Angular to Rectangular

The complete angle for a full turn or perigon of a circle, which is 360 degrees or \(2\pi\) radians, may be divided into sectors in number equal to a base, \(B \). The angle of a sector would be of the perigon a \(B\)th part, that is, 360/\(B\) mesopotamian degrees written decimally or \(2\pi/B\) radians.

A circle may be plotted in a plane of rectangular co-ordinates \(x\) and \(y\), where the co-ordinate axes are perpendicular to each other and of the same scale, letting the centre of the circle be at the origin point \((0, 0)\) of intersection of the co-ordinate axes and the radius \(r\) of the circle be one unit. The circumference of the circle may be subdivided at \(B\) points by the arms of the angles or radial endlines of the sectors. The co-ordinates in rectangular form \((x, y)\) of the points on the circumference may be calculated from the "polar" co-ordinates using the angles \(θ = 2\pi n/B\) radians of the sectors, counting \(n\) of \(B\) sectors from the co-ordinate radius, and the radial distance \(r\) of the points from the origin by \((x, y) = (rcosθ, rsinθ)\).

For certain "polar" form angles \(θ\), the rectangular form co-ordinates \(x\) or \(y\) may be
1. integers of set ℤ, which are positive or negative whole numbers of set ℕ;
2. rational numbers, belonging to the set of numbers denoted by the symbol ℚ, which are those that can be expressed as ratios or fractions or positive or negative whole numbers;
3. real irrational numbers, ℝ\ℚ, which cannot be expressed as ratios of integers.

Both rectangular co-ordinates \(x\) and \(y\) would be rational for angles in the right-angled triangles that have whole number sets of lengths \(x\), \(y\), and \(r\), called Pythagorean triples, of their edges satisfying the equation \(x^2 + y^2 = r^2\). Such angles do not generally divide a rational number of times into the perigon.

Either \(x\) or \(y\) of the rectangular co-ordinates could be rational for angles of a right-angled triangle that has rational lengths for the radius or hypothenuse and one other edge while the length of the third edge is irrational. Only certain such angles divide a rational number of times into the full turn angle. The angles of the first dozenal divisions are of this kind.

For most values for the number \(B\) of sectors, both rectangular co-ordinates \(x\) and \(y\) would be irrational.

The numbers \(B\) of subdivisions into sectors may be ranked by simplicity of the rectangular co-ordinates that their angles produce. The order of simplicity of sets of numbers from simplest to less simple is:
1. Integers ℤ;
2. Rationals ℚ;
3. Irrationals unnested;
4. Irrationals nested, with the simplicity decreasing with the degree of nestings.

Thus, the order from most to less simple of \((x, y)\) is:
1. Both \(x\) and \(y\) integers ℤ;
2. Either one or the other of \(x\) or \(y\) rational ℚ, the other irrational;
3. Both \(x\) and \(y\) irrational unnested;
4. Either one or the other of \(x\) or \(y\) irrational unnested, the other irrational nested;
5. Both \(x\) and \(y\) irrational with nesting once.
6. Both \(x\) and \(y\) irrational with nesting more than once.

The rank of perigonal divisions is as follows:
1. Quarters;
2. Twelfths, sixths, and thirds;
3. Eighths;
4. Twenty-fourths;
5. Twentieths, tenths, and fifths;
6. Square fourths;
7. Sixtieths; thirtieths; fifteenths.

Conclusion
From these rectangular co-ordinates, it is apparent that the most simple and fundamental division of the circle by a base is into its quarters. A base of numeration for this purpose should be capable of representing quarter fractions as simple terminating numbers. This is best achieved if the base is itself divisible by four by having the number four as one of its factors.

Decimal does not represent quarters of a circle well, as the circle would have to be divided into its square a hundred before quarters could be marked.

It would be best to choose a base for division of the circle that is a multiple of the number four. Such bases include octal, four squared, dozenal, the double dozen, and six squared.

