# Rational trigonometry

Rational trigonometry is a proposed reformulation of metrical planar and 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 the motivation for an alternative to traditional trigonometry was to avoid 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.[1] Wildberger draws inspiration from pre set-theory mathematicians like Gauss and Euclid, whom he claims were far more wary of using infinity than modern mathematicians.[1][2] 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 trigonometry problems were subject to dispute by at least one other professional mathematician.[3] (See Notability and criticism below)

## 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. (This also corresponds to a scaled form of the inner product between the lines taken as vectors). 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: one relating the quadrances of three collinear points and one relating the spreads of three concurrent lines, giving the five main laws in the subject.[citation needed]

Rational trigonometry is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair

$(x,y)$

and a line as a general linear equation

$Ax + By + C = 0.\,$

By avoiding calculations that rely on square root operations when taking distances between points, or limiting procedures when evaluating trigonometric functions (and their inverses) as truncations of infinite polynomials, geometry becomes an entirely algebraic subject. There is no assumption, in other words, of the existence of real number solutions, with all results given only over the rational numbers, algebraic field extensions, or over finite field analogs, instead. Following this, it is claimed, many classical results from Euclidean geometry will also be applicable in some form over any field not of characteristic two.[citation needed]

Quadrance (and distance as its square root) both measure separation of points in the Euclidean plane.[4] Following Pythagoras' theorem, the quadrance of two points $A_1=(x_1,y_1)$ and $A_2=(x_2,y_2)$ is therefore defined as the sum of squares of differences in the $x$ and $y$ coordinates:

$Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2.\,$

Unlike addition of distances with line segments, or magnitudes with vectors, adding two quadrances to obtain their combined or resultant magnitude always requires a solution of the associated triangle problem, even in case of collinear segments (a degenerate triangle) where the same calculation made with distances simplifies to addition. In effect, the triangle inequality holds in slightly greater generality under the conditions of rational trigonometry.

Suppose  1 and  2 intersect at the point A. Let C be the foot of the perpendicular from B to  2. Then the spread is s = Q/R.

Spread gives one measure to the separation of two lines as a single dimensionless number in the range $[0,1]$ (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.

• 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.[4]
• Vector: as a rational function of the directions (practically, the slopes) of a pair of lines where they meet.
• Cartesian: as a rational function of the three co-ordinates used to ascribe the two vectors.
• 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.

• Trigonometric

Suppose two lines,  1 and  2, intersect at the point A as shown at right. Choose a point B ≠ A on  1 and let C be the foot of the perpendicular from B to  2. Then the spread s is

$s(\ell_1, \ell_2) = \frac{Q(B, C)}{Q(A, B)} = \frac{Q}{R}.$[4]
• Vector/slope (two-variable)

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

$a_1x + b_1y= constant$ and $a_2x + b_2y= constant$

we may rewrite them as two lines which meet at the origin $(0,0)$ with equations

$a_1x + b_1y= 0$ and $a_2x + b_2y= 0$

In this position the point $(-b_1,a_1)$ satisfies the first equation and $(-b_2,a_2)$ satisfies the second and the three points $(0,0),(-b_1,a_1)$ and $(-b_2,a_2)$ forming the spread will give three quadrances:

$Q_1=(b_1^2+a_1^2),$
$Q_2=(b_2^2+a_2^2),$
$Q_1=(b_1- b_2)^2+(a_1-a_2)^2$

The cross law – see below – in terms of spread is:

$1-s = \frac{(Q_1+Q_2-Q_3)^2}{4Q_1Q_2}.\,$

which becomes:

$1-s=\frac{(a_1^2+a_2^2+b_1^2+b_2^2-(b_1-b_2)^2-(a_1-a_2)^2)^2}{4(a_1^2+b_1^2)(a_2^2+b_2^2)}\,$

This simplifies, in the numerator, to: $(2a_1a_2+2b_1b_2)^2,$ giving:

$1-s=\frac{(a_1a_2+b_1b_2)^2}{(a_1^2+b_1^2)(a_2^2+b_2^2)}\,$

Then, using the important identity due to Fibonacci: $(a_2b_1-a_1b_2)^2+(a_1a_2+b_1b_2)^2=(a_1^2+b_1^2)(a_2^2+b_2^2),$

the standard expression for spread in terms of slopes (or directions) of two lines becomes:

$s = \frac{(a_1 b_2 - a_2 b_1)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}.\,$
• Cartesian (three-variable)
$s = \frac{((y_1 - y_3) (x_2 - x_3) - (y_2 - y_3) (x_1 - x_3))^2}{((y_1 - y_3)^2 + (x_1 - x_3)^2)((y_2 - y_3)^2 + (x_2 - x_3)^2)}.\,$

This uses a general coordinate $(x_3,y_3)$ in place of the origin, and $(x_1,y_1)$ and $(x_2,y_2)$ in place of $(-b_1,a_1)$ and $(-b_2,a_2)$, to specify the endpoints of the vectors.

