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ISSN: 2090-4908
International Journal of Swarm Intelligence and Evolutionary Computation
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The General Sinus

Mostafa Derraz*

Department of Electrical Engineering, Abdelmalik Essaadi Université, Morocco

*Corresponding Author:
Derraz M
Department of Electrical Engineering
Abdelmalik Essaadi Université, Morocco
Tel: +212665104683
E-mail: [email protected]

Received date: January 05, 2017; Accepted date: May 24, 2017; Published date: May 31, 2017

Citation: Derraz M (2017) The General Sinus. Int J Swarm Intel Evol Comput 6:155. doi:10.4172/2090-4908.1000155

Copyright: © 2017 Derraz M. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, we will generalize the Sinus function (than the all Trigonometry function), the general Sinus is defined by two parameter Sin (x, y), which it's be able to use in n-gon not necessary in rectangle. We will apply the general trigonometry function in n-gon, in order to determine the all intrinsic properties of n-gon, using a minimum and reasonable amount of data and no condition apply in n-gon nature, based in construability of polygon we determine it's in this mount n-2 segments and n angles. Finally we will prove this formula is the must generalized in Euclidian geometry.

Keywords

Trigonometry function; Geometry; Polygons; Sinus

Historical Comments

To present the History in a correct way, we will devise it in two parts. In the first one, we will expose a short history of the trigonometrically functions and law of Sinus, and in the second one will be reserved to the history of polygons and the works of some scientist who tried to end a general formula in the geometry polygon.

Trigonometry function and law of sinus

Let’s start by the lexicon of the word trigonometry, it’s comes from the Greek “Trine” (triangle) and “Metron” (measure). The word “Sinus” comes from a confusion of translation from Arabic to Latin. In Arabic, “gabr” means both the opening at the chest of a garment and the half string double angle (our current sinus). When Latinists mathematicians have tried to translate this term, they have translated it as Sinus which means “fold of the toga” in Latin. The “Co sinus” of an angle is the complementary to the sinus of this angle, hence the term “Co sinus” [1,2].

Let’s back in the first use. In 2000 years before our era, the Babylonians applied geometry only in astronomy, using tables of calculations that divide a circle made form 360º into angles of 60º [3]. Moving on to the seventh century, using a half rope, the old Indian Aryabhata first gave us the first tables of the sinus of an angle, then addressed. The trigonometry into a circle [4]. Later on, in the ninth century Mohammed al Battani introduced a table of tangent of an angle [5]. Moving on to the thirteenth century, the Persian al Tusi, separated geometry from astronomy [6]. Then in the tenth century astronomer and mathematician Johann Miller, introduced the systematic use of the term sinus. As a result, trigonometry developed as an independent branch of mathematics [7]. A century later, the French Francois Vieta, counsellor of Henri IV, will evolve trigonometry, to give it the character we know today. Trigonometry became a wide range of applications in a variety of sciences, physical acoustics, optics, electronics, statistics, economics, biology, chemistry, medicine, meteorology, geodesy, geography, cartography, cryptography [8].

One of the important formula in the Euclidean geometry is the law of sinus [9]. Finally, in 2011 Russell generalized the law of sinus and applied it to a simplex space of any dimension with constant (Gaussian curvature) [10]. Let us work up to that initially in two dimension simplex (line segment) with content (length) S in space of constant (Gaussian curvature) K as:

image

Geometry of the polygon

Gauss had discovered that besides the regular polygons of 2n.3, 2n.4, 2n.5 and 2n.15 sides, there were a number of other constructible polygons, including the 17-gon. Gauss secured priority to his discovery by publishing an announcement on June 1, 1796, which appeared in the “Intelligenez Blalt Derallge-meinen Literaturzellung”, the first and only time he published in a journal of advance notices. In his short note he wrote that “This discovery is really only a corollary of a theory with greater content, which is not complete yet, but which will be published as soon as it is complete” [11,12].

The full meaning of Gauss pronouncement came in 1801 with the publication of his monumental Disquisitions arithmetic [13]. Its final two sections (articles 365 and 366) discuss the issue of polygon constructability. Recasting the problem in terms of constructing the n vertices of the polygon on a given circle, Gauss stated his result as follows: In general in order to divide the circle geometrically into n parts, n must be 2 or a higher power of 2 or a prime number of the form 2m +1or the product of several prime number of this form or the product one or several such primes into 2 or a higher power of 2.

This sufficiency of the condition follows readily forms Gauss analysis.

This necessity, however, is not obvious and Gauss never published a proof of this assertion. The first proof is credited to Wantzel [14].