Angular division by base twenty or vigesimal is less simple than by even the double dozen.

After fourths, the next simplest division is by twelfths. These are mildly even simpler than eighths.

Bases containing the prime factor five are relatively far down the list of simplicity and where three is not a factor are impoverished of ability to represent twelfths.

Reference
https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals

Table of Trigonometric Ratios

Division	Angular Radians	Rectangular Co-ordinates
Quarters	0, \( \frac{\pi}{2}, \pi, \frac{3 \pi}{2} \)	\( \left(1, 0\right), \left(0, 1\right), \left(-1, 0\right), \left(0, -1\right) \)
Thirds	\( \frac{2 \pi}{3}, \frac{4 \pi}{3} \)	\( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right), \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)  \)
Sixths	\( \frac{\pi}{3}, \frac{5 \pi}{3} \)	\( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right), \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)  \)
Twelfths	\( \frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{11 \pi}{6} \)	\( \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right), \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right), \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right), \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) \)
Eighths	\( \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4} \)	\( \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right), \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right), \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right), \left(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right) \)
Twenty-fourths	\( \frac{\pi}{12}, \frac{5 \pi}{12}, \frac{7 \pi}{12}, \frac{11 \pi}{12}, \frac{13 \pi}{12}, \frac{17 \pi}{12}, \frac{19 \pi}{12}, \frac{23 \pi}{12} \)	\(   \left(\frac{\sqrt{6} + \sqrt{2}}{4}, \frac{\sqrt{6} - \sqrt{2}}{4}\right), \left(\frac{\sqrt{6} - \sqrt{2}}{4}, \frac{\sqrt{6} + \sqrt{2}}{4}\right), \left(\frac{\sqrt{2} - \sqrt{6}}{4}, \frac{\sqrt{6} + \sqrt{2}}{4}\right), \left(\frac{-\sqrt{6} - \sqrt{2}}{4}, \frac{\sqrt{6} - \sqrt{2}}{4}\right), \left(\frac{-\sqrt{6} - \sqrt{2}}{4}, \frac{\sqrt{2} - \sqrt{6}}{4}\right), \left(\frac{\sqrt{2} - \sqrt{6}}{4}, \frac{-\sqrt{6} - \sqrt{2}}{4}\right), \left(\frac{\sqrt{6} - \sqrt{2}}{4}, \frac{-\sqrt{6} - \sqrt{2}}{4}\right), \left(\frac{\sqrt{6} + \sqrt{2}}{4}, \frac{\sqrt{2} - \sqrt{6}}{4}\right) \)
Square fourths	\( \frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}, \frac{9 \pi}{8}, \frac{11 \pi}{8}, \frac{13 \pi}{8}, \frac{15 \pi}{8} \)	\( \left(\frac{\sqrt{2 + \sqrt{2}}}{2}, \frac{\sqrt{2 - \sqrt{2}}}{2}\right), \left(\frac{\sqrt{2 - \sqrt{2}}}{2}, \frac{\sqrt{2 + \sqrt{2}}}{2}\right), \left(-\frac{\sqrt{2 - \sqrt{2}}}{2}, \frac{\sqrt{2 + \sqrt{2}}}{2}\right), \left(-\frac{\sqrt{2 + \sqrt{2}}}{2}, \frac{\sqrt{2 - \sqrt{2}}}{2}\right),     \left(-\frac{\sqrt{2 + \sqrt{2}}}{2}, -\frac{\sqrt{2 - \sqrt{2}}}{2}\right), \left(-\frac{\sqrt{2 - \sqrt{2}}}{2}, -\frac{\sqrt{2 + \sqrt{2}}}{2}\right), \left(\frac{\sqrt{2 - \sqrt{2}}}{2}, -\frac{\sqrt{2 + \sqrt{2}}}{2}\right), \left(\frac{\sqrt{2 + \sqrt{2}}}{2}, -\frac{\sqrt{2 - \sqrt{2}}}{2}\right), \)

If one were to take a circle, whose diameter is (0,0) to (2,0), and divide this circle into \(n\) parts, the chords from (0,0) to each of these parts, will in product, produce \(n\).