The spread of two lines can be measured in four equivalent positions.

Unlike angle, which can define a relationship between rays emanating from a point, by a circular measure parametrization, and where a pair of lines can be considered as four pairs of rays, forming four angles, 'spread' is a fundamental concept in rational trigonometry, describing two lines with a single measure by a rational function (see above).[4] Being equivalent to the square of a sine, the spread of both an angle and its supplementary angle are equal.

0 0 0
30 (1/6)π 1/4
45 (1/4)π 1/2
60 (1/3)π 3/4
90 (1/2)π 1
120 (2/3)π 3/4
135 (3/4)π 1/2
150 (5/6)π 1/4
180 π 0

Spread is not proportional, however, to the separation between lines as angle would be; with spreads of 1/4, 1/2, 3/4, and 1 corresponding to unevenly spaced angles 30, 45, 60 and 90 degrees.

Instead, (recalling the supplementary property) duplicating (or doubling) a configuration of lines each with spread $s$, to give their new spread value, requires solving the triple spread formula for a triangle with with two spreads of $s$ in terms of its third spread, $r$:

$(r + 2s)^2 = 2(r^2 + 2s^2) + 4s^2r$
$r^2 + 4sr + 4s^2 = 2r^2 +4s^2 + 4s^2r$

giving the quadratic polynomial (in $s$):

$r^2 + 4s^2r - 4sr = 0$
$r^2 - 4s(1-s)r = 0$

and solutions

$r = 0$ (trivial) or
$r = 4s(1-s) = s_2$

Tripling a spread likewise involves the triple spread formula, one spread of $s$, a second spread of $r$ (the previous solution) and obtaining the third spread $t$, expressed again as a polynomial in $s$ , which turns out to be:

$t = s(3-4s)^2 = s_3$

$s_2$ and $s_3$ are the so-called second and third spread polynomials. All multiple configurations of a basic spread of lines will be generated by continuing this procedure.

Any 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:

$\sin^2 (30^\circ/2) = (1-\cos 30^\circ)/2 = (1 - \sqrt{3}/2)/2 = (2-\sqrt{3})/4 \approx 0.0667.$

and thus exists by algebraic extension of the rational numbers.

## Laws of rational trigonometry

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.[4]

In the following five formulas, we have a triangle made of three points A1A2A3, . The spreads of the angles at those points are s1s2s3, , and Q1Q2Q3, are the quadrances of the triangle sides opposite A1A2, and A3, respectively. As in classical trigonometry, if we know three of the six elements s1s2s3, , Q1Q2Q3, and these three are not the three s, then we can compute the other three.

The three points A1A2A3,  are collinear if and only if:

$(Q_1 + Q_2 + Q_3)^2 = 2(Q_1^2 + Q_2^2 + Q_3^2).\,$

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.

### Pythagoras' theorem

The lines A1A3 (of quadrance Q1) and A2A3 (of quadrance Q2) are perpendicular (their spread is 1) if and only if:

$Q_1 + Q_2 = Q_3.\,$

where Q3 is the quadrance between A1 and A2.

This is equivalent to the Pythagorean theorem (and its converse).

There are many classical proofs of Pythagoras' theorem; this one is framed in the terms of rational trigonometry.

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,

$Q(AB) + Q(AC) = Q(BC)\,$

where Q is the "quadrance", i.e. the square of the distance.

For any triangle $\overline{A_{1} A_{2} A_{3}}$ with nonzero quadrances:

$\frac{s_{1}}{Q_{1}}=\frac{s_{2}}{Q_{2}}=\frac{s_{3}}{Q_{3}}.\,$

This is the law of sines, just squared.

### Cross law

For any triangle $\overline{A_{1} A_{2} A_{3}}$,

$(Q_1 + Q_2 - Q_3)^2 = 4Q_1 Q_2 (1-s_3).\,$

This is analogous to the law of cosines. It is called 'cross law' because $(1-s_3)$, the square of the cosine of the angle, is called the 'cross'.

For any triangle $\overline{A_1 A_2 A_3},$

$(s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4s_1 s_ 2 s_ 3 .\,$

This corresponds approximately to the angle sum formulae for sine and cosine (in a triangle, whose angles sum to 180 degrees, sin a = sin(b + c) = (sin b)(cos c) + (sin c)(cos b)). 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.

