binomial_trigo_logs.pdf

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NNNNNNNN NNNN NN N N N NL N NL LNM NMN NML SNLSNNNS NNMLNN NNLNNNNNN NE NMNNNMLN NNNNNNSN NNS SMNMMNNNNNN NMNMNNNLNNMNS NNNNNN NN NNS N NNN NNNN NN N .... ...... . ....... .... .. .. ....... ........ ....... LN NM NL N N NSN N NMMNN4NN 4444

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Page 1: Binomial_Trigo_LogS.pdf

NNNNNNNNN NNNNN NNNNNNLNNL

LNMNMNNML

SNLSNNNSN NNMLNN

NNLNNNNNNN NEN NMNNNMLN

NNNNNNSNN NNSN SMNMMNNNNNN

NMNMNNNLNNMNSN NNNNNNNNN NNS

NNNNNNNNNNN

..........................................................

LNNMNLNNNSN

NNMMNNN4NNN4444

Page 2: Binomial_Trigo_LogS.pdf

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111

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ba1

1 ---

s a s

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00b 0 Þ b 0 00

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ba1

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ba1

---

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1 1

00

00

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4

00

00

0

1

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\ 1 0

1444 0

1 Þ 0 0s1 ]

11 bn0

(1 g ) (1 g g ) (1 g g g 0) 111111 (1 g g g 111111 g 100) g 0 ggn 0 n0 asg 00n0

g g 0 0 n s 0 0 0 ns ...... 1

( ) 4e 0 ( ) 0 0 ( ) 1 0 ( ) 0 0

02n0 2 2n s 0 0 n0 g 0 g 0 0ngs g ns 0 0 1 g

2n s 0 0 n0 g 0 g 0 0ngs g ns 0 0 1 g g 0

0ndn ag dd 2n s 0 0 n0 g 0 g 0 0ngs 00g 0

0 1 g g 111111 g 100 0 0 0 ]

110 11

00 g d 0 0 � g s4q a00 � d 0 4 sn0 q 0

( ) 4 g d4 0 e ( ) 10d =+

( ) 0 g d0 0 ( g d ) ( ) d =+

00 1 g0 a00n0 a00 s b gag n0

0

sn044g s0 q+q- d 0

sn044g s0 q-q-

0

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)4g s1() sn01(4 q+-q-

0 (1 g sn0 q ) � g s q d 0 (1 � sn0 q) � g s q 0 1 g sn0 q g sn0 q d 0 1 � sn0 q g sn0 q

0 (1 g sn0 q) d 0 (1 � sn0 q) ÞÞÞ d =+ ]

0 g0a 2 gg q 0 4

p a00 gn0d ]

114bn0 s d å

=

10

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!k·k s a s

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00 1 � 0 ga gd k! (k g 1 � 1) 222...22].]]2..]]

(k g 1)! � k!

0 0 ( )å=

-+10

1k

!k)!1k( 0 ( ! � 1!) g (0! � !) g 1111111 g (11! � 10!) 0 (11)! � 1 ....]

Page 3: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

11 sp

00 ad bd g ag 0ns n0g sn n g a n0 21,1d 0 a 0 g ssn 0d

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ab

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----++-+-

0 agabbgb

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+--

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ag)ga(bb

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++-

-

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)agb(

+-

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1101 10

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( ) 0s sg s b 0 gn ( ) gn b 0 ns sg s

( ) gn ns sg s ( ) s n a ga

02n0 2 (b g g) 0 bga 222....2]...2]].]]2..]]

(b g g) 0 sn0 bg

\ sn0 0 bg

gb +

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gb + ³ bg Þ b g g ³ bg

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117 bn0

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g

0

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1

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1

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7 0 0

8 Þ 0 - 7 0 48 Þ 0 0 ]

1180 11

00 a0 0 g s01

g s sn00 -

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sn0

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g s)10(

sn0

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sn0

+

Page 4: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

00 1 a0( g ) 0 a0 a01

a0 a0

-+

0

g s01

g s sn00· a01

g s01

g s sn00 a0

--

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0 g s sn00) g s01( g s

g s sn00) g s01( sn0

--

+-

0 ) sn00 g s01( g s

0 sn0 --

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g s)01(

sn0

- ]

11e 7 bn0

( ) 0+0

0 g 0 g g 0 Î a00 Î a00 0 f 0 f 1 0 a 0d 0 - 0 g 0 - ns

( ) a 0a ga 0 db g ( ) a 0 a n n0 g

( ) a gnd 0 db g ( ) ag ngga n 0a 0 db g

02n0 2 0 � 0 g 0 � 0 � 0(1 � 0) � (1 � 0) 0 � 0( g 0)(1 � 0)]

g g g 0 0 ( g 0 )0 a00 1 � 0 0 0 � 0 ( � 0 )0

0g 0 � 0 g 0 � 0 � 1 0s1 ]

1110 10

0sn0 g gna0 ng g 0 as s g01 00 0 f a f p 0 0 db g 0 n0 ga a s 0 g ns

( ) 0 ( ) 0

( ) 4 ( )

00 1 g s 0 17·1e·

g)17()1e( -+ 0 0 222..].2]].]]2..]]

\ g f (1e) g (17)

g f (18 g 1) g (18 � 1)

g f (18 g 1)

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g 0 a00 g f 0

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4 f g f 070 1111(0)

\ g ns 0d 4d d 1111111 d

\ 0 db g 0 n0 ga a s 0 g ns 0 0s1 ]

1111 1

0 ga s n 0 0 s a n 0 a0 a g 0 a0 a 0 1 ns n 0 bd 2

( ) a 0 0p

( ) a 0 ( 0 g 1) p

( ) a 0 (00 g 1) p1

( ) a 0 0 p1

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a0a 0

41 0 +±- 0 0 ±- 0 0 - g � ( )0 +

g d a0aÞ0 0 - 0 a0(p 1 )

a 0 0p g p 1

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Page 5: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

111 8 sp

s d n00n0n d 0 s gn s 1

1 g

1

1 + g1

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1

1 + g1

1 0+ + g111111

011111111140 1

1 0 +++++

= 0 )10(0

+ 0 úû

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-10

1

0

1 Þ 0¥ 0 ]

11104 11

cn 0 a g a g g s g p ( )a +F

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Î

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00 1 (ag1) g g s g p pa

+F

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g (ag1) g g p pa

+F

HGIKJ 0 0

0g d n 0 0 ] 00 a 0 �1 Þ a0 p 0 0 Þ Î 0 ]

1114 1 bn0

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1

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1

1

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1

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111 0 10

00 a0 ag a0 0 gna0 d n0 d a00 C ag 0 0 g 00ng ags 0g d d a00

0 d a00 g s g n d1 2 ns n0 gs g n 0 0 a00 1 00 sn0 0 0 a00 0 0ed

0 0 2 ns

( ) 4 ( ) 48 ( ) ( ) 4

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n 0 sn0 0 0 g s 0 4

a 0 0e

\ sn0

a 0

0

·0e 0 Þ 0 0

\ 2 0 0 ·

4 0 0s1 ]

11104 10

gna0 as sn0 s 0d 7d 81 n0 g n s n0g 0 g aga s g s sn0 ns 0gag0 d

g g sn0 s a , a00 11 0 0 s d 0 ,1 ns

( )

1 ( )

4

1 ( )

7

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e

00

Page 6: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

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s D 0

g· 1 ( sn0 D 0 g·s)

\

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·0 0 0 Þ

g 0

7

0 g ,1 a00 ag sndn nag

g - 0

0

,1Þ

g1- 0

0

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7

1- 0

0

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7

0

0

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7

00 0s1 ] 0s1 ]

7777bn0 00 a0sn 0 0 (1 g g g 1111111 g 7)(1 g g g 111111 g 14) d g 00ngn 0 0 8 ns

( ) 1e ( ) 4 ( ) 078 ( ) 40

00 1 0g 2 00ngn 0 0 g n0 (1 � )� 0d 0 Î ns 0gg�1 g1 222..].22]..]2..]]

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1

1

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1 8

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2 0g g 00ngn 0 0 8 (1 � � 8)(1 � )�0 � · g 00ngn 0 0 10 n0 (1 � )�0 � 1

0 00 � · 1

� 1 0 40 � 10 � 1 0 4 ... ]

111811 11

00 0 sn0

7

p g sn0

4

7

p g sn0

8

7

p a00 0 g s

7

p g g s

4

7

p g g s

8

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g s q 0 4

14 -+ 0

4

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1

0g g s q ns dn0nd d n0 0 0

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0

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p 0s1 ]

Page 7: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

11 00 bn0

00 g 0s a0 gd 0 bn0 dna a0sn 0

0

1 ÷

ø

öçè

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g

1÷ø

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0 g g 0s a0 gd g 0 01111(1)

\ gd 0 g g · g (�1)g 0 � 100

0g g d s b 001

2 0g g gk g 0 0

� 0 0 · 0 0 � 100

\ g 0 0 Þ 0 0 0 0s1 ] 222..].2]].]]2..]]

