euler's identity with substitutions from the difference in circumferences of two circles...

8/14/2019 Euler's Identity With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorea…

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Euler's Identity With Substitutions from the

Difference in Circumferences of Two Circles Applied

to the Pythagorean Theorem

© Parker Emmerson November 2009

Disclaimer : The knowledge gianed by reading the material in this document may be dangerous and hazardous. This is experimen-

tal research done through thought. The author does not condone any physical experiment by humans or other sentient beings

conducted using any form of high physical energy dynamics to exploit these structures. Experiments can be conducted using

energy dynamics within virtual, computational, or thought systems.

This paper uses Euler' s Equation to deliver more expressions for an

angular section of a circle. This first attempt to organize the multitude of

solutions will begin with only the first solution to theta from a difference in

circumferences of two circles applied to Pythagorean Theorem. The theorem,2 p r - 2 p r1 = q r is provable and delivers the expression for theta

q = 2 p ± p2

- p2 Sin@bD

2.

‰ ^HÂ qL = Â Sin@qD + Cos@qD

q = 2 p + p 2

- p 2Sin@bD

2

ü The Forms

‰ ^ Â 2 p + p 2

- p 2 Sin@bD2 = Â Sin@qD + Cos@qD

Printed by Mathematica for Students



‰ ^ Â 2 p + p 2

- p 2 Sin@bD2 = Â SinB2 p + p

2- p

2 Sin@bD2 F + Cos@qD‰ ^ Â 2 p + p

2- p

2 Sin@bD2 = Â Sin@qD + CosB2 p + p 2

- p 2 Sin@bD2 F

‰ ^

HÂ q

L== Â Sin

B2 p + p

2- p

2 Sin

@b

D2

F+ Cos

@q

D‰ ^HÂ qL == Â Sin@qD + CosB2 p + p

2- p

2 Sin@bD2 F‰ ^HÂ qL == Â SinB2 p + p

2- p

2 Sin@bD2 F + CosB2 p + p 2

- p 2 Sin@bD2 F

ü The Solutions

SolveB‰ ^ Â 2 p + p 2

- p 2 Sin@bD2 == Â Sin@qD + Cos@qD, qF

::q Ø - ArcCos

B1

2

‰-2 Â p 1-Sin@bD2 1 + ‰4 Â p 1-Sin@bD2

F>,

:q Ø ArcCosB 1

2‰-2 Â p 1-Sin@bD2 1 + ‰

4 Â p 1-Sin@bD2 F>>

RevolutionPlot3DB- ArcCosB 1

2‰-2 Â p 1-Sin@bD2 1 + ‰

4 Â p 1-Sin@bD2 F, 8b, - p , p <F

2

Euler's Identity With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmer




RevolutionPlot3DB ArcCosB 1

2‰

-2 Â p 1-Sin@bD2 1 + ‰4 Â p 1-Sin@bD2 F, 8b, - p , p <F

RevolutionPlot3DB:- ArcCosB1

2‰

-2 Â p 1-Sin@bD2 1 + ‰4 Â p 1-Sin@bD2 F,

ArcCosB 1

2‰

-2 Â p 1-Sin@bD2 1 + ‰4 Â p 1-Sin@bD2 F>, 8b, - p , p <F

ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb

3




RevolutionPlot3DB:- ArcCosB1

2‰

-2 Â p 1-Sin@bD2 1 + ‰4 Â p 1-Sin@bD2 F,

ArcCosB 1

2‰

-2 Â p 1-Sin@bD2 1 + ‰4 Â p 1-Sin@bD2 F>, 8b, - 2 p , 2 p <F

Solve

B‰ ^ Â 2 p + p

2- p

2 Sin

@b

D2

== Â Sin

@q

D+ Cos

@q

D, b

F::b Ø - ArcSinB

p2 + LogB- Cos@qD + Â Sin@qD F2p

F>,

:b Ø ArcSinBp2 + LogB- Cos@qD + Â Sin@qD F2

pF>,

:b Ø - ArcSinB 4 p2 + Log@Cos@qD + Â Sin@qDD22 p

F>, :b Ø ArcSinB 4 p2 + Log@Cos@qD + Â Sin@qDD22 p

F>>

4





RevolutionPlot3DB- ArcSinBp 2 + LogB- Cos@qD + Â Sin@qD F2

p F, 8q, - 4 p , 4 p <F

RevolutionPlot3DB ArcSinBp 2 + LogB- Cos@qD + Â Sin@qD F2

p F, 8q, - 4 p , 4 p <F


5




RevolutionPlot3DB- ArcSinB 4 p 2 + Log@Cos@qD + Â Sin@qDD22 p

F, 8q, - 4 p , 4 p <F

RevolutionPlot3DB ArcSinB 4 p 2 + Log@Cos@qD + Â Sin@qDD22 p

