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lwm pl:ún nig Esn Bisidæ

sñaédeá£Bum<pßay

esovePAKNitviTüa ½ 1-dMeNa£RsaylMhatKNitviTüa eá£Bum< qñaM2000 ( sRmabeRtomRbLgcUlsaklviTüal&y nig GaharUbkrN_ ) 2-BiPBsVIútcMnYnBit ( sRmabfñak´TI11 nig sisßBUEkKNitviTüa ) 3- GnuKmn_RtIekaNmaRt ( sRmabfñak´TI11 nig sisßBUEkKNitviTüa )

4- dMeNa£RsayKMrU cMnYnkMupøic lImIt edrIev ( sRmabfñak´TI12) 5- segçbrUbmnþKNitviTüa ( sRmabfñak´TI11-12 ) 6- KMrUsikßaGnuKmn_ ( sRmabfñak´TI11-12 ) 7- kMEnlMhatKNitviTüafñakTI10km μviFIsikßafμI ( PaK1 qñaM2008 ) 8- 151 KNnalImIt ( sRmabfñakTI11-12 ) 9- KNitviTüaCMuvijBiPBelak PaK1- PaK2 nig PaK3 ( sRmabsisßBUEkKNitviTüafñakTI11 nig 12 ) 10- kMEnlMhatKNitviTüafñakTI11 kMritmUldæan km μviFIsikßafμI ( PaK1 qñaM2009 ) 11-sIVúténcMnYnBit sRmab´fñakTI11 kMritx<s´ nig kMritmUldæan

GñkcUlrYmRtYtBinitübec©keTs

elak lwm qun elak Esn Bisidæ elakRs I Tuy rINa elak Titü emg elak RBwm sunitü elak pl b‘unqay

GñkRtYtBinitüGkçraviruTæ elak lwm miKþsir

karIkMuBüÚT&r kBaØa lI KuNÑaka

GñkniBnæ nig eroberog elak lwm plþún nig elak Esn Bisidæ

GarmÖkfa esovePA GnuKmn_ Gics,:ÚNgEsül elakarIt nig RtIekaNmaRt sRmabfñak´TI11 kMritmUldæan nig kMritx<s´ EdlGñksikßakMBugkanénAkñúgéden£ xMJúáTáneroberog niBnæ eLIgkñúgeKalbMngTukCaÉksarsRmabGñksikßaEdlman bMngcgýldwgGMBIemeronsIVúténcMnYnBiten¼[kanÉtc,as´ nigmüageTotedIm,I CaCMnYydlGñksikßaEdlmanbMngeRtomRbLgsisßBUEkEpñkKNitviTüa nig eRtomRbLgykGaharUbkrN_nana . enAkñúgesovePAen£eyIgxMJúánxitxMRsavRCaveRCIserIsyklMhatýa¨g sRmaMgbMputykmkeFIVdMeNa£Rsayyagek,a£k,ayEdlGac[elakGñk gayylqabcgcaMGMBIsil,£énkareda£RsayTaMgGsén£ . büEnþeTa£Caya¨gNak¾eday kgV£xat nig kMhusqþgedayGectnaRákd CaekItmanCaBMuxan TaMgbec©keTs nig GkçraviruTæ . GaRs&yehtuen£ eyIgxMJúCaGñkeroberogrgćaMedayrIkrayCanic©nUvmtiri£Kn´ EbbsSabnaBIsMNak´GñksikßakñúgRKbmCÄdæanedIm,ICYyEklMG esovePAen£ [ánkanÉtsuRkiRtPaBEfmeTot . CaTIbBa©bén£eyIgxMJúGñkeroberogsUmeKarBCUnBrdlGñksikßaTaMgGs´ [mansuxPaBmaMmYn nig TTYlC&yCMn£RKbParkic© .

átdMbgéf¶TI 12 kk;da 2009 GñkniBnæ lwm plþún Tel : (017) 768 246

Éksareyag 1> esovePAKNitviTüafñak;TI11 kMritmUldæan nig kMritx<s;rbs;RksYg Gb;rM qñaM 2009 2> Gius,Ú:Ng;Esül nig elakarIt rbs;elak lwm yUesg qñaM1999 3>103 Trigonometry Problems ( Titu Andreescu and Zuming Feng )

4> 360 Problems for Mathematical Contests ( Titu Andreescu and Dorin Andrica )

5> Mathematical Olympiad Treasures ( Titu Andreescu and Boddan Enescu )

6> International Mathematical Olympiads 1959-1977 ( Samuel L. Greitzer )

eroberogeday lwm pl:ún nig Esn Bisidæ

- 1 -


- 2 -

EpñkRbFanlMhat 1111----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ 22 3243 22)( xxxxxf −−−−++++++++−−−− ++++==== k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf . . . . x> rktémø x> rktémø x> rktémø x> rktémø x edIm,I[ edIm,I[ edIm,I[ edIm,I[ 20)( ====xf . . . . 2222----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_

22

22)(

1

++++++++====

++++

x

x

xf

k> k> k> k> cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit m nig nig nig nig n ebI ebI ebI ebI 1====++++ nm enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa 3)()( ====++++ nfmf . . . . x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk ∑∑∑∑

====

++++====

p

kp p

kfS

1 1 . . . .