11 114 sp

0 a g a0 n s a0 00 0 db g 0 s 0 s ag 0 a n0 g a s 0 10 d1 s s 0 s a b

ass db 0 ag 00 dn00 s 0 1 gs 0 ga0 gaggd 0 d 0 s 0 a a nd 1 da0 gaggn 0

8 b s ag n0 gn s 0 n0 dn00 d gaggdn0 s 0 s n0 s gg ssn 0d g bd g gn0 a 0ns a0g 0 418

kd1 0 0 db g 0 s 0 s ns

( ) 1 ( ) e ( ) 01 ( ) 0

02n0 2 g b 0 g 1 s 0 s n1 1 0 s 0 s 0 ag sn0 0 dn00 s 0 1 da0 gn g 0 0 dd ngk

ngk 0ngs s 0 a00 g g0d 40 d1 0 g s g 00 s 0 a00 s 01 0 g 0s

(0 ) ( � 0 g (0 � 1) 0) 0 100(0 g 1) d gs ngk s 0 s 0 0 sn0 d a00 0g

0 0(0 g 1) d d ngk a s 0 s1

\ 00 (0 g 1) 0 4800 d g 0 0 1 1

\ g ag 0 g 1 0 01 s 0 s ]

11 e 1

db g 0 gn0gn a s n 0(s) 0 s a n 0d

sn00 sn0 10·4

= d ns

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222...]]]]]2..]]]

00 1 sn00 sn04

=+ Þ g 4sn0 0 0 sn0 Þ sn0 � 0 sn0 g 1 0 0

Þ ( sn0 � 1)(sn0 � 1) 0 0

\ sn0 0

1 g sn0 0 1

\ 0 0

pd 0

pd

p 0s1 ]

11 00 10

gna0 ns gn a0 0 a 1 n0 s , a00 1 ag 0 d 0 s s g a

, 0 ,1 0 1 1 00 , 0 0 a00 1 0 4 0 0 ns s a

( ) 7 ( ) 00 ( ) 4 ( ) 4

00 1 00 D , 22]]22]2..]].22....2]..]

e 0 g g � g g s b g s 0 0

g

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g

Page 8: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

e 0 g 0

g 1111(1) xxx d 10 0 g

0

b 1111( )

(1) g ( )

0 g 0

1(b g g ) 0 g 0

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7744 11

s 0 a0 s b g 0 0 a00 p sa ns0dn0 s a n 0 4 g s qÞ- g s qÞ- 1 0 0 ns

( ) p p p p1

1

1e

1

0

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RST

UVW

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1

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10

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1

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7

1

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UVW

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g sq 0 8

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4

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4

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1

0

1

1

p=

p-p

p=qÞ

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g sq 0 g s(p� p 1 ) g s(pg p 1 )q 0 7p 1 17p 1 Þ ( ) ]

11 00 bn0

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� a

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1 g a

� a

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222.....].2]2..]]. ]...]

11 07

b a gna0 gn a0 0 a 1 a 0 a ·a

a a

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1

a

1

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a

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1

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++

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a

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Page 9: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]

11 700 bn0

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s

0

èçç

ö

ø÷÷

7

010 d g ns a gd sndn ag sd 0 a gd ns s a

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Þ g 0 Þ 0 0 10 Þ g 001 0 s 0 10 0 ]

11 810 sp

00 0 0 1 g 0 g g 1111111 g (ee) 0 a 0 s d g 4 g 0 g 1111111 g (100) ns

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a00 0 g 4 g 0 g 1111111 g (100)

\ ( � 0) 0 ( � 1 ) g (4 � 0 ) g 1111111 g (100 � ee )

0 (1 g ) g (0 g 4) g 11111111 g (ee g 100) 0 1 g g 0 g 4 g 111111 g ee g 100

0 0 0

\ 0 0 g 0 0 0s1 ]

11 e04 11

00 a gna0 d a0 f a0 a00 a s 0 a00 sa ns0d s a n 0

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11

0

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Page 10: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

00 1 00 D (d

0 ?) 222...]]]2.].]]..22...2]]2.]2...]

°+108sn0

d 0

°00sn0

d + 0

°°

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108sn01111(1)

1 g

d 0

°°

00sn0

7 sn0 0 g s 00°

d 0

1 + � 1 0

1 -

d

0

)1 )(1 (

)1 (

+-

+ 0

1 + ... ]

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s d 0 0 0 g · 0

0 g 0 · 0

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0 ns s a

( ) 1 g · 0 ( ) 1 g 1 ( ) 1 g e · 0 ( ) 0

00 1 0 0 0 g · 0

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01111(1)

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0 (0 0) 0 b sn0 s 0 s a n 0 (1)

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g 0 ·

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0

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00 a0 1,1 gn 0ngs gd a a00 g dd 0 0n00 g 0g 0 (ad 0 ¹ 0)d ga n Þ 0 s d 0

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Page 11: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]

110417 10

00 D n0 a 0 8d b 0 ed g 0 10d 0 a 0 sn0

a0 ns

( ) e

0 ( )

7

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4

1( )

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sn01 22.].2]2...].222]]

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sn0 0

b

g 0

e

10 sn0 g sn0 ag g s 0

ab

gba -+ 0 10

\ sn0

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g s

sn0

sn0 0

e

10 ·

10 0

e

0 0s1 ]

110 1 1

0 db g 0 s n 0 0 s a n 0d å=

1g

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110708 bn0

0 db g 0 a s 0 g sa ns0dn0 s a n 0 d 0e 0 1

0e g g- - 0

0e

1

0e0

g g-- ns

( ) 1 ( ) ( ) 0 ( ) 4

02n0 2 g 0 0 g g 0 0 ns 0 ssnb ]

11081e 10

00 a gna0 d ns bns g g 0 a0 1 00 g s

as a

1

0 a00 l ( ) 0 0d 0

1 1

a b+

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Page 12: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ,]

02n0 2 D 0 D1 g D

0

1

ab sn0 0 ab sn0

g s

0 1

0b sn0

g

1

0a sn0

ÞÞÞÞ

1 1 1

ea b+ = ]

110ee0 10

ad bd g b g sn0 s 0 a gna0 0 s a0ga ng s a n 0

b g (b g g � a ) g g 0 0 as

( ) b nda n0agd g s ( ) b sn n g s

( ) b 0 a n g s ( ) 0 sn n a00 0 0 a n g s1

00 1 b g (b g g � a ) g g 0 0

2 b g g � a 0 bg g s (0g d g sn0 g )

0 ( ) 0 b g ( bg g s ) g g 0 0 a s Î (0d p) n0 a gna0 \ g s Î (� 1d 1) bg g s Î (� bgd bg)

Þ 0 ( bg g s ) � 4b g 0 4b g 3 33 21

)1 (g s

-

-

Þ f 0 Þ ( ) ns g gg g ] 22]..].2.].]]2..]].]]2]

11408 bn0

00 080 g 880 ns 0n n0 0 bd 4ed 0 g dan00 g ns

( ) 0 ( ) ( ) 1 ( ) 0

00 1 (1 g 7)80 g (7 � 1)80 0 (1 g 7)80 � (1 � 7)80 222...3.2]........]2]]]]2..]]]

0 080 1·7 g 80

0·70 g 1111111111g 80

80·780] 0 ( · 7 · 80) g 4e0 g g 0 a0 n0 g

\ 14 2 80 0 110

4e

110 0

4e

0 0

\ g dan00 g ns 0 0s1 ]

114107 11

00 a sn0 g b g s 0 1 a00 a g b 0 1 (ad b 0 0)d 0 g 0sn0 g 0 gn0 s a d 0 s2

I sn0 0 a II a0 0 a b III a0 0 b

( ) 0 d 000 ns 0a s ( ) 0 d 0 ns g

( ) 0 0d 00d 000 d s b g ( ) 0 0 0d 00 g 000 ns g gg g 1

00 1 b g s 0 (1 � a sn0 ) 222..].2.].]]2..]]

b (1 � sn0 ) 0 1 g a sn0 � a sn0

(a g b ) sn0 � a sn0 g 1 � b 0 0

sn0 � a sn0 g a 0 0

(a � sn0 ) 0 0 sn0 0 a Þ a0 0 ba 0s1 ]

114 71 bn0

gd n00 00 0 0 n0 a0sn 0 0 e

1

0

18

èç

ö

ø÷ d 0

0d ns a nd s g gg s 00n0

bn0 dna g 1 00ngn 0 1 0 a ns

( ) 0 ( ) 1

0( ) -

1

0( ) 1

Page 13: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

00 1 g g 1

0 18 g · (e )18 � g · (�1)g · g

g

0

) ( -

g g 1

0 18 g · e18 � g ·

g

0

)1(- · 18 ·

g0

\ 18 �

g0 0 0 Þ g 0 1

18 1 · 1

0

0

e 0 a · 18

Þ a 0 1]

114000 sp

agn d ng d a0 0 0n0 0 db gs n0 n 0 s ed eed eeed 1111111 eeeeeeeee} ns a e 0n n

0 db g d a g s 0n n s ag 0ns n0g 1 0 db g 0 s 0 g 0 an0 0n n

( ) 0 ( ) ( ) ( ) e

02n0 2 0 e

1 e g ee g eee g 1111 g eeeeeeeee} 0 1 g 11 g 111 g 1111111g1111111111 .....222...]]]]]2..]]]