F, 8q, - 4 p , 4 p <F

Solve

B‰ ^ Â 2 p + p

2- p

2 Sin

@b

D2

== Â Sin

B2 p + p

2- p

2 Sin

@b

D2

F+ Cos

@q

D, q

F::q Ø - ArcCosB‰2 Â p 1-Sin@bD2 - Â SinB2 p + p

2- p

2 Sin@bD2 FF>,

:q Ø ArcCosB‰2 Â p 1-Sin@bD2

- Â SinB2 p + p2

- p2 Sin@bD2 FF>>

6





RevolutionPlot3DB- ArcCosB‰2 Â p 1-Sin@bD2 - Â SinB2 p + p

2- p

2 Sin@bD2 FF, 8b, - p , p <F

RevolutionPlot3DB ArcCosB‰2 Â p 1-Sin@bD2

- Â SinB2 p + p 2

- p 2 Sin@bD2 FF, 8b, - p , p <F


- p 2 Sin@bD2 == Â SinB2 p + p

2- p

2 Sin@bD2 F + Cos@qD, bF

::b Ø - ArcSinB 1 -

ArcCosB-1+Cos@qD

2

F2p2

F>, :b Ø ArcSinB 1 -

ArcCosB-1+Cos@qD

2

F2p2

F>,

:b Ø - ArcSinB 1 -

ArcCosB 1+Cos@qD

2

F2p2

F>, :b Ø ArcSinB 1 -

ArcCosB 1+Cos@qD

2

F2p2

F>>


7




RevolutionPlot3DB- ArcSinB 1 -

ArcCosB-1+Cos@qD

2

F2

p 2F, 8q, - 4 p , 4 p <F

RevolutionPlot3DB ArcSinB 1 -

ArcCosB-1+Cos@qD

2

F2

p 2F, 8q, - 4 p , 4 p <F

8





RevolutionPlot3DB- ArcSinB 1 -

ArcCosB 1+Cos@qD

2

F2

p 2F, 8q, - 4 p , 4 p <F


- p 2 Sin@bD2 == Â Sin@qD + CosB2 p + p

2- p

2 Sin@bD2 F, qF::q Ø - Â ArcSinhB‰

2 Â p 1-Sin@bD2- CosB2 p + p

2- p

2 Sin@bD2 FF>>

RevolutionPlot3DB- Â ArcSinhB‰2 Â p 1-Sin@bD2 - CosB2 p + p

2- p

2 Sin@bD2 FF, 8b, - 2 p , 2 p <F


9




RevolutionPlot3DB

- ArcSinB 1 -

ArcCosBCsc@qD -1- 1-Sin@qD2

2

-1-Sin@qD2 1- 1-Sin@qD2

2

F2

p 2 F, 8q, - 2 p , 2 p <F

RevolutionPlot3DB

ArcSinB 1 -

ArcCosBCsc@qD -1- 1-Sin@qD2

2

-1-Sin@qD2 1- 1-Sin@qD2

2

F2

p 2F, 8q, - 2 p , 2 p <F


11




RevolutionPlot3DB

- ArcSinB 1 -

ArcCosBCsc@qD 1- 1-Sin@qD2

2

+1-Sin@qD2 1- 1-Sin@qD2

2

F2

p 2 F, 8q, - 2 p , 2 p <F

RevolutionPlot3DB

ArcSinB 1 -

ArcCosBCsc@qD 1- 1-Sin@qD2

2

+1-Sin@qD2 1- 1-Sin@qD2

2

F2

p 2F, 8q, - 2 p , 2 p <F

12





RevolutionPlot3DB- ArcSinB

1 -1

p 2 ArcCosBCsc@qD 1

2+

1

21 - Sin@qD2 - 1 - Sin@qD2 1

2+

1

21 - Sin@qD2 F

2

F, 8q,

- 2 p , 2 p <F

RevolutionPlot3DB ArcSinB

1 -1

p 2 ArcCosBCsc@qD 1

2+

1

21 - Sin@qD2 - 1 - Sin@qD2 1

2+

1

21 - Sin@qD2 F

2

F, 8q,

- 2 p , 2 p <F


13




RevolutionPlot3DB- ArcSinB

1 -1

p 2 ArcCosBCsc@qD -

1

2+

1

21 - Sin@qD2 + 1 - Sin@qD2 1

2+

1

21 - Sin@qD2 F

2

F,

8q, - 2 p , 2 p <F

14





SphericalPlot3DB

ArcSinB 1 -1

p 2 ArcCosBCscB 4 p

3- I- 4 p

2+ 12 p

2 Sin@bD2M ì 6 - p 3

+ 18 p 3 Sin@bD2 + 3 3

- p 6 Sin@bD2 + 11 p

6 Sin@bD4 + p 6 Sin@bD6

1

ê3

+ 2

3- p

3+ 18 p

3 Sin@bD2 +

3 3 - p 6 Sin@bD2 + 11 p

6 Sin@bD4 + p 6 Sin@bD6 1ê3F -

1

2+

1

21 - Sin@qD2 +

1 - Sin@qD2 1

2+

1

21 - Sin@qD2 F

2

F, 8b, - p , p <, 8q, - 2 p , 2 p <F


15




RevolutionPlot3DB ‰-2 Â b I1 + ‰2 Â bM2 p , 8b, - 2 p , 2 p <F

RevolutionPlot3DB- Â LogB- ‰-Â ‰-2 Â b I1+‰2 Â bM2 p F, 8b, - p , p <F


17




RevolutionPlot3DB:- Â LogB- ‰-Â ‰-2 Â b I1+‰2 Â bM2 p F, ‰-2 Â b I1 + ‰

2 Â bM2 p >, 8b, - 2 p , 2 p <F

SolveB‰ ^HÂ qL == Â SinB2 p + p 2

- p 2 Sin@bD2 F + Cos@qD, bF

::b Ø - ArcSinB 1

2-

4 Â ArcSinhA‰Â q - Cos@qDEp

+ ArcSinhA‰Â q - Cos@qDE2

p2

F>,

:b Ø ArcSinB 1

2-


+

ArcSinhA‰Â q - Cos@qDE2p2

F>>

18





RevolutionPlot3DB

- ArcSinB 1

2-


+

ArcSinhA‰Â q - Cos@qDE2p 2

F, 8q, - 4 p , 4 p <F


19




RevolutionPlot3DB ArcSinB 1

2-


+

ArcSinhA‰Â q - Cos@qDE2p 2