The span of chords of any given polygon form a closed set to multiplication, and as such the set formed by said span intersects with the rationals to produce the integers, ie \( \mathbb Zn \in \mathbb Q = \mathbb Zn \in \mathbb Z\), where \( \mathbb Z_n \) is the span of chords of a polygon of \(n\) sides. Further, it is not possible to reach the centre of a polygon, in terms of unit steps parallel to the edges, only when the polygon is a prime or prime-power.

Gauss demonstrated that the polygon's chords (and hence the sines and cosines), are in the set \(\mathbb G_2\), [that is, the closure of integers to division and square roots], if the number of sides is a divisor of \(2^n \cdot 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537 \). [the last two translate 2.17 and 4.66.17].

On the other hand, chordal stone-boards can be used for any set \(\mathbb Zn\), but these are used mainly to establish the calculation table.

Plympton 332 shows us that the Sumerians used right triangles of integers, sorted by increasing tangent, to provide numeric values for the trignometric functions. These then connect with their use of reckoner tables to avoid division, by multiplying by the inverse.

The only trignometric value i use is the chord of an angle, since this is a direct measure of the curvature and angle of the apex of an isocelese triangle. In practice, the square of the chord is used, which corresponds to the double-versine or something. The difference here is that I go to values greater than 2 for the chord.

wendy.krieger wrote:The only trignometric value i use is the chord of an angle,

The specification of angles by the ratio of the chord to the radius was used in Greek trigonometry.

Letting the length of the chord be say \(d\) and the radius be \(r\), the angle may be obtained by the ratio \(d/r\) of the chord as
$$\cos{θ} = 1 - \frac{(d/r)^2}{2}.$$
I derived this formula nine years ago and used it then to tabulate trigonometric ratios of chords to the radius for multiples of a tenth of the perigon as follows.

Table of Chord to Radius Ratios for Tenths Angles

Angle in radians	Chord to Radius
\( \frac{\pi}{5} \)	\( \frac{\sqrt{5} - 1}{2} \)
\( \frac{2 \pi}{5} \)	\( \frac{\sqrt{5 - \sqrt{5}}}{\sqrt{2}} \)
\( \frac{3 \pi}{5} \)	\( \frac{1 + \sqrt{5}}{2} \)

wendy.krieger wrote:If one were to take a circle, whose diameter is (0,0) to (2,0), and divide this circle into \(n\) parts, the chords from (0,0) to each of these parts, will in product, produce \(n\).

I gather by this is meant
$$∏_{m=1}^{m=n-1} \sqrt{2 \left(1-\cos{\frac{2 \pi m}{n}}\right)} = n$$

wendy.krieger wrote:the Sumerians used right triangles of integers, sorted by increasing tangent, to provide numeric values for the trignometric functions.

In April of the year 2013, I made the following table of angles in right-angled triangles having rational lengths of sides, where \(a\) is the length of the side adjacent to the angle \(θ\), and \(h\) is the hypothenuse.

Table of Angles of Right Angled Triangles

\(a/h\)	\(θ\) in degrees where \( \cos{θ} = a/h \)
3/5	53.1301°
5/13	67.3801°
8/17	61.9275°
7/25	73.7398°
20/29	46.3972°
12/37	71.0754°
9/41	77.3196°
28/53	58.1092°
33/65	59.4898°
16/65	75.7500°
48/73	48.8879°
36/85	64.9424°
39/89	64.0108°
65/97	47.9250°

The idea was to select some triangle with an angle close to the size desired.

The table did not include the right-angled triangles and angles formed by the Pythagorean triples 61, 60, 11 and 85, 84, 13 as I must have deemed their angles to be too far from the value I was seeking.