0 1 0 4 0 7 8 e Þ ( ) ]

1144 1 10

�n s a 0 a n 0sd n0 a gna0 d a g s( � ) g b g s( � ) g g g s( � ) ns s a

( )

abg( ) 4

abg( )

abg4( )

abg

00 1 , a 0 sn0 g1 222..].22]..]2..]].]]2]

0 0sn0 g sn0 g sn0 g sn0 g sn0 g sn0 ]

0 (sn0 g sn0 g sn0 ]

0 8 ·

a ·

b ·

g 0

abg 0s1 0s1

g0a n d2 a 0 sn0 0 sn0( g )

0( sn0( g ) g s( � ) g 11111111 g 111111111111]

0(sn0 g sn0 ) g 111111111 g 11111111111

(sn0 g sn0 g sn0 )

8 sn0 · sn0 · sn0 0

abg 0s1 ]

114 78 bn0

cg a s gd n0 bn0 dna a0sn 0 0 (a g )e

g 0

a 0 1 p 0

1

0 ns 2

( ) 0g0 p 4 ( ) 4 p ( ) 0 d 4 ( ) 0 d

00 1

g

g

+ 1 0

e e

e1

e 1 1

e

a

gg g

gg g

-

-- + -

1 ( )

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e 1- +gg

a 0

e 1- +gg

0 0

10 - g

g

0

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0

g n0

10 - g

g

0 0 1 n1 1 0

0

g n1 1 g f 4

2 0g 0 g a s 0 g

0 d

g g 1 f

g

g 0 g 0 4 d g g 1

0 g n1 1

4 0

p b 0 d ag g a s gds ]

Page 14: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

114040 11

00 g s

0g s 0

0

1 0 g s d a0 d 0 £ £

pd 0 a 0

sn0

0sn0 0 g s d d ns

( ) 0

7( )

0

( ) 1 ( )

0

00 1 0sn0 gd sn0

0sn0 �

g s

0g s 0

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g s sn0

sn0 0 ·

sn0

sn0 0

s sn0

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0

1 0

g sn0

0sn0 0 g

0

1 0

0

7 0s1 ] 222...2]]..]2...]

11471 bn0

s d 0 bn0 dna g 00ngn 0 s 0 1

0

ëêù

ûú ns s a 01 g 0s a0 gd n0 a0sn 0

ns

( ) 11 0 ( ) 110 ( ) 1 10 ( ) 0 0

02n0 2 0 g

1 g

g 111111111

0 0 0 Þ 0 0 8 Þ 0 0 8 ]

114800 sp

00 0 g a0 1,1 a1 d a

d a

0 d1111 d a

0 d1111

a1 g a

0 g a

0 � 1 a00 a

1 a a0 0 8

0 a 0 a g a

4 g a

0 s a s

( ) � 1 ( ) � 10 ( ) � 18 ( ) � 1

0 2n0 2 1s gds 0 1,1 ag

a � 0 d a � 0 d a d a g 0 d a g 0

0 g a1 g a

0 g a

0 � 1

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a s a11 a

1 a

0 0 8

(a � 0) ( a � 0)a 0 8

� 4 (� 4 � 0) (� 4 � 0) 0 8 Þ 0 0 � 0

2 0g 1,1 ns d � 1 d � 4d � 7d � 10d � 10 d111111

2 0g a g a

4 g a

0 0 � 1 ]

114e 10

In s a 0 a n 0s n0 a gna0 d ( 0 01 ) · ( 0 0

) · ( 0 0

0 ) as a s a

( ) g ( ) g ( ) 4 g ( ) 10 g

02n0 2 0 01 ns a gdg ng s a0gn a ga gn 0 0

1 as 0nad g 2.2]........]]]22]2..]]]

a s Ð 01 0

-

p

a dn0 sn0 ag n0 01

100

g s

a=

\ 0 01 0

g s

g s·

sn0 ·

0 4 sn0

\ Õ 100 0 04 0 sn0

sn0

sn0

0 10 g ( sn0 4 sn0

· sn0

· sn0

0 g) ]

Page 15: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

11 010 1

0 ga s n 0 0 gn 0 d gng s a n 0

a0 g a0 g a0 0 0 a0 · a0 · a0 0 ns

( ) 0 0p ( ) 0p 0

p( ) 0 0p ( ) 0

0

0p

g g 0 Î 000 1 I k0 g a

a0 · a0 · a0 0 0 a0 0 � a0 � a0

0g a0 g a0 g a0 0 0 a0 0 � a0 � a0 Þ a0 g a0 0 0

\ a0 0 a0 (� )

0 0p �

0 0

0pd 0 Î 0 ]

11 10

s 0 a s 0 sa ns0dn0 snd a0 s d n0 s a n n s ( ) ( )

( )( ) - -

-

8

1010 107

³ 0 a00

- 0 - 01 0 0 ns 2( ) a 0n s ( ) a0 d d s

( ) a0 n00n0n s ( ) a s g 0sns n0 0 ag d g d 0 s 1

02n0 2 a g f 0d 0g n 0 n0 s a n d (1) ns g 0 d n0 g ³ 0n1 1 ( � 8)( � ) f 0 a00 � 0 0 01

n1 1 £ £ 8 a00 � 0 ³ 01 0 d 0 8 sa ns0n s b n0 s a n d ]

11 8 bn0

�n 0 (1 � g � 10 0) (1 g )0 0 1 g a1 g a

g 1111 a00 a

1a 0 a 0 a 0 0 ns

( ) 0 ( ) ( ) ( ) 0

02n0 2 ( 1 � g g 10 0) 0 0 g

1 g

g 1111] 0 1 g a

1 g a

g 11111

a1 0 0 � a00 a

0 0

)10(0+-

-

1a 0 a

(0 � ) 0 0 (0 � 1) � 40 g 10

0 � 40 g 4 0 0 � 0 g 10

0 0 0 0s ]

11 04 11

ga s 0 d 0 sn0 d d 0 g s d d 0 a0 a00 d 0 g s g ag 0gag0 0 sad a s 0g d 0

p 1 g nga n0 ns 0gag0 g n0 g g ga s 0 d 0 g s a00 d 0 a0 gg ssdn0 gs g n0 g g ga s a n0 s a00 1 0 0 n0 s d 0 ns2

( ) 1 ( )

1 -

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1 +

00 1 cn 0 a0 0 g s 2]..]]

g sn0 0 g s 0 1 � sn0 1111(1)

p p 4 1

1 Ö

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g s

a0

1

0 gd g s g � sn0 0 sn0

sn01 - 0 1 (0g d (1) ) ]

Page 16: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

11 4 0 10

s g g g g 0 g 0 ga a0 q ns g 0s g g 0 n0 a gngg gn g 0 g g a00 0 ga0n s 01 ga0n s 0 gngg a ns gngg dsggnb 0 ab gna0 g d ns

( ) 0 g s

q( ) 0 s g

q( ) 0 (g s

q g ) ( ) 0 s g

q

00 1 0 D4

abg D 0 qsn00·0·

1 0 18 sn0 q 222..].2]].]]2..]]

a 0 b 0 0

sn0

q 0

1

gÞ g 0 1

sn0

q

g

g00 q 0

g

0 qq

sn0·18·4

) sn0(1 ·00 0 0 s g

q ...

:.......:.::. g 0 0 g

0

0 a0 ( ) q

g 0 0 g 0 a0 ( ) q 0 0 s g ( ) q Þ g 0 0 s g ( ) q

4g 0 00 s g ( ) q Þ g 0 0 s g ( ) q ... ]

11 bn0

a0sn 0 0 (1 g )0 as 0 g 0s g n gds gn g 00ngn 0 s n0 ga n 1 2 2 0 a00 ga0 b

ggn 0 n0 0 gd 0 k 0

k g 1 2 0

k g 1 s d 0 a ssnb a s 0 (0 g k) ns

( ) 18 ( ) 1 ( ) 8 ( ) 0

00 11k

0k

0

+ 0

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1222..]..]]22]2..]].]]2]

g k0

1k

-+

0

1

k g 0 0 � k

0 � 0k 0 1111(1)

xxx d0

k0

1k0

=+

+

)!1k0()!1k(

!0

--+ · !0

)! k0()! k( --+ 0

0

1k0

k

--+

0 0

0k g 0 0 0 � k �

0 � k 0 8 1111( )

Cg d (1) a00 ( ) 0 0 14 a00 k 0 4

\ 0 g k 0 18 ... ]

Page 17: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

11 0 e bn0

db g 0 ga n 0a gds n0 a0sn 0 0 ( ) 04100

+ ns 2

( ) ( ) 0 ( ) 7 ( ) 8

02n0 2 g g 1

0 100 g

100

- g

1 0g 4

Þ g d s b 0 a00 0n nsnb bd 4 Þ g 0 0d 4d 8d 111111 d 100 ]

11 7 7 10

a £ b £ g b 0 s 0 sn0 s 0 a gna0 1 00 a g b f g 0 g ng 0 0 0 gn0 d s b g ?

