambda\ + A_x</math> and <math>C_y = -a\lambda\ + A_y.\,</math>
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| Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of ''λ'':
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| :<math>\begin{align}Q(AB) & \equiv (B_x - A_x)^2 + (B_y - A_y)^2 \\ & = b^2 + (-a)^2 \\ & = a^2 + b^2\end{align}</math>
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| :<math>\begin{align}Q(BC) & \equiv (C_x - B_x)^2 + (C_y - B_y)^2 \\ & = ((b\lambda\ + A_x) -B_x)^2 + ((-a\lambda\ + A_y) - B_y)^2 \\ & = (b\lambda\ + (A_x -B_x))^2 + (-a\lambda\ + (A_y - B_y))^2\\ & = (b\lambda\ + (-b))^2 + (-a\lambda\ + a)^2\\ & = b^2(\lambda\ - 1)^2 + a^2(-\lambda\ + 1)^2 \\ & = b^2(\lambda\ - 1)^2 + a^2(\lambda\ - 1)^2\\ & = (a^2 + b^2)(\lambda\ - 1)^2\end{align}</math>
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| :<math>\begin{align}Q(AC) & \equiv (C_x - A_x)^2 + (C_y - A_y)^2 \\ & = ((b\lambda\ + A_x) - A_x)^2 + ((-a\lambda\ + A_y) - A_y)^2 \\ & = (b\lambda\ + A_x - A_x)^2 + (-a\lambda\ + A_y - A_y)^2\\ & = (b\lambda)^2 + (-a\lambda)^2\\ & = b^2\lambda^2 + (-a)^2\lambda^2\\ & = b^2\lambda^2 + a^2\lambda^2\\ & = (a^2 + b^2)\lambda^2\end{align}</math>
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| where use was made of the fact that <math>(-\lambda\ + 1)^2 = (\lambda\ - 1)^2</math>.
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| Substituting these quadrances into the equation to be proved:
| |
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| :<math>(Q(AB) + Q(BC) + Q(AC))^2 = 2(Q(AB)^{2} + Q(BC)^{2} + Q(AC)^{2})\,</math>
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| :<math>((a^2 + b^2) + (a^2 + b^2)(\lambda\ - 1)^2 + (a^2 + b^2)\lambda^2)^2 = 2((a^2 + b^2)^2 + ((a^2 + b^2)(\lambda\ - 1)^2)^2 + ((a^2 + b^2)\lambda^2)^2)\,</math>
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| :<math>(a^2 + b^2)^2(1 + (\lambda\ - 1)^2 + \lambda^2)^2 = 2(a^2 + b^2)^2(1 + ((\lambda\ - 1)^2)^2 + (\lambda^2)^2)\,</math>
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| Now, if <math>A\,</math> and <math>B\,</math> represent distinct points, such that <math>(a^2 + b^2)\,</math> is not zero,
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| we may divide both sides by <math>Q(AB)^2 = (a^2 + b^2)^2\,</math>:
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| :<math>(1 + \lambda^2 -2\lambda\ + 1 + \lambda^2)^2 = 2(1 + (\lambda^2 -2\lambda\ + 1)^2 + \lambda^4)\,</math>
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| :<math>(2\lambda^2 - 2\lambda\ + 2)^2 = 2(1 + \lambda^4 - 2\lambda^3 + \lambda^2 - 2\lambda^3 + 4\lambda^2 - 2\lambda + \lambda^2 - 2\lambda + 1 + \lambda^4)\,</math>
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| :<math>4(\lambda^2 - \lambda\ + 1)^2 = 2(2\lambda^4 - 4\lambda^3 + 6\lambda^2 - 4\lambda + 2)\,</math>
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| :<math>4(\lambda^4 - \lambda^3 + \lambda^2 - \lambda^3 + \lambda^2 - \lambda + \lambda^2 - \lambda + 1) = 4(\lambda^4 - 2\lambda^3 + 3\lambda^2 - 2\lambda + 1)\,</math>
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| :<math>\lambda^4 - 2\lambda^3 + 3\lambda^2 - 2\lambda + 1 = \lambda^4 - 2\lambda^3 + 3\lambda^2 - 2\lambda + 1\,</math>
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| {{hidden end}}
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| === Pythagoras' theorem ===
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| The lines ''A''<sub>1</sub>''A''<sub>3</sub> (of quadrance ''Q''<sub>1</sub>) and ''A''<sub>2</sub>''A''<sub>3</sub> (of quadrance ''Q''<sub>2</sub>) are perpendicular (their spread is 1) if and only if:
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| : <math>Q_1 + Q_2 = Q_3.\,</math>
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| where ''Q''<sub>3</sub> is the quadrance between ''A''<sub>1</sub> and ''A''<sub>2</sub>.