Primes of the form2m+1, and therefore necessarily of the form Fk=22k are the Fermat primes and he knew that Euler had shown F5 to be composite. Even today no other Fermat primes have been found, although the smallest unsettled cases are F22, F24 and F26 (The primarily status of Fermat numbers as of 1983 can be found in Keller [15] and a shorter but more recent table is included in Young and Buell [16]).

Beyond the brilliant theoretical breakthrough of Gauss there is still an intriguing puzzle:

How can one devise a sequence of steps with straightedge and compass which constructs a regular n-gon? Gauss expressed interest in this problem, although he did not offer explicit geometric constructions. Throughout the last century, and even into present times, numerous constructions have been contrived for the 17-gon [11,17]. Richelots's constructions of the 275-gon required a total of 194 pages, published in four parts in "Grelles" journal in 1832. A professor "Hermes" laboured ten years on the construction and associated algebra of the 65537- gon. The work filled a trunk which was donated to the Mathematical Institute at Gottingen.

Nearly a century later it remains stored in an attic there, in all likelihood having never been read. Fallowing to Gauss, we found that "Roark" worked on regular polygons and found a series of equations that defined by segment, radius of interior circle, radius of exterior circle [18]. And in 1979 “Court” found out an equation gathering all the properties of a polygon vertex. [19]

Another scientist worked on polygon is “{Ptolemy}”: “Ptolemy's Theorem” is a relation among these lengths in a cyclic quadrilateral. In Euclidean geometry Ptolemy's Theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. "Ptolemy" used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. [20]

Results and Discussion

In fact this part, we will discuss two themes. In the first one we will generalize the Trigonometrically Function (based on the law of sinus) [9] in the second one we will use the latest results (General trigonometrically function) to find the general formula .This formula can determinate all the propriety of a polygon based on the minimal amount of data, and we will prove that it's the most generalized.

The general Sinus

In this paragraph we will discuss the general trigonometrically function based on the projections of a point on a line and the law of sinus. We know that the principal trigonometrically function is the sinus function. So that what we are going to start with:

If we observe the definition (formula) of the sinus image we found that the formula calculate the opposite of the proportion distanceimage between the vertices (point A2) of the angle (angle α1) and a point included in it (point A1) and the distance image between the point (point A1) and it's verticals projection on the other segment (point A3) of the studied (angle α1) (Figure 1).

swarm-intelligence-evolutionary-Dominance-relation

Figure 1: The triangle.

Remark 1

We notice that the formula (actual) of sinus is just a special case not a general formula, because it's defined by the vertical projection, which is just a special case of the projection.

If we joined these points we will get a triangle (exactly rectangle) A1A2A3 . So the use of the sinus (actual) is limited in rectangles, as result it can calculate the segment image in function of another segment image and the studied angle α1.

image

So if we want to calculate the distance image in a triangle which is not necessarily rectangle (the general use of sinus), we must generalize the function sinus, it means to generalize the projection (not necessarily vertical projection), so we can redefine the formula of the sinus with another variable (the projection angle), in fact the (actual) sinus is defined by two variables not one, because it defined by vertical projection (in fact we must see projection fallowing angle image), it means the (actual) sinus defined by two angles ,the first one is the studied angle and the second one the particulaire projection angle image.

Finally the general Sinus must be defined by two angles:

1. The studied angle.

2. The projection angle.

Remark 2

According the conjunction in definition between sinus and Co sinus, so we will use the same analyze to generalize the Co sinus, then the function Tangent.

We announced in previous paragraph that the key to generalize the sinus is to find the function which is able to calculate the image

in a triangle (not necessarily rectangle).

These propose is served, if we use the law of sinus in this triangle. According to this law, we can calculate this segment, as a result we will find the general formula of the sinus.

The generalizing of the trigonometrically functions, necessity to create a new function to calculate the projection angle, which is defined by the two segmentimage

and image

one angle (studied angle α1).

We named the new function (direction), it means in Latin (direction) [21].

Theorem 1

The general sinus: In the triangle A1A2A3 (not necessarily rectangle), where:

α1=A1 A2 A3

α2=A1 A2 A3 Projection angle (Figure 2).

swarm-intelligence-evolutionary-vertical-projection

Figure 2: The vertical projection.

The condition to form a triangle:image

The general sinus:

image

Theorem2

The general trigonometrically function: In the triangle A1A2A3 (not necessarily rectangle), where:

α1=A1 A2 A3.

α2=A1 A2 A3 Projection angle (Figure 2).

