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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
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‰ ^ 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
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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
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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
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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
Euler's Identity With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmer
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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
SolveB‰ ^ Â 2 p + p 2
- 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>>
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
7
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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
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RevolutionPlot3DB- ArcSinB 1 -
ArcCosB 1+Cos@qD
2
F2
p 2F, 8q, - 4 p , 4 p <F
SolveB‰ ^ Â 2 p + p 2
- 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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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
Euler's Identity With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmer
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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  ArcSinhA‰Â q - Cos@qDEp
+ ArcSinhA‰Â q - Cos@qDE2
p2
F>,
:b Ø ArcSinB 1
2-
4  ArcSinhA‰Â q - Cos@qDEp
+
ArcSinhA‰Â q - Cos@qDE2p2
F>>
18
Euler's Identity With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmer
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RevolutionPlot3DB
- ArcSinB 1
2-
4  ArcSinhA‰Â q - Cos@qDEp
+
ArcSinhA‰Â q - Cos@qDE2p 2
F, 8q, - 4 p , 4 p <F
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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RevolutionPlot3DB ArcSinB 1
2-
4  ArcSinhA‰Â q - Cos@qDEp
+
ArcSinhA‰Â q - Cos@qDE2p 2
F, 8q, - 4 p , 4 p <F
SolveB‰ ^HÂ qL == Â SinB2 p + p 2
- 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
Euler's Identity With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmer
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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SolveB‰ ^HÂ qL == Â SinB2 p + p 2
- 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 -
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
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>,
: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
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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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
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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
ty With Substitutions from the Difference in Circumferences of Two Circles Applied to the Pythagorean Theorem ©Parker Emmerson 2009.nb
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