( ) 0 a0 s 0 ag ag 1 ( ) 0 d a0 0 ns b s 1

( ) g0 a0 0 ns a gn a0 1 ( ) s g gna0 ga0 ns 1

02n0 2 a g b f g 222....2]...]2]]22]2..]]]

a g b f a g b � ab g s

\ g s f 0

Þ ns b s Þ ( ) ]

11 84e 11

00

0

pp< < d 0 a 0 g ssn 0

1 1

1 1

- + +

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ns

( ) �g

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( ) a0

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sn01 sn01

x g sx sn01 sn01

---+++-

0 ( )

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x g sx1

-+

0 ) (sn0

g s1

--

Þ ( ) ]

11 e8 bn0

g 00ngn 0 0 dn00 gd n0 bn0 dna a0sn 0 n0 g gs 0 0 (1 g a )4 a00 0(1 � a )0 ns sad n0 a s a s

( ) � 0

( )

0

10( ) �

10

0( )

0

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a

4 n0 (1 � a )0 0 0

0(�a )0 0 � 0

0a0 0

0g 4 a 0 � 0

0a0

a 0 � 0

0

4

a 0 �

0

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10

0 0s1 ]

1100108 sp å

=÷÷ø

öççè

æ

+++

000

1k k)1k(1kk

1 ns ga n 0 g g a n gnd sn n n0 gs m a00 n1

a 0 (m g n) ns s a

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00 1 k 0

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1

+++ 0 1kk

k1k

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k

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\ 0 0 å=

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+-

000

1k 1k

1

k

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001

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1e

11- 0

1e

18

\ d g 0 0 07 ... ]

Page 18: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

110147 sp

00 Î d 0 db gs ( 1g g 1 - )d a d ( g � ) 0 gd a0 1,1 0 a d s n n0 n0 g a ( ) 01d ] ( ) 0 d ] ( ) 0 d 1 ] ( ) 01 d ¥)

00 1 1g g 1 - d a d g �

÷ø

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ø

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1

1

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ada

0 ¥ Þ 01 d ¥) 0s ]110

10 bn0 00

0d

1d

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0 0 1 10

1 0 11+ + + +111111 ns s a

( )

11

11

( ) 1

11

11 -( )

0

11

11

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11

11 -

02n0 2 00 ga a0sn 0 0 (1 g )10 b sn0 b g 0 0 a00 11 ]

1100 8 10

gna0 b a0 ns sg s gna0 gn 0 1 0 s a a0 bns g g 0 n s a0

d s sn0 a a n0 a00 a 0 g 1 as g 0 a0 n0 0 g sd ns

( ) 80 ( ) 100 ( ) 110 ( ) 100

00 1 0 g 222..]..2]22]2..]]

a 0 g s

g

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s

sn0 0n0 ag n0 DÞ a00 n0 DÞ

sn0

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4sn0

sn0 Þ

4sn0· sn0

sn0· sn0 4sn0· 0sn0 - 0 1

sn0 0 · sn0 4 � sn0 · sn0 0 sn0 · sn0 4

(g s � g s 7 ) � (g s � g s 0 ) 0 g s � g s 0

g s 0 � g s 7 0 g s � g s 0

sn0 · sn0 0 sn0 4 · sn0

as sn0 ¹ 0 0g sn0 0 sn0 4

g 4 0 180° Þ 0 0° 0g Ð 0 180° � 80° 0 100° ... ]

1104 sp

00 s d 0 0ngs 11 gds 0 a0 agn d nga g g ssn 0 s a s a 0 0ngs 1e gdsd 0

s d 0 n s 0ngs 00 gdsd ns

( ) s a 0 ( ) s a � 1 ( ) s a 1 ( ) 0 0 0ns

00 1

110 a g 100 ] 0

1e0 a g 180] 22]........]2]].]]2..]]]

11 · (a g 0) 0 1e · (a g e0)

11 a g 0 0 1ea g 1710

8a g 1100 0 0 Þ a g e0 0 0

Þ 000 0 0 Þ ( ) ]

Page 19: Binomial_Trigo_LogS.pdf

f f , f ] ]] ] ]]

110 10 bn0

g dan00 gd g 0 (1 0 g 0 0) ns 0n n0 0 bd 1ed ns

( ) 4 ( ) 1 ( ) 0 ( ) 18

02n0 2 0 (1e � 4) 0 g (1e g 4) 0

0 01e 0 g 0 · 1e 1 · 4 g 1111111111 g 0

· 1e · 4 ]

0 · 1e 01e g 0 · 1e 0 · 4 g 111111111g 0

· 4 ]

Þ ns 0n nsnb bd 1e Þ dan00 g 0 0 0s1 ]

1100 0 11

s s g0 n0 0n g ns a n 0 0 a00

g 0 g 0 d s gngg dgngg 0 D a , g g , 0 1 0ndn ag d 1 0 d a00 C 0 t1 00 ad bd g g s g n d

0 0 s sn0 s d a00 0 t

g

d

b

a++

as a s a

( ) a0 g a0 g a0 ( ) g g g g g

( ) g s g g s g g s ( ) g s g g g s g g g s g

02n0 2 0 a0 n0 D, a00 0 a0 0 g D, \ g 0 a 0 ( a0 g a0 )

a 0 a0 g a0

ÞÞÞÞÞÞÞ g s ]

110701 10

00 a D d a 0 a b g

a b g

g s g s g s+ ++ +

ns s a 2

( ) g

( )

g ( )

g( )

g

02n0 2 20 [ ][ ]

sn0 sn0 sn0

sn0 sn0 sn0

+ +

+ + 0

4

4

sn0 sn0 sn0

1 g s g s g s

0 4 sn0

sn0

sn0

0

g

]

1108 1

c 0 ga s n 0 0 s a n 0

s g 0 1 g g s g g s g g s0 g 1111111111 ¥d ns

( ) 0p g 0

p( ) 0p

0

p( ) 0p

0

p( ) 0p g

0

p

g g n ns a0 n0 g1

00 1 g s

1 0

g s1

1

-Þ g s 0

1 0 g s

0

p 0 0p

0

p 0s1 ] 2]..:....2..]]

110e0 11

ag a 0 e0 80 0 0

0 0 110

sn0 sn0 sn0

sn0 sn0 sn0

° ° °° + ° + °

ns s a

( ) 1 ( ) 4 ( ) �1 ( ) 48

Page 20: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

02n0 2 å sn0 0 4Õ

g s n0 g1 as g g 0 p ]

11700 10

In s a 0 a n 0 n0 a D d n0 0 k ( ) ( ) ( )g g g g g g

g g g g g g

1 0 0 1

1 0 0 1

+ + +

+ + g g k as a s a

( ) 1 ( ) ( ) 1 4 ( ) 4

02n0 2 d ga g 0 20 0 k s 1 4 a00 0 0 dn0a g 0 20 0 s ( sn0 g1 0 s a0

g1)

Þ 0 d ga g

0 0 a gdn0 0 20 0 k · 4 0 20 0 Þ k 0

1

4]

117118 bn0

00 0 Î p 0 ns 0 d 0 1

1 1

1

0 0

1

1

1 11 ( ) ! ! ( ) ! ! ( ) !111111

( ) ! !0 0 0 0-+

-+

-+ +

- 0

( ) 0 ( ) 10

0

-

!( ) 0 0 ! ( ) 0 0 0 s

02n0 2 0 g 0

)!g0(!g

!0

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g0

0 )!g0(!g

1

- g 0 1d 0d d 11111111 a00 a00

Þ 20 0 !0

100

1 g 0

0 g 0

g 111111] 0

!0

10-

0s1]

117 07

00 010( � 1) f

010e ( � 1) d 0 n s n0 n0 g a

( ) ( d ¥) ( ) (1 d ) ( ) (1d ¥) ( ) 0 0 0 s

0 00 8 d ]

02n0 2 cn 0 010( � 1) f

010e( � 1)

010 ( - 1) f (1 )

010 ( � 1) g

010 ( - 1) f

010( � 1)1

� 1 0 1 -a00 ( �1) 0 � 1 ÞÞÞ( � ) ( � 1) 0 0 ÞÞÞÞ 0 ]

117007 11

a 0 g g g (00t g ) g g (1 0t g ) ns s a 2

( ) g 0 ( ) a0 0 ( ) 0 a0 0 ( ) 0 e

0

0

-

-

a0

a0 a0

00 1 g g )00 sn0(

)00 g s(

) 00sn0(

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--

+++

0 )00 sn0()00 sn0(

) sn0(

sn0

g s

-++

0 0 sn04

g s sn08

sn0

g s -

+ 0 sn00 sn04

g s sn08 g s0 g s sn040

-

+-

0 sn0

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0- 0 0 g 0 Þ

a0 a00

] a001000

-

- 0s ]

Page 21: Binomial_Trigo_LogS.pdf

f f , f ] ]] ], ]

117488 bn0

g dan00 gd n0 1 g g g 0 g 111111g 1eee ns 0n n0 0 bd ns

( ) 0 ( ) 1 ( ) ( ) 0

02n0 21

)1 (1 000 - 0 000 � 1

( � 1)1000 � 1 0 (1 � )1000 � 1

1 � 1000 1· g 1000

· g 11111111g 1000

1000· 1000 � 1

I ng ns 0n nsnb bd 1 ] 222.3.2]...]2]].]2..]]]