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| This is equivalent to the [[Pythagorean theorem]] (and its converse).
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| There are many classical proofs of [[Pythagoras's theorem|Pythagoras' theorem]]; this one is framed in the terms of rational trigonometry.
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| The ''spread'' of an angle is the square of its [[sine]]. Given the triangle ''ABC'' with a spread of 1 between sides ''AB'' and ''AC'',
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| :<math>Q(AB) + Q(AC) = Q(BC)\,</math>
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| where ''Q'' is the "quadrance", i.e. the square of the distance.
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| {{Hidden begin|title=Proof}}
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| [[Image:PythagoreanTheoremProofIllustration.svg|thumbnail|250px|right|Illustration of nomenclature used in the proof.]]
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| Construct a line ''AD'' dividing the spread of 1, with the point ''D'' on line ''BC'', and making a spread of 1 with ''DB'' and ''DC''. The triangles ''ABC'', ''DBA'' and ''DAC'' are similar (have the same spreads but not the same quadrances).
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| This leads to two equations in ratios, based on the spreads of the sides of the triangle:
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| :<math>s_C = \frac{Q(AB)}{Q(BC)} = \frac{Q(BD)}{Q(AB)} = \frac{Q(AD)}{Q(AC)}.</math>
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| :<math>s_B = \frac{Q(AC)}{Q(BC)} = \frac{Q(DC)}{Q(AC)} = \frac{Q(AD)}{Q(AB)}.</math>
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| Now in general, the two spreads resulting from dividing a spread into two parts, as line ''AD'' does for spread ''CAB'', do not add up to the original spread since spread is a non-linear function. So we first prove that dividing a spread of 1, results in two spreads that do add up to the original spread of 1.
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| For convenience, but with no loss of generality, we orient the lines intersecting with a spread of 1 to the coordinate axes, and label the dividing line with coordinates <math>(x_1, y_1)</math> and <math>(x_2, y_2)</math>. Then the two spreads are given by:
| |
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| :<math>s_1 = \frac{(x_2 - x_2)^2 + (y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}
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| = \frac{(y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2},</math>
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| :<math>s_2 = \frac{(x_2 - x_1)^2 + (y_2 - y_2)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}
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| = \frac{(x_2 - x_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}.</math>
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| Hence:
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| :<math>s_1 + s_ 2 = \frac{(x_2 - x_1)^2 + (y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}
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| = 1.\,</math>
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| So that:
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| :<math>s_C + s_B = 1.\,</math>
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| Using the first two ratios from the first set of equations, this can be rewritten:
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| :<math>\frac{Q(AB)}{Q(BC)} + \frac{Q(AC)}{Q(BC)} = 1.\,</math>
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| Multiplying both sides by <math>Q(BC)</math>:
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| :<math>Q(AB) + Q(AC) = Q(BC).\,</math>
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| [[Q.E.D.]]
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| {{hidden end}}
| |
| | |
| === Spread law ===
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| For any triangle <math>\overline{A_{1} A_{2} A_{3}}</math> with nonzero quadrances:
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| : <math>\frac{s_{1}}{Q_{1}}=\frac{s_{2}}{Q_{2}}=\frac{s_{3}}{Q_{3}}.\,</math>
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| This is the [[law of sines]], just squared.
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| === Cross law ===
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| For any triangle <math>\overline{A_{1} A_{2} A_{3}}</math>,
| |
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| : <math>(Q_1 + Q_2 - Q_3)^2 = 4Q_1 Q_2 (1-s_3).\,</math>
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| This is analogous to the [[law of cosines]]. It is called 'cross law' because
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| <math>(1-s_3)</math>, the square of the cosine of the angle, is called the 'cross'.
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| === Triple spread formula ===
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| For any triangle <math>\overline{A_1 A_2 A_3},</math>
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| : <math>(s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4s_1 s_ 2 s_ 3 .\,</math>
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| This relation can be derived from the formula for the [[Trigonometric identity#Angle sum and difference identities|sine of a compound angle]]: in a triangle (whose three angles sum to 180°) we have,
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| :<math>\sin (a)=\sin (b+c)=\sin (b)\cos (c) + \sin (c)\cos (b)</math>.
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| Equivalently, it describes the relationship between the spreads of three concurrent lines, as spread (like angle) is unaffected when the sides of a triangle are moved parallel to themselves to meet in a common point.
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| Knowing two spreads allows the third spread to be calculated by solving the associated quadratic formula but, because two solutions are possible, further ''triangle spread rules'' must be used to select the appropriate one. (The relative complexity of this process contrasts with the much simpler method of obtaining a supplementary angle of two others.)