The condition to form a triangle: α1α2 ≠ nπ, n?N

The general formula of Trigonometrical Function:

image

Theorem 3

The function direction: In the triangle A1A2A3 (not necessarily rectangle), where:

α1=A1 A2 A3

α2=A1 A2 A3 Projection angle (Figure 2).

 

The condition to form a triangle: α1α2 ≠ nπ, nòN

The function direction represent the projection angle α2 defined by the angle α1 and its segments image and image

image

Definition 1

Trigonometrically function value’s e and general unit circle: Fallowing the theorem (1) and theorem (2) we will build the value’s table of usual angles in case the general Trigonometrical Function (Table 1).

0 image image image image
0 1 1 1 1 1
image 0 image image image image
image 0 image image image image
image 0 image image 1 2
image 0 image 1 image image

Table 1: Trigonometrically function value’s.

We create the new conjuration (the value circle) to reduce and present the results of the last table like the value circle of the actual sinus, with a difference, this conjuration is defined by two circles not one, the first represents the studied angle (the same in the actual sinus) and the second represents the projection angle (Figures 3 and 4).

swarm-intelligence-evolutionary-The-rectangle

Figure 3:The rectangle.

swarm-intelligence-evolutionary-circle-presentation

Figure 4:The value circle presentation.

The value of start (0) in the second circle is the studied angle of the first circle. To use this conjuration we must fixe (first) the studied angle in the first circle and we will get the value of start (0) in the second circle then we fix the projection angle in second circle (see an example how to use it in the paragraph: Analogy between the actual Trigonometrically Function and general form) (Figures 5-7).

swarm-intelligence-evolutionary-circle-example

Figure 5:The value circle example 1.

swarm-intelligence-evolutionary-circle-example

Figure 6:The value circle example 2.

swarm-intelligence-evolutionary-circle-proof

Figure 7:The value circle proof.

The graphic presentation

The graph of Sinus is 3D view is a surface in space reference by two variables (two angles) and other one in amplitude (segment), also Direction function (Figures 8 and 9).

swarm-intelligence-evolutionary-sinus-graphic

Figure 8:The sinus graphic presentation.

swarm-intelligence-evolutionary-direction-graphic

Figure 9:The direction graphic presentation.

Proof 1

This proof we will be based on the previous discussion, we reduce the discussion into the following

The key to generalize the sinus, is to find a function which can calculate the A1A3 in a triangle (not necessarily rectangle).

This purpose was resolved, if we used the law of sinus in this triangle. According to this law, we can calculate this segment. Consequently we will find the general formula of the sinus.

Let's comeback to Sinus law (Figure 2):

image

We find that the function in equation (7) calculate the Sinus side of the angle α1 and the function in equation (8) calculate the adjunct segment of the angle α1, so this function has the same purpose of the (actual) function sinus and Co sinus with a simple different, these functions are used in the general form of a triangle (not necessarily rectangle), so we can deduct that those functions are the general form of the actual sinus and Co sinus, also they general form is available in any triangle (see detail in the next paragraph: Analogy between actual function and general form ).

In what comes next, we based on the law of sinus and we will create a new concept, function direction, as we announced in previous discussion, this function presents the angle of projection (α2). To create the formula of direction, we will consider a triangle with the parameters.

Angle (A3A2A11): Two segment of studied angle A1A22 and A1A31 and we want to find the angle α2, which is the angle of projection.

Back to the formula of the Sinus law (equation 7 and equation 8), we will resolve this equation where α2 is variable and (α1, a1 and a2) are the parameters. We find:

image

We note

image

Analogy between the actual trigonometrically function and general form

In this part we will compare between the actual trigonometrically function and general form.

Rectangle

We will begin by applying the general sinus and Co sinus in a rectangle. We consider a rectangle A1 A2 A3 and we will apply the general sinus and Co sinus in this rectangle, so we got these results (Figure 3).

image

N.B following the law of sinus notation

In this case we notice:

image

So, the same formula of the actual sinus.

We announce also:

image

N.B following our notation

image

So, the same formula of the actual Co sinus.

We will compare the general tangent with the actual form. We announce the general form:

image

So, the same formula of the actual tangent.

The derivative as a function. We will compare the derive function. Derive function of the general form:

image

image

the link of the general form and actual form), we get:

image

So, the same formula of the actual Sinus and Co sinus.

The table of usual angles and the value's circle

will compare the value's table of the usual angles between actual form and general form of the trigonometrically function (Table 1).

α Represent the studied angle (the line) α2 represent the projection angle (the colonel).

We notice if we have α12=π/2. We will get the same table of the actual sinus.

If we have α12=π/2, the Sinus axe is a vertical (Figure 5).

So this we notice the same configuration of actual form.