117 7 sp

s1 d s

d s

0 1111111 a00

1 d

d

0 1111111 ag g agn d ng s s 0g s s g a s

1 0

1 ¹ 0 s

0

a00 å=

10

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=

1

1nn 1 0 a 0

1

1

ss

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( ) 8 0 ( ) 0 ( ) 1e 8 ( )

00 1 cn 0 s1 g s

s

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10 0

1 g

g

0 g 11111 g

1 22]..].2.]22]2..]]

1s s s 0g ns

a1d a

1 g 0

1d a

1 g 0

1d 111111111

a00 00 ns a1 d a

1 g 0

d a

1 g 0

d 1111111 (sn0g s

1 0

1)

n 0 s 0

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1 0 (a

1 g 0

)

\ a1 0 0

1 � 0

1111(1)

g a 0n00 1

1

ss

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1

0

0 0 ?

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100 a

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1] 0

1 0 a

1 g 140

]

ns n s a1 0 e0

1 � 10

1111( )

0g d (1) a00 ( )

1

0

0 0

8

1e 0s1 ]

1170 0 bn0

00 a0sn 0 0

0

4

4

00÷÷

ø

ö

çç

è

æ+

- s d 0 bn0 dna g 00ngn 0 s ns 04 a00 gd gn

g a s bn0 dna g 00ngn 0 g 0s ng0 gd bd (0 � 1)d 0 a 0 d s b

( ) 1 ( ) ( ) 0 ( ) � 1

02n0 2 0 g

1 g 111111 g

0Þ 0 0 0 Þ 0 0 0 ]

117774 11

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Page 22: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],,]

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Page 23: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

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Page 24: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

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Page 25: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

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00 0 0400 0

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Page 26: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

2 x sn0 g g s x � x sn0 � g s x

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C g ad bd g 0 01t g d g a 0ns n0g d s a n 0d (a g b ) - b (a g g) g b g g 0 0 as 0 01t g g a g s 1 g0 0 s g s ns a s g 0 s a n 0 2

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Page 27: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

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a 0 g s gp18 � 0 s g

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Page 28: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

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Page 29: Binomial_Trigo_LogS.pdf

f f , f ] ]] ],]]

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Page 30: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

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Page 31: Binomial_Trigo_LogS.pdf

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êê

ù

û

úú ns a d0 dna n0 0 0 g

( ) 7 ( ) 0 ( ) 4 ( ) 0

02n0 2 4 1 + 0 d a00 a00 d

17 0 ( 7

1 d g 7

0 d0 g 7

d g 7

7 d7 )] Þ 0 g ns 0]

1111 111 sp

s d å¥

=÷ø

öçè

æ+10

4 40

0 ns s a

( ) 1 4 ( ) 1 0 ( ) 0 8 ( ) 1

00 1 0 0 4 04)4040(

0

-++ 0 )0 () 0(

0

-+ 0 )0 0)(0 0(

0 -+++

0 0 ú

û

ùêë

é

+-++

+--++

) 0 0)(0 0(

) 0 0()0 0(

4

1

0 úû

ùêë

é

++-

+- 1)10(

1

1)10(

1

4

1 222..]..]].]]2..]].]]2]

00 0 å

¥

=100

0 8

0 ... ]

1111071 sp

00 a ¹ 1 a00 l0 a g (l0 a ) g (l0 a )0 g 1111111 0 0 ( )1111111)a0()a0()a0(a0 40 ++++ llll 0 a ns

s a

( ) 1 ( ) ( ) 0 ( ) 4

00 1

a01

a0

l

l

- 0

a01

a00

l

l

- ®a0 1

a0

l

l

- 0 a01

a00

l

l

- ® (l0 a) � (l0 a) 0 0 l0 a � 0(l0 a)

® 4(l0 a) � l0 a 0 0 ® l0 a 0 0 g 4

11 sd a 0 1d 1 4 ] 222...]2]]]]2...]]

11114 e 1

n b s 0 a s n 0s s a n 0 g s · sn0 ÷ø

öçè

æ +

1 0 01 db g 0 g a 0 db gs

g 0 an0 0 bd n n0 n0 g a (0 f f p)d ns( ) 0 ( ) 1 ( ) ( ) d g a0

00 1 g s · sn0 ÷ø

öçè

æ +

1 0 0 22]..]..]].]]2...].]]2]

g s 0 0 Þ 0 p

sn0 ÷ø

öçè

æ +

1 0 0 Þ

1 + 0 0pd 0 Î 0

n0 Î (0d 1) 0

1 + Î ( d ¥) 0 g 0 0

2 0g g ag n00n0n s n 0 ]

Page 32: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]],]

1111 8e 11

00 D d dn0nd d a 0

Õ

å

g

g 1

g

ns

( ) 1 ( ) ( ) 0 ( ) 0 0 ns 0

02n0 2 0 a0

g a0

g a0

0 g g 0sn0 g

a0

a0 ÷

ø

öçè

æ - ³ 0 g a00 a00 g s 1 ]

11110 4 10

d 0na0s 0 a D ag e gdd 1 gd a00 1 gd g s g n d 1 0 ag a 0 gna0 ns( ) e0 ss gd ( ) 84 ss gd ( ) 7 ss gd ( ) 00 ss gd

02n0 2 ,g 0 g d 0na0 s g a

c 0 1 C n0 a00 1 g c

ns a aga gad 1 a sn0 s 0

D c ag 0d 8d 10 ÞÞÞÞÐc 0 e0t

g a 0

g a 0

D

D

= =

= =

ù

û

úú

1 8

0 8

48

1

1

1

ÞÞ g a 0 D 0 00

ÞÞÞ g a 0 D 0 7 gd ]

1111704 bn0

00 s g 00 gd 0 a0sn 0 aa

a

0

1 10

1

ëêê

ù

ûúú-

ns 14a 0 a 0 0

0

0

ns

( ) 4 ( ) 0 ( ) 1 ( ) 0

02n0 2 0 0

1 ( ) 1010 1a-· aa 0 14a

g 0 · 10

10

a

-

0 14a

0 · 10

140

a

-

0 14

0g 10

140 - 0 0 Þ 0 0 14

0 g

140

14

0

!11·!0

!14

!14

! ·! 0

0

1 0 4 0s1 ]

11118e0 11

a 0 g 71

0

g a0 071

0

� g 071

0

� a071

0

ns 2

( ) a ga n 0a 0 db g ( ) ngga n 0a 0 db g ( ) (0 g 0 ) ( ) (0 � 0 )

Page 33: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

1111e 0 10

00 g1d g d g0 b ga0nn 0 gngg s 0 gna0 d 0

å

å

1

1

gg

g

ns s a 2

( ) å

g ( )

g

g å ( ) å

a0 ( ) Õ

a0

02n0 2 s

a0s å

0 å

a0 Þ ]

111 00e bn0

a 0 (4 · 0 1 g 4 · 0

g 40 · 0

0 g 111111 g 40) ns

( ) 0 ( ) 0 g 1 ( ) 0 ( ) 0 - 10 2n0 2 cn 0 0 4 1 0

1 g 4 1 0

g 40 1 0

0 g 111111 g 40 1 0

0

0 (1 g 4)0 - 1 Þ 0 - 1 Þ ]

111 1e0 11

00 m a00 n ag sn n n0 gs sa ns0dn0

1 g g s q g g s 4q g g s 0q g g s 8q g g s 10q 0 q

qqsn0

0sn0·dg s

0 (m g n) ns s a

( ) e ( ) 10 ( ) 11 ( ) 1

00 1 0 0 g s 0° g g s q g g s 4q g 1111111111 g g s 10q 22]...].]]]2...]

sn0q · 0 0 sn0q 0g s0 g g s q g 1111111111 g g s 10q] 0 sn0 q g sn0 q 0 sn00q � sn0 q 0 sn0 q � sn00q 0 sn07q � sn0 q 0 sn0eq � sn07q 0 sn011q � sn0eq���������

sn0q · 0 0 sn011q g sn0q sn0q · 0 0 sn00q · sn0 q

0 0 q

qqsn0

g s0sn0 0

qqq

sn0

dg s0sn0

Þ 0 0 0 a00 d 0 0s1 ]

111 8 sp

s a ns1111111111111111101810141

71 1011

810141

1011

0141

011

41

1¥++++

( ) 4

1( )

0

1( )

1( ) 1

00 1 0 0

) 0 (0 1111111111110141

)10 111111111( 1011

+-

0 0 g ) � ( 0 g 1)]

0 0

0 1111111111110141

)10 111111111( 1011 - �

) 0 (0 1111111111110141

)10 )(10 111111111( 1011

++-

\ 00 0å 0 0

1 �

) 0 (0 1111111111110141

)10 111111111( 1011

++

a 0¥ 0

1]

Page 34: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

111 0e8 11

dn0nd d a 0 g ssn 0 sn0

4 sn0 e + 0 g Î (0d p) ns

( ) 0

10( ) 0 ( ) 1 ( )

0

8

00 1 0 e sn0 g sn0

400 a sn0 0 0 n0 (0d p) ] 222...]]]]]2..]]] 2]..]]