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| == Trigonometry over arbitrary fields ==
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| As the laws of rational trigonometry give algebraic (and not transcendental) relations, they apply in generality to algebraic number fields beyond the rational numbers. Specifically, any finite field which does not have [[characteristic (algebra)|characteristic]] 2 reproduces a form of these laws, and thus a finite field geometry.<ref>{{Cite journal | arxiv = arXiv:0807.2692 | postscript = | title = Explicit tough Ramsey graphs | author = Le Anh Vinh, Dang Phuong Dung | date = July 17, 2008 }}, page 1. Another version of this article is at Le Anh Vinh, Dang Phuong Dung (2008), "[http://www.math.harvard.edu/~vinh/rogics_final.pdf Explicit tough Ramsey Graphs]", ''Proceedings of International Conference on Relations, Orders and Graphs: Interaction with Computer Science 2008'', Nouha Editions, 139–146.</ref> The 'plane' formed by a finite field <math>F_p</math> is the [[cartesian product]] <math>F_p\times F_p</math> of all ordered pairs of field elements, with opposite edges identified forming the surface of a discrete [[Torus (mathematics)|torus]]. Individual elements correspond to standard 'points' whereas 'lines' are sets of no more than <math>p</math> points related by incidence (an initial point) plus direction or slope given in lowest terms (say all points '2 over and 1 up') that 'wrap' the plane before repeating.
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| ===Example: (verify the spread law in ''F''<sub>13</sub>)===
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| The figure (right) shows such a 'triangle' of three lines in this finite field setting (''F''<sub>13</sub> × ''F''<sub>13</sub>).
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| Each line has a separate symbol and the intersections of lines ('vertices') is marked by the appearance of two symbols together at the point[[Image:Triangle in Z13.svg|thumb|right|A Triangle through the points (2, 8), (9, 9), and (10, 0) of the [[finite field]]-plane F<sub>13</sub> × F<sub>13</sub>.|278x278px]]
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| s: (2,8), (9,9) and (10,0).
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| Using ''Pythagoras' theorem'' (with arithmetic [[modular arithmetic|modulo]] 13) we find these sides have quadrances of:
| |
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| : (9 − 2)<sup>2</sup> + (9 − 8)<sup>2</sup> = 50 ≡ 11 mod 13
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| : (9 − 10)<sup>2</sup> + (9 − 0)<sup>2</sup> = 82 ≡ 4 mod 13
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| : (10 − 2)<sup>2</sup> + (0 − 8)<sup>2</sup> = 128 ≡ 11 mod 13
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| Then, using manipulation of the ''Cross law'' – see below – to give an expression in '''s''', the three (opposite) spreads of the triangle are found to be:
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| : 1 − (4 + 11 − 11)<sup>2</sup>/(4.4.11) = 1 − 3/7 ≡ 8 mod 13
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| : 1 − (11 + 11 − 4)<sup>2</sup>/(4.11.11) = 1 − 12/3 ≡ 10 mod 13
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|
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| : 1 − (4 + 11 − 11)<sup>2</sup>/(4.4.11) = 1 − 3/7 ≡ 8 mod 13
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| In turn we see that these ratios are all equal as per the ''Spread law'' – see below (at least in mod 13):
| |
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| : 8/11 : 10/4 : 8/11
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| Since first and last ratios match (the triangle is 'isosceles') we just need to cross multiply and take differences to show equality with the middle ratio:
| |
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| : (11)(10) − (8)(4) ≡ 78 (0 mod 13)
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| Otherwise, the standard Euclidean plane is taken to consist of just rational points, <math>\Q\times\Q</math>, omitting any non-algebraic numbers from consideration as solutions. Properties like the geometric incidence of objects (representing the solutions or 'content' of geometric theorems) may therefore follow a number theoretic approach that is more subtle than one relying on an assumption of real numbers. For instance only certain of the lines passing through the centre of a circle are considered capable of being to be incident with the circle at its circumference. To be incident, such a line must be of the form <math>ax + by = 0, a^2 + b^2 = c^2 (a,b,c \in \Q)</math> and so meet the circle in a rational point.
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| == Computation – complexity and efficiency ==
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| Rational trigonometry makes nearly all problems solvable with only addition, subtraction, multiplication or division, as trigonometric functions (of angle) are purposively avoided in favour of trigonometric ratios in quadratic form.<ref name="horizons">{{Cite journal | first = Norman J. | last = Wildberger | title = A Rational Approach to Trigonometry | journal = Math Horizons | volume = November 2007 | pages = 16–20| publisher = Mathematical Association of America| location = Washington, DC| year = 2007| issn = 1072-4117| postscript =}}</ref> At most, therefore, results required as distance (or angle) can be approximated from an exact-valued rational equivalent of quadrance (or spread) after these simpler operations have been carried out. To make use of this advantage however, each problem must either be given, or set up, in terms of prior quadrances and spreads, which entails additional work.<ref>Olga Kosheleva (2008), "[http://web.maths.unsw.edu.au/~norman/papers/Kosheleva.pdf Rational trigonometry: computational viewpoint]", Geombinatorics, Vol. 1, No. 1, pp. 18–25.</ref>
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| The laws of rational trigonometry, being algebraic and 'exact-valued', introduce subtleties into the solutions of problems, such as the non-additivity of quadrances of collinear points (in the case of the triple quad formula) or the spreads of concurrent lines (in the case of the triple spread formula) absent from the classical subject, where linearity is incorporated into distance and circular measure of angles, albeit 'transcendental' techniques, necessitating approximation in results. Additional complexity is also introduced by the need to have 'rules' to handle the dual solutions these quadratic relations generate.