Knowing two spreads allows the third spread to be calculated by solving the associated quadratic formula but, as 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 complementary angle.)

## Computation – complexity and efficiency

Rational trigonometry makes some problems solvable with only addition, subtraction, multiplication, and division, with fewer uses of other functions such as square roots, sine, and cosine compared to classical trigonometry. Such algorithms execute more efficiently on most computers, for problems such as solving triangles. Other computations, however, do involve more computations than their classical analogues; such computations include determining the quadrance of a line segment given the quadrance of two collinear line segments which compose it, or such as computing the spread of the sum of two angles with known spreads.[5]

## Trigonometry over arbitrary fields

Rational trigonometry makes it possible to work with the same standard 'laws' in almost any arithmetic setting (fields of characteristic '2' being excluded for technical reasons) whether finite or infinite. {.[6] The real numbers receive no treatment at all in order to avoid questions of irrationality arising[citation needed], so rational numbers fulfil the role of a more limited form of continuum in the geometry. In certain circumstances, results having no rational number solutions, such as: finding if a line at 45 degrees (spread one-half) to a coordinate axis passing through the origin, and a circle of unit quadrance centred there, 'intersect' (if they did it would be at (½√2,½√2) ) can be interpreted differently to the usual scalar assumption that they will. But problems involving roots of non-square rational numbers as intermediate results can still be considered within the framework of Rational trigonometry. This can be achieved through extending the field (to introduce algebraic numbers) without requiring scalar evaluation (e.g. of the real numbers): all results having 'exact' algebraic expressions.

Over a finite field, the 'plane' corresponds to the cartesian product of its ordered pairs. With opposite edges identified, this region forms the surface of a (discrete) torus; individual elements match to 'points' and 'lines', each consisting of an initial point plus all integer multiples of the 'vector' (say '2 over and 1 up') specifying a direction or slope in lowest terms, 'wrap around' it.

### Example: (verify the spread law in F13)

The figure (right) shows such a 'triangle' of three lines in this finite field setting (F13 × F13).

Each line has a separate symbol and the intersections of lines ('vertices') is marked by the appearance of two symbols together at the point

A Triangle through the points (2, 8), (9, 9), and (10, 0) of the finite field-plane F13 × F13.

s: (2,8), (9,9) and (10,0).

Using Pythagoras' theorem (with arithmetic modulo 13) we find these sides have quadrances of:

(9 − 2)2 + (9 − 8)2 = 50 ≡ 11 mod 13
(9 − 10)2 + (9 − 0)2 = 82 ≡ 4 mod 13
(10 − 2)2 + (0 − 8)2 = 128 ≡ 11 mod 13

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:

1 − (4 + 11 − 11)2/(4.4.11) = 1 − 3/7 ≡ 8 mod 13
1 − (11 + 11 − 4)2/(4.11.11) = 1 − 12/3 ≡ 10 mod 13
1 − (4 + 11 − 11)2/(4.4.11) = 1 − 3/7 ≡ 8 mod 13

In turn we see that these ratios are all equal as per the Spread law – see below (at least in mod 13):

8/11 : 10/4 : 8/11

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:

(11)(10) − (8)(4) ≡ 78 (0 mod 13)

## Notability and criticism

Rational trigonometry is mentioned in a modest number of mainstream mathematical publications, in addition to Wildberger's own articles and book. Divine Proportions was dismissed by reviewer Paul J. Campbell in Mathematics Magazine, who wrote: "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." Meanwhile, William Baker, Isaac Henry Wing Professor of Mathematics at Bowdoin College, also writing for the MAA, concluded: "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" [7] New Scientist's Gefter described the approach of Wildberger as an example of finitism.[1] A glowing review by Arlinghaus raises doubts is to the possibility of penetrating rigid institutional frameworks.[8]

An analysis by mathematician Michael Gilsdorf found Wildberger's claim that rational trigonometry takes fewer steps to calculate some problems compared to the classical method was false, by using the same example Wildberger claimed was made easier by using rational trigonometry. It was shown that Wildberger's calculation using classical trigonometry added in extra steps which were unnecessary, such as taking a Taylor series for an angle already known exactly. By objectively comparing both, it was found that Wildberger's method took significantly more steps.[3] So far, he has failed to refute this paper or make suggestions about other examples for which rational trigonometry will take fewer steps. Wildberger's claim that rational trigonometry would be easy to teach to children has only been asserted without any evidence, and the fact that quadrance is not linear like distance removes much of the intuition normally present in geometry.