0

sn0

sn0 0 ÷

ø

öçè

æ- g 1

\ dn0 0 1 g ng gg gs g 0 0 sn0 0 Þ sn0 0 0]

0 a sn0 ns g 0 n0 s a 0 0 a00 a an0s a p g ng ns g a g a0 0 a 0 p d 0g n d s ak 0 n0 (0d p) ]

111 44 bn0

s d 0 g - 00ngn 0 s 0 a 0 g gs 0 n0 a0sn 0 0 ( - 0 g 1)11 ns( ) 1 010 ( ) 0 1 010 ( ) 011 ( ) 0 0

02n0 2 ( - 0 g 1)11 0 a0 g a

1 g a

g 111111 g a

0n00 a

0 g a

g a

4 g 11111 g a

?

0 g 0

0 0 , (1) 0 0 - (1) g g ,( ) 0 ( � 0 g 1)11

0 - 0

0 0 , (- 1) 0 011 - ( )Þ 0

0 011 Þ 0

0 0 1 010 ]

111 117 sp

s d å= ++

100

1k 4 1kk

k ns s a

( ) 10101

4e 0( )

10101

0 0( )

10101

1 1( ) 0 0

00 1 k 0 k)1k(

k

-+ 0 )k1k)(k1k(

k -+++ 222..].2]].2]2...].]]2]

0

1÷÷ø

öççè

æ

-+++

-+-++

)k1k)(k1k(

)k1k()k1k(

0 ÷ø

öçè

æ++

--+ k1k

1

k1k

1

1

0 0 ÷÷ø

öççè

æ

++-

-+å= k1k

1

k1k

1

1

0

1k

0 ÷ø

öçè

æ -0

1

1

1

1

g ÷ø

öçè

æ -7

1

0

1

1

M M

g ÷ø

öçè

æ

++-

-+ 010

1

010

1

1

00 0 ÷

ø

öçè

æ++

-010

11

1 0 ÷

÷ø

öççè

æ

++

-++

010

1010

1

0 )100(

)10(0 ++

+ 0 0 100

10101

0 0 ...]

Page 35: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

111 0 7 10

00 d d a00 t ag 0ns a0g s 0 n0g 0 g 0g d g ng s 0 gna0 g s g n d 0

td

gba ns s a

( ) Õ

a0 ( ) å

g ( ) å

a0 ( ) å

sn0

00 1 0 g g s g

a 0 g ÷ø

öçè

æ +

g

g

sn01

g

g

a÷ø

öçè

æ += 0

sn01

sn0

g s1

sn0

\

sn01

sn0

sn0

g s1

g s1

g s

dt

abg= 0

g 1

g 1

g

00 a gna0 Õ

g 0 å

g ]

111 7 bn0

as g 0n n s 0 0 db g 0 7100 � 0100 ag

( ) 100 ( ) 000 ( ) 00 ( ) 000

02n0 2 g 0sn0 g ( g )100 � ( � )100

0 0100 1 ee · g 100

0 e7 · 0 g 1111111 g 100

ee · ee]

0 01000 · e8 g 100

0 e4 g 1111111 g 1000 · e8]

Þ dn0nd d 000 as as g 0n n s1 Þ ( ) ]

111 887 sp

gngg 0 ga0n s g ns n0sggnb 0 n0 a ss ag 1 dn0 n0 s 0 sn0 s 0 ss ag a b 0

g 00 g 0 bd n0 s d 0 a00 a 0 g ss ag g s 01 sn0 s 0 g s n0 ss ag g g a s

g 00 g 0 bd s d 0 s s a a 0 g ss ag gas b an0 0 a00 s 0d 0 ga0n s 0 gngg

n0sggnb 0 n0 0 ss ag ns

( ) g

01

úúû

ù

êêë

é -

( ) g

000

úúû

ù

êêë

é -

( ) g

0

úúû

ù

êêë

é -

( ) g

00

úúû

ù

êêë

é --

00 1 0n0 0 ss ag 01 0 g

sn0 0 ss ag 0 0 g (a g a 0 4g Þ a 0 g )

0

1

1

g -

÷ø

öçè

æ 0

1

1g

-

úû

ùêë

é

sn0 0 ss ag 00 0 g

10

1-

÷ø

öçè

æ0 g

1÷ø

öçè

æ

a00 s 0 d

Page 36: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

sn0 0 ss ag 00 0

10

1g

-

÷ø

öçè

æ \ ga0n s 0

10

1

g

--

÷÷

ø

ö

çç

è

æ 0 ÷

÷

ø

ö

çç

è

æ -

01

g

a00 s 0d

sn0 0 ss ag 00 0 ( ) 10 1 g

-- 0 ÷÷

ø

ö

çç

è

æ -

01

g ]

111 e04 10

00 n0 a D d g s ·g s g sn0 sn0 sn0 0 1 0d s a d 0 g ng ns n0g gg g d ns( ) D ns ns sg s b 0 gn a0 0 ( ) D ns ag a0 0

( ) D ns gn a0 0 ( ) as a0 0 gna0 ns p4

02n0 2 sn0 0 1

1-

£g s g s

sn0 sn0

1 g g g 0 0 0

0

8

p a00 0

p4

1 £ g s ( � ) Þ g s( � ) 0 1 Þ 0 ]

1110007 10

g 0 g 0 agn d ng d a0 0 0 s 0 sn0 s 0 a gna0 a00 agd 0ng d a0 0

0 s 0 a n 0 s 0 gna0 ns s a 2

( ) D ( ) D ( ) 0 D ( ) 4 D0 g g D ns ag a 0 gna0 ]

0 2n0 2 a 1 0 b

0 g

0 0 D Þ

1 1 1

1 0 + + 0

a b g+ + D

Þ D=++

++

1

1

1

0

gba

0 1

]

1110100 1

00 sn0 g 7 g s 0 e as a as 0 s n 0 0 d s b

( ) a0 00 n0 g ( ) a0 0 n0 g

( ) a ga n 0a 0 db g ( ) a0 ngga n 0a 0 db g

00 1 sn0 g 7 g s 0 e ns ssnb 0 d n0 sn0 0 1 a00 g s 0 1 222..]..]].]]2..]].]]2]

0 (40 g 1)

pa00 0 dp Þ 0

d p(dd 0 Î 0)

\ (40 g 1)

p 0

d pÞ 0

104

d4

+\ Î ga n 0a ... ]

1110 88 10

00 a gna0 d Ð 0 1 0°d 0 0 a00 0 41 00 g 00ng ag g 0s g g 0 0 sn0 a a00 sn0 a d s a 0 ns s a

( ) 0 ( ) 0

08( ) ( )

0

010

00 1 00 a00 d a ,1 22]..]...].]]2..]]

0 a D , ns gn a0 as s g01

0 g a0 00° 0 10

0, ns 00°100°1e0° gna0 d 0g , 0 ( )(0) 0 0]

d

b

40

1 0°00° 00°

00°

q

q

,

0

\ 0 0

10 0

0

010 0s1 ]

Page 37: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

111008e sp

00 abg0 0 1 g g ad bd gd 0 ag sn n g a s 0 dn0nd d a 0

a g b g g g 0 g ab g ag g a0 g bg g b0 g g0 ns

( ) 0 ( ) 10 ( ) 1 ( ) 0

22]........]2]]]]2..]]]