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| ==Notability and criticism==
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| Rational trigonometry is mentioned in only a modest number of mathematical publications, besides Wildberger's own articles and book. ''Divine Proportions'' was dismissed by reviewer Paul J. Campbell, writing in [[Mathematics Magazine]]: "the author claims that this new theory will take 'less than half the usual time to learn'; but I doubt it. and it would still have to be interfaced with the traditional concepts and notation." Reviewer, William Barker, Isaac Henry Wing Professor of Mathematics at [[Bowdoin College]], also writing for the [[Mathematical Association of America|MAA]], was more approving: "''Divine Proportions'' is unquestionably a valuable addition to the mathematics literature. It carefully develops a thought provoking, clever, and useful alternate approach to trigonometry and Euclidean geometry. It would not be surprising if some of its methods ultimately seep into the standard development of these subjects. However, unless there is an unexpected shift in the accepted views of the foundations of mathematics, there is not a strong case for rational trigonometry to replace the classical theory" <ref>http://www.maa.org/publications/maa-reviews/divine-proportions-rational-trigonometry-to-universal-geometry</ref> New Scientist's Gefter described the approach of Wildberger as an example of [[finitism]].<ref name=gefter1/> A glowing review by Arlinghaus raises doubts as to the possibility of penetrating rigid institutional frameworks.<ref>See http://141.213.232.243/handle/2027.42/60314</ref>
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| | |
| An analysis by mathematician Michael Gilsdorf of the same example trigonometric problems used by the author in an earlier paper, found the claim that rational trigonometry takes fewer steps to solve most problems compared to classical methods could be untrue, if free selection of classical methods is available for optimal solution of a given problem; like using the cross product formula for the area of a triangle from the coordinates of its vertices, or applying Stewart's theorem directly to (and in the special case of) the median of a triangle. Concerning pedagogy, and whether the quadratic measures introduced by rational trigonometry offered real benefits over traditional teaching and learning of the subject, the analysis made further observations that classical trigonometry was not based on the use of Taylor series to approximate angles, but rather on measurements of 'chord' (twice the sine of an angle), so with a proper understanding students could reap advantages from continued use of linear measurement without the claimed logical inconsistencies when circular parametrization of angles is subsequently introduced.<ref name="web.maths.unsw.edu.au"/>
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| == See also ==
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| * [[Finitism]]
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| * [[Hyperbolic geometry#Universal hyperbolic geometry|Universal hyperbolic geometry]]
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| == Notes ==
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| * [http://web.maths.unsw.edu.au/~norman/Rational1.htm Wildberger's rational trigonometry site], including downloadable papers and sections of his book
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| * [http://web.maths.unsw.edu.au/%7Enorman/papers/TrigComparison.pdf A comparison of classical and rational trigonometry]
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| * [http://web.ist.utl.pt/ist152027/content/tfc/files/paper.pdf Rational Trigonometry Applied to Robotics], by João Pequito Almeida
| |
| * [http://web.maths.unsw.edu.au/~norman/papers/Trisection.pdf The Impossibility of Trisecting and Angle with Straightedge and Compass: An Approach Using Rational Trigonometry], by David G. Poole
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| * [http://www.austms.org.au/Publ/Gazette/2007/Sep07/Gazette34(4)WebVersion.pdf#page=33 How to multiply and divide triangles], by Maurice Craig
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| * {{citation | author = N J Wildberger| year = 2005| title = Divine Proportions: Rational Trigonometry to Universal Geometry| publisher = Wild Egg}}
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| * Wildberger, N.J., ''Divine Proportions : Rational Trigonometry to Universal Geometry'', Wild Egg Books, Sydney, 2005
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| ==References==
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| {{Reflist}}
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| == External links ==
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| *[http://arxiv.org/abs/0911.1025 Spread polynomials, rotations and the butterfly effect]
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| *[http://euler.rene-grothmann.de/Programs/Examples/Rational%20Trigonometry.html Euler Math Toolbox implementation of Rational Trigonometry]
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| [[Category:Trigonometry]]
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| [[Category:Articles containing proofs]]
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| [[Category:University of New South Wales faculty]]
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