F, 8q, - 4 p , 4 p <F


- p 2 Sin@bD2 F + CosB2 p + p

2- p

2 Sin@bD2 F, qF::q Ø - Â LogBCosB2 p + p

2- p

2 Sin@bD2 F + Â SinB2 p + p2

- p2 Sin@bD2 FF>>

20





RevolutionPlot3DB- Â LogBCosB2 p + p

2- p

2 Sin@bD2 F + Â SinB2 p + p 2

- p 2 Sin@bD2 FF, 8b, - 2 p , 2 p <F


21





- p 2 Sin@bD2 F + CosB2 p + p

2- p

2 Sin@bD2 F, bF::b Ø - ArcSinB

. 1 -1

p2

ArcCosBKIÂ ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O + IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD -

2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O -

ISin@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD OOì H- 1 + Cos@qD + Â Sin@qDLF2 F>,

:b Ø ArcSinB. 1 -1

p2

ArcCosBKIÂ ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD +

Sin@qD2MMì K2 Cos@qD + Â Sin@qD O + IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 +

2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O -


:b Ø - ArcSinB. 1 -1

p2

ArcCosBK-IÂ ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD +

Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O - IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 +

2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O +


:b Ø ArcSinB. 1 -1

p2

ArcCosBK-IÂ ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD +

Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O - IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 +

2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD O +

ISin@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM ì K2 Cos@qD + Â Sin@qD OOì H- 1 + Cos@qD + Â Sin@qDLF2 F>>

22





RevolutionPlot3DB- ArcSinB. 1 -

1

p 2 ArcCosBJIÂ

,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD N + IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD -

2 Â Cos@qD Sin@qD + Sin@qD2MMí J2 Cos@qD + Â Sin@qD N -

ISin@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD NN í H- 1 + Cos@qD + Â Sin@qDLF2 F, 8q, - 2 p , 2 p <F


23




RevolutionPlot3DB ArcSinB. 1 -

1

p 2 ArcCosBJIÂ

,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD N + IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 +

2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MMí J2 Cos@qD + Â Sin@qD N -

ISin@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD NN í H- 1 + Cos@qD + Â Sin@qDLF2 F, 8q, - 2 p , 2 p <F

24





RevolutionPlot3DB: ArcSinB. 1 -

1

p 2 ArcCosBJ-IÂ

,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD N - IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 +

2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MMí J2 Cos@qD + Â Sin@qD N +

ISin@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD NN í H- 1 + Cos@qD + Â Sin@qDLF2 F,

- ArcSinB. 1 -1

p 2 ArcCosBJ-IÂ

,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD +

Sin@qD2MMí J2 Cos@qD + Â Sin@qD N - IÂ Cos@qD ,I- 1 + 2 Cos@qD - Cos@qD2 +

2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MMí J2 Cos@qD + Â Sin@qD N +

ISin@qD ,I- 1 + 2 Cos@qD - Cos@qD2 + 2 Â Sin@qD - 2 Â Cos@qD Sin@qD + Sin@qD2MM í J2 Cos@qD + Â Sin@qD NN í H- 1 + Cos@qD + Â Sin@qDLF

2

F>, 8q, - 2 p , 2 p <F


25

euler's identity with substitutions from the difference in circumferences of two circles...

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