02n0 2 [s ³ c b g 0 n 0 10 0 db gs n1 1 a d b d g d 0 d abd agd a0d bgd b0d a00 g0 ]

1110470 10

gna0 as bas 10 gd 0 a00 bas a0 s 0 0° a00 70°1 00 gnd g 0 gna0

n s

g d g s t° g g t Î (0d e0) 0 a 0 g d g t s a s( ) 00 ( ) ( ) 0 ( ) 40

00 1 cn 0 , 0 g d g s t° 222....]....]]..22]2]2..]]

s , 0 10 g b g g 1111(1)

g sn0 sn0 ag

°00sn0

10 0

° 0sn0

b 0

°70sn0

g

\ b 0 °°

00sn0

0sn010 a00 g 0

°°

00sn0

70sn010

b g g 0 °00sn0

10(sn0 0° g sn0 70°)

0 °00sn0

100 sn0 00° · g s 10°] 0 0 g s 10°

\ , 0 10 g 0 g s 10°

\ 0 10 d 0 0 t 0 10

g d g t 0 40 0s1 ]

1110 48 bn0

sn n a 0 a s a g 00ngn 0 0 ns s a a 0 1 n0 a0sn 0 0

10

0

a ÷

ø

öçè

æ + ns

( ) 1

0( )

1

0( ) 1 ( ) 0

11100e0 sp

s s 0g 0 s n a ga gna0 s ns 0gag01 a n 0 0 ag ns 0 nd s a n 0 0

g g 0n0 gna0 d 0n00 g 0g b g 0 ag a 0 0ngs gna0 a00 sn gna0 ns e08 0

ss ag 0n 1 gnd g 0 0ngs gna0 ns

( ) 10 ( ) 1 ( ) 10 ( ) 18

00 1 a n 0 0 1s D 0 222..2]..]]]]2...]

a n 0 0 00 D 0 0

0g0 D 0 0 g1

0 g sn0 0 1s D ns a 0 0

\ a 0 0·

0

0

Page 38: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

a0 0 0

xxx d a0 0 18

0 g0

4

0) 18·(

4

0 - 0 e08 0

81 � 0

0 e08 Þ

0

4

0 e08Þ 0 1 Þ 0 0

\ a 0 0 ·0

0 4

gnd g 0 1 0s1 ]

11107ee 10

b a gna0 gn Ð 0 0

p a00 0 s g a ( )( ) 0 11 00 agn s 0

0 s ssnb 0 0 a0 bns g g s a s

( ) 1 0 ( ) 1 ( ) 0 ( ) 0

00 1 0 d 0

g s

gb

bg

+ 0

b

b

+(as g 0 ) 222..].2.].2]2...].]]2]

b b 0 1 Þ b 0

1

\ d 0

1

1

1

+=

+

dda

0

1 sn0g dn0nd d a 0 0 0 dn0a g ns n0 0 0 Þ ] ]. ]

11108e0 sp

00 ad b a00 g ag g g 0s g n sn n gds 0 a c1,1 0 ga 0 d 0 a g b g g ns

( ) a g g a n0 gs g s 1a ns a g 0ns n0g n0 s1

( ) 0 ng d b g 1a ns1

( ) 0 ng d ab 1a ns1

( ) a0 0 1a ns1

0 a0a n 02 cn 0 b 0 ag

0 b � 4ag 0 b � 4b 0 � 0b f 0 a s a 0 0 Þ ( ) ] 2]..:....2..]]

1110e4e bn0

0 0 a 0 g 0 g g ng d 18 g - g 1

18 g - 1 g

18 g ³ 0

10 g 0 an0s 2

( ) 4 d 0 s ( ) d 0 s

( ) 7 d 0 s ( ) 10 d 0 s

02n0 2 20 0 g ³ 0

10 Þ 0

g ³ 0

7 Þ g 0 7d 8d ed 10d 11d 1 d 10 Þ ]

11140110 sp

C g g ng sn n n0 gs n ns ga n d

å

å

=

=0

1k

0

1k

k

k

a0 n0 g?

( ) 00 n 0 d ( ) 0 n 0 d

( ) 0 0 1 g 0k 0 dd g g k ³ 0 a00 k Î 0 ( ) 0 0 1 g 0kd n0 g k ³ 0

Page 39: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

02n0 2)10(0·0

·)10 )(10(0

+++

d s b a0 n0 g 222..].22]2.]2..]].]]2]

0

10 + d s b a0 n0 g Þ ( 0 g 1) ns 0n nsnb bd 0

Þ 0 Î 1d 4d 7d 10d 1111111d 0 ns 0 0 gd 0 (0k g 1)d k ³ 0d k Î 0]

]]]]]]]]]]]]]]] ]]]]]

].............]........]....2]2....2]]

0 a n 0 a00 a bns g g ag 0gag0 n0 gna0 0g d g 1 0 ns k0 g0 a

0 0 sn0 0 1d a00 da 0n 0 s 0 a0 s d d d 0 gd a0 agn d ng

g g ssn 01

11141410 10

ag a 0 gngg gngg dsggnbn0 D ns

( ) 8

p( )

4

p( )

p( ) p

1114 414 10

g b gngg dg 0 g 0 D d ga0n s 0 gngg n0sggnb 0 n0 D g ns

( ) 08

1( )

04

1( )

0

1( )

1

1114041 10

b nda 0 n0 gn g s g sn0 0 D d 0 0 ns s a

( ) 4

0( )

4

( )

1( )

0

00 1 0 s d d a00 ag n0 1,1 222..].2.]22]2..]].]]2]

0 a � 0b 0 a � b 0 a g b

a00 0 a g 0b0 gd a � 0b 0 (a g 0b) g (a g b) 0 sn0 gn g a0 g d]

a bg0 ab

�0

a bga bg

a b�

Þ a 0 �7b

\ b 0 � 4

pd a 0

4

7p

a00 Cg d D

0

p=b+a+b-a

b g b 0

pÞ a g b 0

4

p

p66 p63

00

0

00

g

1

\ Ð 0 (a g b) 0

pd

0

p=Ð d

0

p=Ð

Þ ns 00°�e0°�00° gna0

].] g a 0 gngg gngg dsggnbn0 D 0 p

1÷ø

öçè

æ 04

p

]..] D g ns s n a ga Þ g 0 s

D 0

04

1

0

1

1

4

0

=

÷øö

çèæ

÷ø

öçè

æ

Page 40: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

]...] 0 g sn00

p 0

0sn0

1 p 0

4

0

\ 0 0

0 ]

] ] ]66 6 6 6 6 6 ]] ]]

0sn0 g bn0 dna a0sn 0 0 (1 g )0 0 0 g 0d g g 0 ns n0 ga ag 0 a00 0 ns

0gag n 0a ag 0 d 0 Î 1 s s d 0 g 00ngn 0 s 0 ns 0 011

11144bn0 a 0 (0 g � 0) 0 g 0

1 s a s

( ) 7 ( ) 8 ( ) e ( ) 10

1114 bn0 00 n gds ns g a s gd 0 g 0

1d 0 n s a s

( ) 4 ( ) ( ) 0 ( ) 7

11140bn0 00 k gds ns a n0 g a s g 00ngn 0 0 s d 0 a ssnb a (s) 0 k ns

( ) 0 ( ) 7 ( ) 11 ( ) 10

00 1 0 (1 g )0 222..]..]]22]2..]].]]2]

0 1 s d 0 a g 00ngn 0 s

\ 00 0 0 01 0 08 Þ 0 0 8

].] 0 g 0

1 0 ( )81 +

g 0sn0 g ( ) ( ) ( ) n0 g 011111111 1 1 8

0

8

000

88=úû

ùêëé +=-++

++333 3333 21

sn0g 0 ns n0 g Þ 0 g 0 d s b a0 n0 g

b 0 f 0 g 0 f Þ 0 g 0 0 1 Þ 0 � 0 1 � 0

0 g 0 g � 0

0 g (1 � 0) 0 ( ) ( )001 ·1 8 -++ 0 8 g 1 0 e ...

]..] g g 1

n0 (1 g )8 0 8 g( )g 0 8

g g 0 0

1

0 g gg1 ³

g

1

g

1g<>

+Þ 1

1g8

g8

<>-

g1g <>+

1!8

)!ge()!1g(·

)!g8(!g

!8³

---

(e � g) ³ g Þ e ³ g0 g g 0 1d d 0d 4 ns ns g

n1 1 0

4

b 0 g g 0 0 f

Þ ns g a s gd Þ ] ]

Page 41: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]] ]

(nnn) a an0 k g1

0 8 k · k · k

k 0 8

k � 1 · k � 1 · k � 1

k � 1

0 8 k �

· k � · k �

g ga0 0n00 gd a n0 g a s g 00ngn 0

\ k � 1 · 8 k � 1

0 k · 8 k

11111(1)

a00 k � 1 · 8 k � 1

0 k � · 8 k �

11111( )

0g d (1) )!k8(!k

!8·

)!ke()!1k(

·!8 k1k

->

--

-

Þ k

)ke(

1>

- Þ k 0 18 � 0k Þ k 0 0

a an0 k � 1 · 8 k � 1

0 k � · 8 k�

)!k10()! k(

!8·

)!ke()!1k(

·!8 k1k

-->

--

--

Þ k10

1

1k

->

-

Þ 0 � k 0 k � 1 Þ 1 0 0k Þ k f 7

Þ 0 f k f 7 Þ 0 a00

7 gd as g a s g 00ngn 0

Þ k 0 0 g 7 Þ s d 0 0 g 7 0 10 .....]

