Pi Polygon Method
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Transcript of Pi Polygon Method
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8/11/2019 Pi Polygon Method
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AOC OD AOC COOA
=CDDA
C OD AO
E
C
D
A O
E
AOC
mDOA= mCE A
mCOD+mDOA+mCOE= 180
mOCE+mOEC+mCOE= 180
mCOD+mDOA= mOCE+mOEC
COE CE A DOA CA
AE = DA
AO CA
DA= AE
AO CA = C D + DA
AE= AO+OE
CD +DADA
= AO+OEAO
CD
DA =
OE
AO
CD
DA =
CO
OA
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C
D
E
A O B
F
AB O AC A
AOC 6 CF = 2 CA CF OA
AC
= CircumferenceDiameter
O
C
A
D
F
OAAC
OCCA
AOC OD OAAD
OAAD
ODDA
OCCA
n
AOC OD D AC CDDA
= COOA
CD+DADA
=CO+OAOA
= CADA
CO+OACA
= OADA
COCA
+ OAAC
= OAAD
ODDA
OA2 +AD2 =OD2
OA2
AD2+ 1 =
OD2
DA2
OA2
AD2+ 1 =
OD
DA
mAOC
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OAAC
> 265153 OAAC
=cot(6 ).
OAAC
=
3
3 > 265153
ACOA
153265
13
OCCA
= 21 = 306153
csc(6 ) = 1sin(
6)
306
153+265
153
349450153
349450> 591 18
ODDA
> 591 1
8
153
ODDA
=csc( 12).
ODDA
OAAD
AOD OE OA
AE = OD
DA+ OA
AD OA
AE >
591 18
153 + 571153 =
1162 18
153
AC CO OA
mAOC
AOE ODDA
OAAD
OAAE
OAAE
OEEA
= 24 AEOA
3 17
OAAC
cot(6 )
6
OAAD cot( 12) 12
ntan(n
)
v1, v2, v3...vn
vi vi+1
vn v1 O
viOvi+1 0 i n 1 vnOv1 mviOvi+1 =
2n
0 i n 1 mvnOv1 = 2n tan(
n)
=
n
ntan(
n) =
ntan(n
) = U Bn
OCCA
+ OAAC
= OAAD
csc() +cot() = cot(
2)
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Ov1
v2
pi/n
pi/n
v3
v4
v5v6
OA2
AD2+ 1 = OD
DA
cot2() + 1 = csc()
=nsin(n
)
v1, v2, v3...vn
vi
vi+1
vn v1 O
mviOvi+1 = 2n
0 i n 1 mvnOv1 = 2n viO D
O
v1
v2
v3
v4
v5
v6
v7
v8
v1
v2
v3
v4
v5
v6
v7
O D
mviOvi+12 = mviDvi+1 =
n
viD Dvi+1vi
sin(n
) = vivi+1viD
=
nsin(
n) =
nsin(n
) = LBn
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A
C
dD
O
E
B
AB O C CAB
6 ACB ACD
mCOB= 2mCAB
ACCB
ABBC
cot(mCAB) csc(mCAB ) cot(mCAB2 ) cot(mCAB)
csc(mCAB) csc(mCAB2 )
cot2(mCAB2 ) + 1
csc(mCAB) csc(mCAB2 ) nsin(n
) mCAB
n
n
csc(
n) +cot(
n) = cot(
2n)
cot2(
2n) + 1 = csc(
2n)
csc(n
) cot(n
)
nsin(n
) ntan(n
) LBn U Bn
csc(
n) +cot(
n) =cot(
2n)
1
sin(n
)+
1
tan(n
)=
1
tan( 2n)
12nsin(
n)
+ 12ntan(
n)
= 12ntan( 2n)
1
2LBn+
1
2U Bn=
1
U B2nU Bn+LBn2LBnU Bn
= 1
U B2n
U B2n= 2LBnU BnU Bn+LBn
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cot2(
2n) + 1 =csc(
2n)
1
tan2( 2n)+ 1 =
1
sin( 2n)tan2( 2n) + 1
tan2( 2n ) =
1
sin( 2n) tan2( 2n )
tan2( 2n) + 1=sin(
2n)
tan2( 2n)
sec2( 2n) =sin(
2n)
tan2(
2n)cos2(
2n) =sin(
2n)
tan(
2n)sin(
2n)cos(
2n) =sin(
2n)
4n2tan(
2n)
sin(n
)
2 =2nsin(
2n)
2ntan(
2n)nsin(
n) =2nsin(
2n)
U B2nLBn= LB2n
U B2n LB2n U Bm LBm m
U Bm
LBm
624 629
6 227
35 262
U B4n LB4n 110(U Bn LBn) U B2n LB2n 110(U Bn LBn)
LB2n LBn
n
U Bn LBm
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i U B62i , LB62i
U B62i LB62i i
U B2nLBn LB2n
LBn
U B2n LB2n U B2n LBnU B2n LBn= 2LBnU Bn
U Bn+LBn LBn
= 2LBnU BnU Bn+LBn
LBn(U Bn+LBn)U Bn+LBn
=LBnU Bn LB2n
U Bn+LBn
=LBn(U Bn LBn)
U Bn+LBnU B2n LB2n
U Bn LBn LBn
U Bn+LBn
U Bn
LBn
LBnUBn+LBn
12
U B2n LB2nU Bn LBn
1
2
110
< 12
errori= O(2i) i
errori= U Ba2i LBa2i a
12
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errori = O(2i)
log210 Nlog210 O(N2) O(N) O(N2) O(N2)
f(x) = x2 y O(log2N) 2N N O(2N2) O(N)
12
O(N)
Nlog210
O
N2
+O
N
+O
N2
+O
N2
+O
log2N
O(2N2) +O(N) +O(N)
N O(N3log2N)
O( )
314159265358979323846264338327950288100000000000000000000000000000000000
314159265358979323846264338327950289100000000000000000000000000000000000
O(N) O(N2) O(1) O(2N2) O(N)
12 O(N) Nlog210
O(N2)
Nlog210
O
N
+O
1
+O
N2
+O
log2N
O(2N2) +O(N) +O(N)
+O(N2)
N O(N3log2N)
N = 35 3544200
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