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The Compelling Arc in Calculus

Beauty in geometry shining through the science of physic

gives light and understands in defining all the circular functions.

Binary Polygon scripts provide the technology to define how to engineer

in calculus; truly a breakthrough in mathematics.

The Giuseppe Stagno Arc Theory proves that the Circle

is a Binary Polygon and has 2^55 Sides.

The maximum arc length is equal at π  *  r; the π is a constant of

the circle and r is the radius.

The Technology of the Arc uses the small Polygon Side in

constructing the length of the Arc  with

the methods of Pythagoreas Right Triangle Theorem and

Giuseppe Stagno Arc Theorem.

Triangle Theory

Picture a circle centred with (x, y) axis

at (0, 0)   and  defined by     x^2 + y^2 = 1 = 1^2 = r^2

The layout tells us that the point cos (t), sin (t)  is on the unit circle.

Moreover, this approach leads to a definition of cos (t) and sin (t) for

all real (t).

t = Degrees varying from 90 ° to 0 ° Pythagoras Theorem             x^2 + y^2 = 1 = 1^2            y^2 = 1-x^2

Sine       y = (1-x^2) ^(1/2)  = sin(t) varying from 1 to 0

Cosine     x = cos(t) varying from 0 to 1

Tangent = y/x

Now let picture Geometrically how to define the Polygon Side function.

The symbol is: "S" a straight line that links two points on a circle or a curve is called;

Polygon Side or Side. The Side angle is the length between two points on a circle separated by that angle.

It is easily related to the sine function by taking one of the points to be zero.

y^2 = 1-x^2       (1-x)^2 = 1-2x+x^2

S^2 = y^2+(1-x)^2 = 2-2x Therefore S =(2-2X)^(1/2).

Arc Theory

In drawing isosceles triangles on the circular sector, it has shown the area

between the Arc and the Side; all the triangles are varying in size smaller

near the arc until they cover the total area. The small Arc length at this stage is equal at the small Side being best for

completing the total length of the Arc;Therefore, At this state the small Side is used to calculate the length of the Arc.

Calculus An applied approach to the mathematics of change; focused on

Limitis, Functions, Derivatives, Integrals, Infinite series and Incremental.

Giuseppe Stagno Theory

Integral y = (1-x^2)^(1/2) Derivative dy = S/2

This is Giuseppe Stagno dy = [(1-x)/2]^(1/2)

Stagno Theorem  state:      dy^2+dx^2 = 1 = 1^2 = r^2 = Y^2+X^2 Area

(x^2+y^2)^(1/2) = (dy^2+dx^2)^(1/2) = r = 1 Linear

dx^2 = 1-dy^2      thi is Giuseppe Stagno dx= [(1+x)/2]^(1/2)

Side       2dy = S = (2–2x)^(1/2)

Incremental          2dx = 2*[(1+x)/2]^(1/2)

I = Increment = (2+?)^(1/2)     ? is the variable 2x

I1 = (2+2x)^(1/2)

this equation is Giuseppe Stagno Incremental Theorem

Now let deal with distance and time

y = distance x = time

y = (1-x^2)^(1/2)

Tangent = y/x = [(1-x^2)^(1/2)]/x = velocity

Therefore dy/dx = derivative Tangent = velocity = [(1=x)/(1+x)]^(1/2).

Polygon Sides

From now on let expand the Side to construct the Arc step by step to

see the full expansion with the input at x = 0.

S = (2-2x)^(1/2)            this Polygon has four Sides

S has one Side varying from 2^(1/2) to 0

Let start with the first expansion S = (2-2x)^(1/2)                           dS = [2-(2+2x)^(1/2)]^(1/2)

2dS = S1

This Polygon has eight Sides

S1 = (2-I1)^(1/2) *2^1                                       I1 = (2+2x)^(1/2)

at First expansion varying from 1.53 to 0       S1 has two Sides.

This equation is Stagno Polygon Sides Theorem

this polygon has sixteen sides

S2 = (2-I2)^(1/2) *2^2                       I2 = [2+(2+2x)^(1/2)]^(1/2)

at second expansion varying from 1.56 to 0   S2 has four sides.

this polygon has thirty two sides

S3 = (2-I3)^(1/2)*2^3                   I3 = {2+[2+(2+2x)^(1/2)]^(1/2)}^(1/2)

at third expansion varying from 1.568 to 0   S3 has eight sides.

n is last Increment = last

Sn = [(2-In)^(1/2)]*2^n  at last expansion varying from π/2 to o  Sn has a lot of small sides.

The half-angle formula is sin(t/2)

S = 2*sin(t/2)=(2-2X)^(1/2)

S1 = 2^2*sin(t/2^2)=(2-(2+2x)^(1/2))^(1/2)=(2-I1)^(1/2)*2^1 I1 = (2+2x)^(1/2)

S2 = 2^3*sin(t/2^3)=(2-I2)^(1/2)*2^2 = {2-[2+(2+2x)^(1/2)]^(1/2)}^(1/2)

I2 = [2+(2+2x)^(1/2)]^(1/2)

Therefore

arc length = 2^54*sin(t/2^54) = pi/2 at 90 degree. pi/2 at 90 degree = 2^54*sin(t/2^54) = S53 = [(2-I53)^(1/2)]*2^53

S53 = [(2-I53)^(1/2)]*2^53 = arc length = [(2-In)^(1/2)]*2^n = Sn.

Those are Giuseppe Stagno arc length formulas

Expander is varying from 1 to 53 and is called n.

[(2-In) ^(1/2)] is equal at the small side and at this stage  is equal

at the small arc length also

[(2-In) ^(1/2)] = [(2-I53) ^(1/2)].

Full expanded sides or arc length is equal at

S53 = [(2-I53) ^(1/2)]*2^53 This is Giuseppe Stagno arc length theorem.

Conclusion In this article I have considered the theoretical and numerical value of the expanded polygon sides; the arc length now is proven and calculates with algebraic equations.

Sn = arc length value in radians the value of the input "x"  can vary from 0 to 1 Full expanded polygon sides is = [(2-I53)^(1/2)]*2^53=S3 pi/2 at 90 degree = 2^54*sin(t/2^54) =S3= [(2-I53)^(1/2)]*2^53

at 90 degree and has 2^53 sides.

Tell that the circle has 2^55  sides and is a regular polygon.

pi=2^54*sin(90/2^54)*2=2*S3=[(2-I53)^(1/2)]*2^53*2 Sincerely

Author Giuseppe Stagno

Born 8 December 1946 on the island of Sicily town Delia

email       gstagno31@gmail.com