].............]........]....2]]....2].

ad (d 0 1d d 1111111d ) b ssnb n0 ga a s 0 a 0 g g ng ga s 0

f ( ) 0 a g b g b a00 ( ) 0 � 0b � a d s a s d n0 0 g a g a a s 0 b1

g 0 Õ

=

-

1dd)ag( a00 0

0 0 å

=

0

1gg d 0 Î 1

11147401 sp

dn0nd d ssnb a 0 a ns

( )

1( )

0

( )

08

0( )

40

11148 s d 0 a s 0 0 0 g g ng 00 a0ns s ns

( ) 8 ( ) e ( ) 10 ( ) 1

1114e a 0 å¥

= g g

1 ns s a

( ) 0

1( )

0

1( )

1

1( )

18

1

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Page 42: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]],]

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Page 43: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

a0 0 1

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111 0004 sp

.........]2. 00 7 abg ³ (a g b g g)0 a00 0a g 4b g g 0 1 0 a

1 g 0b

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00 1 cn 0 (abg)1 0 ³ 0

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\ 0a g 4b g g 0 1 Þ a 0 b 0 g 0 1 222..]..]]..]2..]].]]2]

\ a

1 g 0b

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111 400 10

b a0 ag gna0 g s g g 0 g ns a 21 n 0 0g d ns g 0 g 0 d

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Page 44: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

111 00 sp

.........]2. 0n00 g 0g b g 0 s d 0 0ngs 100 0 0a ga 0 db gs a00

s d 0 0ngs 100 00 0a ga 0 db gs ns 1001

e......

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n 00 0a ga 0 db gs ns n1

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( ) 0 a d 0 11 ns g d s a d 0 1 ns 0a s 1

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111 0007 10

b a0 ag a0 gna0 a00 d d C ag 0 0 g 00ng ag 0g d d d

sn0 s d a00 g s g n d1

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111 7008 10

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1

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d a

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Page 45: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

00 1 a1 g (a

1 g 0)d (a

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1000(a

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100)] 0

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0 0 )0a(

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d 0 7sn0 0sn0 sn0 4sn0 0sn0 sn0 sn0

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++++++++++++

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Page 46: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

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00 d 0 7�a ( g g a g 0) ns 0 0n0 0 " Î d 0 ssnb n0 ga a (s) 0 a ns ag

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n1 1 4 � 8(a g 0) f 0 Þ 1 � a � 0 f 0 Þ � a f Þ a 0 �

1111(1)

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00 sn0( - d)d sn0 a00 sn0 ( g d) ag n0 21,1d 0 sn0 1 s gd

0

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d

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d

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d

0 ]

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sn0 g 0 g s d 0 0 a00 0 sn0 d g g s 0 4 0

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( ) a00 d ga0 b g 0 0 gn a0 s 0 a 0141 gna0 gn d 0 1

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4 g e g 1 sn0( g d) 0 Þ sn0( g d) 0 1 0 sn0

p

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p 0 (40 g 1)

p 0 Î 0 Þ ] ]

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pÞ d 0

p �

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4

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00 sn0 b 0 sn0 a g s a 0 g s b as a s a 2

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Page 47: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

00 1 sn0 b 0 sn0 a g s a

sn0

g s1 a=

b-

g s b 0 1� sn0 a Þ ( )

0 1 � g s(p � a) 0 sn0 ÷ø

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Þ ( ) ]

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a

s b s g

a

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-- -

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g 111111)

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g 1 0

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2 0g 0 0 1

1 0 0

1

0 0

g g ng ag n0 1,1 ]

Page 48: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

11170 11 11

g aga g g0s ag 0gag0 0 sad sn0 0 g 0 g 0 a gngg 0 ga0n s 1 0 ns 0 00

a d s b 00 a0 a0 0 Þq a00 q a g 0 g 0 gngg 1 g 00ng ag 0ns a0g b g 0 g g0s ns

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q

( ) 1

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4

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q4

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q

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4

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s a n 0 sn0 a 0 a0(a � b) g g s a · a0 b 0s g n0( ) a 0 0p g b ( ) a 0 0p g b( ) a 0 0p a00 bÞÎ ( ) b 0 0p a00 aÞÎ (n ns a0 n0 g)

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bba-ba

g s

sn0g sg ssn0 0

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ag �ag 0 4Þ ag (g�1) 0 4 1111( )

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a d ag d ag a d ag d ag

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00 g s 0 s a n 0d 0 g g s � 1 0 0 0 gd a0 n0gg asn0 �1,1 g g a00 s ag g a d

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Page 49: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]]]]

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g a g g a g ag 0 � 1111(1)

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1 g g 1 g g 0 � 1111(4)

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Page 50: Binomial_Trigo_LogS.pdf

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1111(1)

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+

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ba

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sn0 Þ ])] ]

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gna0 as 0 a g d gba

g sg g sb g sa

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sn0

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0 0sn0 g sn0 g sn0 ] 0 sn0 g sn0 g sn0 ( sn0 a 0 sn0 g1)

Þ 4 sn0 sn0 sn0 0 sn0 g sn0 g sn0

\ 0 8

abg4 0

gba ++

Þ

abg 0 (a g b g g) 0 s

4D 0 s úû

ùêë

é =D

4

abgÞ

s

D 0 Þ g 0 Þ 2 ]

s g 0 Þ gna0 ns s n a ga Þ 2)] ]

11177 07 1

00 g s 0q 0 g s 0a 0 a 0 sn0 q ga0 b n 0 bd

( ) sn0 a ( ) sn0 ÷ø

öçè

æ a±p0

( ) sn0 ÷ø

öçè

æ a+p0

( ) sn0 ÷

ø

öçè

æ a-p0

Page 51: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]] ]

00 1 g s 0q 0 g s 0a 22]..].2.].]]2..]].]]2]

0 0 0d 1

0q 0 0p 0a\ 0q 0 0a g � 0a g p g 0a g p � 0a

q 0 a g � a g a+p0

g a-

p0

Þ ( )d ( )d ( ) ag g gg g

n0 0 0 � 1

0q 0 � p 0a

q 0 � a±p0

sn0 q 0 sn0 ÷ø

öçè

æ a±p

-0

0 � sn0 ÷

ø

öçè

æ a±p0

0 � sn0 ÷

ø

öçè

æ a±p

-p0

0 � sn0 ÷÷ø

öççè

æ÷ø

öçè

æ a±p

-p0

0 � sn0 ÷ø

öçè

æ a±p0

0g ( ) ns 0 g gg g 1 ]

11178 17 sp

00 gn s ad bd g a00 ( a � b)d ( b � 0g)d ( 0g � a) ag n0

agn d ng g g ssn 0 0

( ) 18(a g b g g) 0 18(a g b g g ) g ab ( ) ad bd g ag n0 c1,1

( ) ad bd 0g ag n0 21,1 ( ) ad bd g ga0 b 0 s 0 sn0 s 0 a gna0

( ss d a agn dng gds b 0 0n0 0)

00 1 ad bd g ag n0 1,1 222..]..]]..]2..]].]]2]

Þ b 0 a g g

\ b 0 ag 1111(1)

Þ ad bd g ag n0 c1,1 Þ ] ]

a s n 0 ( a � b)d ( b � 0g)d ( 0g � a) ag n0 1,1

Þ ( b � 0g) 0 0g � b

Þ 0 b 0 0 0g

\ b 0 0g 1111( ) Þ 4b 0 eg 1111(0)

0g d (1) a00 (0)

4ag 0 eg Þ a 0 4

ge a00 b 0

g0

a 0 4

ge b 0

g0 a00 g 0 g

\ ad bd g 0 gds sn0 s 0 gna0 Þ ]]]

a g b 0 g b g g 0 a g g a 0 b

b ad b a00 0g ag 0 n0 21,1

xxx d gn0d ] ] ]

1117e 08 bn0

00 ( )e 80+

0

0 1 g 0 g g 0

d 0 ag n0 gs a00 0 f 0 f 1d 0

( ) 0 ns a0 00 n0 g ( ) 0 ns a0 0 n0 g

( ) (0 g 0) (1 - 0) 0

1 ( ) 1 - 0

0

( )e 80-0

00 1 ( )0 4e + 0 0 g 0

( )0 4e - 0 0

Page 52: Binomial_Trigo_LogS.pdf

f f , f ] ]] ]],]

0 g 0 g 0 0 ( )0 4e + g ( )0 4e - 0 0 n0 g

0 f 0 g 0 f Þ 0 g 0 0 1

\ 0 ns a0 00 n0 g Þ ( ) ]

11180 1 sp

1d

ag g s 0 s a n 0 � 0 g 0 0

0 d

4 ag g s 0 s a n 0

� 1 g 0 0d s g a 1d

d

0d

4 0 gd a0 n0gg asn0 c1,1d 0

( ) 0 ( ) 0 0 ( ) 1 g

0 0 ( )

g

4 0 10

0 1 1 g

0 0

0 g

4 0 1 222..]..]].2]2...].]]2]

1 0

0 4 0

1 0 a

0 ag

0 0 ag d

4 0 ag0

a g ag 0 0 ag (1 g g) 0 1

a g 0 g (0) 0 1

1d d 4d 8 g 0 4

0 1 g

0 0 4 g 0 a 0 1

0 0 g

4 0 10 ]