Remark 3

So, fallowing the previous discussion we can notice the formula in the theorem (1) is the general Sinus, also they formula in the theorem (2) is the general Trigonometrical Function.

General formula in n-gon

In this part we are going to discuss the general trigonometrically function in n-gon, based on:

1. The notion constructability to polygons to discuss and justify the amount of data (that was announced in the abstract) which we think is minimal and reasonable. Based on small hypotheses: the amount of data is minimal and reason- able if it is sufficient to form a unique polygon. (See detail in paragraph: Is there any Formula more general than this one?)

2. The result of the previous part (General trigonometrically function).

3. Decomposition a polygon in (n-2) triangle.

The formula is denied by parameters that we must be present as data: n-2 successive segments and n angles (In fact we need just n-1 angles, but, according to Euclid's formula, the lack of data is not important).

Definition 2

Notation and terminology: To simplify and explain this formula, we add new symbols and concepts:

Let's π a plane, we have n points named A1, A2, A3…An.

When we join those points, we must follow this order: Ai→Ai+1 as a result we will get the polygon of n-segment (Figure 10).

swarm-intelligence-evolutionary-gon-presentation

Figure 10:n-gon presentation.

Sinus side a: The Sinus side is the segment in polygon, the sinus side is an. In case the triangle, sinus side is a3. So we notice if we consider the angle a3 is the studies angle, the segment a3 is Sinus side in actual sinus.

Adjunct side: The Adjection side is the segment in polygon; the adjunct side is an-1. In case the triangle, sinus side a2. So we notice if we considered the angle a3 is the studie angle, the segment a2 is adjunct side in actual co sinus.

Sous sinus side: In a polygon π plane, if we joined the A1 point to (n-1) points (to every other point), except two segments image andimage we will get (n-3) sous sinus side, also we decompose this polygon in (n-3) triangles, where a sous sinus side becomes a sinus side in triangle.

We note scsini (sous side sinus in Latin linguistic: Sous costula sinus).

image

Remark 4

If i = 0, image we notice scsin0 becomes a polygon segment. And ifimage

we notice SCSinn-2 becomes a sinus side.

Direction angle

We will take angles defined by A2 A1 An. If we divide the polygon in the point A1, this angle will be also divided and the partial angles will be the angle of projection in every triangle. This angle is noted as directi and defined by (A2+iA1A3+i), i ? (0, 1, 2, 3…n)

So directi= (A1+iA1A2+i).

Remark 5

If i=1, direct0=(A2A1A3)

• Complementary direction angle: Codirecti: represent the complementary of the direction angle the other i.

Accumulation Direction angle: Accudircti: represent the accumulation of the direction angle other i.

Theorem 4

The general theorem in n-gon: We consider a n-gon, must know (n-1) segment and n angles. We can calculate two segment sinus side and ajection said and n-3 (all) sous side sinus, also (n-3) (all) direction angles (Figure 10).

image

Remark 6

This formula can be used in all forms of polygons, due to its independent of the polygons nature. However, the difference is that a general method of application does not exist. So, the user chose the appropriate parameters.

Is there any formula more general than this one?

We think that the result we had is the most general in Euclidean geometry because any unreasonable decrease in the amount of data will lead to infinity of possible studied objects.

Our thesis is based on simple and traditional fundamentals depending on efficiency and we are aware that this concept ‘efficiency’ is used only on applied sciences and by that we suggest this concept (efficiency) in Mathematics to evaluate and discuss theories.

According to this concept we can evaluate a mathematical theories and formulas by discussing the amount of results and compare it to the amount of data. We mean by results, the properties of the studied objects, so the maximum result we can find is to find all the object’s intrinsic properties, this mean that the number of results is limited and we can’t improve the efficiency of our Formula unless we try to reduce the amount of data which is the case in our theory.

Our formula is based on the study of a simple triangle, so to explain that the formula can use the minimal and reasonable amount of data we will come back to the triangle (Figure 11).

swarm-intelligence-evolutionary-point-presentation

Figure 11:The unicité of point presentation.

Let's suppose to have two points A1 and A2 of a plane (if we define the position of A1 and A2 we will fix a segment image if we want to form a unique triangle using this amount of data image.

we will find an infinity of intersections (the point A3) of the lines passing through A1 and others passing through A2 (in one single point passes an infinity of lines) where the triangle A1A2A3 is formed.

In order to form a unique triangle we must fixe the position of the point A3, so that mean we should fix the angle A3A1A21) and A1A2A32). After fixing the segment image

the only parameter that can influence the position of the point A3 is the angle, so the minimal and reasonable amount of data is the segment image and the two angles A3A1A21) and A1A2A32).

The unicity of the triangle is proven by the next proof:

Let's: y1=a1.x3+b1 Cartesian equation of the adroit passing in A1(x1+b1) and y2=a2.x2+b2 Cartesian equation of the adroit passing in A2 (x2, y2).

So A3 (x3, y3) verify this system

image

According to equation (27) we find:

image

So,

image

Remark 7

In Equation (30) you are going to notice that a1 and a2 must be under a condition: a1-a2 ≠ 0.

Based on Equation (33), tanα1-tanα1 ≠ 0, so α1 and α1 are conditioned, that means our first statement is the Abstract concerning n-gon nature was wrong. In fact this condition does not make our statement false because it is necessary to form a triangle.

It mean a depend on α so x3 depend que α because b just constant.

Finlay position of point A3 (x3y3) depend on α1, α2 and image.

In the quadrilateral we need to fixe three points (fixing two segments. In order to form the quadrilateral we must fixe the position of the fourth point (fixing the angles: the ones who define the proportionality factor).

As for the Pentagon we need to fixe four points (fixing three segments). In order to form the Pentagon we must fixe the position of the fifth point (fixing the angles: the ones who define the proportionality factor).

Following the exact same logic, in the n-gon we notice the need to fixe “n-1” points (fixing (n-2)) segments), in order to form the n-gon we must fixe the position of the last point.

Using the Mathematical induction we can prove that the minimal and reasonable amount of data is in function of “n” (n: number of segments and number of angles) and are expressed by this formula.

(n-1) angles and (n-2) segments.

Proof 2

In this proof, will be based on the theorem (2) and theorem (3). To generalize this results and applies it in the polygon, we will work in this plan:

We will apply theorem (2) and theorem (3) in a quadrilateral, a pentagon and a Hexagon.

2. After noticing the results of the previous forms we can use the mathematical induction to write the all formula

So we will begin by:

Quadrilateral: We consider a quadrilateral A1A2A3A4 (Figure 12). In triangle A2A1A3

swarm-intelligence-evolutionary-quadrilateral-presentation

Figure 12:The quadrilateral presentation.

image

Where image

image

=direct1=directio(a2a1α1)=directio(a2, scsin0, codirect1)

Finally

image

image

So we get:

image

So

image

Pentagon: We notice that the equations (32) and (33) (triangle A2A1A4) are the same in a Pentagon, the rest of the proof is based on the previous equations (32) and (33).

We will find (Figure 13):

swarm-intelligence-evolutionary-pentagon-presentation

Figure 13:The pentagon presentation.

In triangle A3A1A4

image

image

Where:

image

So:

image

Hexagon: We notice that the equations (32), (33), (37) and (38) (triangle A2A1A3 and triangle A3A1A4 ) are the same in a Hexagon, the rest of the proof is based on the previous equations (Figure 14).

swarm-intelligence-evolutionary-hexagon-presentation

Figure 14:The hexagon presentation.

In triangle A4A1A1

image

In triangle A5A1A6

image

Where:

image

And

image

image

After noticing that if we write the final results (in function of n ), we find the same results in (quadrilateral, pentagon and hexagon). We conclude that this formula follows a logical law, so we will use mathematical induction to confirm the generality of the formula

Mathematical induction

We suppose that the equation (21) is true and we will prove that:

image

If we add a point in plane, two segments will be added to the polygon (forming by n points). ‘an’ Will be now an interior segment in the new polygon (forming by n+1 points), so an = scsinn-3 (Figure 15).

swarm-intelligence-evolutionary-induction-png

Figure 15:The mathematic induction png presentation.

We notice that:

Direct2And scin2 are the same.

In (n=4, quadrilateral) direct2, direct3 and scin2, scin3 are the same.

Generally, the term scini and directi = i ? (1, 2, 3….n) remain the same, so what makes an+2 special is co-directn and accu-directn and if we apply (21), we will find a (47) and according to principle of math induction, we conclude that the formula is correct and generalized.

Conclusion

We tried in this paper to discuss the generalized trigonometrical function and all the properties of an object based on the amount of data we have (we think it's the reasonable and minimal amount of data). To open the door of discussion we will propose a series of problematic in order to exploit the maximum from these results.

1. We think if we generalized the trigonometrical function, we will get much more opportunity to evaluate it in many fields of science, for example we think that these results might be a solution to the problem of the retaliatory signals. We think also that our results can improve the kinetical study in the mechanics field, and we think may be evaluating the rayonement study.

2. Can we improve the general law of Sinus to find the same results while studying an object in the space based on a new minimal amount of data?

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