3333----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xx

xx

ee

eexf −−−−

−−−−

++++−−−−====)(

k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit a nig nig nig nig b cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³ (((( ))))

)()(1)()(

bfafbfaf

baf++++

++++====++++


- 3 -

4444----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ (((( )))) (((( ))))xxxf 3232)( −−−−++++++++====

k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 2)( ≥≥≥≥xf . . . . K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar 14)( ====xf . . . . 5555----edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> 02433.49 2 ====++++−−−− ++++xx x> x> x> x> 01282.94 2 ====++++−−−− ++++xx K> K> K> K> 026.53 32132 ====++++−−−− ++++++++++++ xxx X> X> X> X> (((( )))) (((( )))) 102323 ====−−−−++++++++ xx g> g> g> g> (((( )))) (((( )))) 143232 ====++++++++−−−−

xx 6666----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³

x

xx

x

xx

x

xx

x

xx

6

6

5

5

4

4

3

32222 −−−−−−−−−−−−−−−−

====++++++++ 7777----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 882264

2)1(2)1(2)1( ++++++++====++++ −−−−−−−−−−−− xxx 8888----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ (((( )))) 222210114 6333 xxxxx −−−−====++++++++ −−−−−−−−−−−−


- 4 -

9999----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 04882)14(4 ====++++−−−−−−−−++++ xx xx 10101010----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 0

421

215

7443

2.34.9 ====−−−−++++

−−−−−−−− −−−−−−−− xxxx

11111111----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³ 324232

3331

3 −−−−++++−−−−−−−− ++++====++++ xxxxx 12121212----edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar ³³³³

====

====

4321

43

721

32

yx

yx

13131313----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³

323.2 11

2 ====++++−−−−

−−−− xx

x 14141414----eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday 10 ====a nig TMnak;TMng nig TMnak;TMng nig TMnak;TMng nig TMnak;TMng kMeNIn kMeNIn kMeNIn kMeNIn (((( ))))nana

na ++++++++ ++++==== 1

21 24log Edl Edl Edl Edl ...;2;1;0====n . . . . k>eKtag k>eKtag k>eKtag k>eKtag na

nb 21++++==== . . . .


- 5 -

bgðajfa bgðajfa bgðajfa bgðajfa 21 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n ....

x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa n

nb 23==== . . . . K>TajrktY K>TajrktY K>TajrktY K>TajrktY na énsIVút énsIVút énsIVút énsIVút (((( ))))na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . 15151515----eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday 10 ====a nigTMnak;TMng nigTMnak;TMng nigTMnak;TMng nigTMnak;TMng kMeNIn kMeNIn kMeNIn kMeNIn (((( ))))nbnana

na ++++++++++++ ++++++++==== 121


nb 31++++==== . . . . bgðajfa bgðajfa bgðajfa bgðajfa 3

1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n .... x>KNna x>KNna x>KNna x>KNna nb nig nig nig nig na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . 16161616----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_

++++−−−−====

xx

xf11

lg)(

k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit 11 <<<<<<<<−−−− a nig nig nig nig 11 <<<<<<<<−−−− b cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa (((( )))) (((( ))))

++++++++====++++abba

fbfaf1

. . . .

17171717----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit (((( )))) 0≥≥≥≥nnu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³ au ====0 nig nig nig nig (((( )))) nu

nn uu 2log1 ====++++ Edl Edl Edl Edl 2>>>>a . . . .

k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n . . . .


- 6 -

x>eKtag x>eKtag x>eKtag x>eKtag nn uV 2log==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa 21 nn VV ====++++ . . . .

K>eKtag K>eKtag K>eKtag K>eKtag nn VW 2log==== . . . . bgðajfa bgðajfa bgðajfa bgðajfa (((( ))))nW CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt X>KNna X>KNna X>KNna X>KNna nn VW ; nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . 18181818----eK[ eK[ eK[ eK[ x nig nig nig nig y CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; 12.544 −−−−++++====++++ yxyx . . . . cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa

++++++++====++++3

22log21 2

yx

yx

19191919----edaHRsaysmIkarxedaHRsaysmIkarxedaHRsaysmIkarxedaHRsaysmIkarxageRkam ageRkam ageRkam ageRkam ³³³³ k> k> k> k> (((( )))) 0

4loglog 2

22 ====

++++ xx

x> x> x> x> (((( )))) 04

loglog3

5,02

2 ====

++++ x

x

K> K> K> K> 4log1log2 xx ++++==== X> X> X> X> (((( ))))(((( )))) (((( )))) 0212log312log 3

2

3 ====++++++++−−−−++++ xx g> g> g> g> 03437.87 2log12

2log ====++++−−−− ++++ xx 20202020----edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) (((( )))) 06log5log 2

22 ====−−−−++++−−−−++++ xxxx


- 7 -

x> x> x> x> (((( )))) 0216log)5(2log 32

3 ====++++−−−−−−−−++++ xxxx

K> K> K> K> (((( )))) 044log3log21

2

21 ====−−−−++++++++++++

xxxx

21212121----k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa AaBa BA loglog ==== Edl Edl Edl Edl BAa ;; CabIcMnYnBitviC¢man nig xusBI CabIcMnYnBitviC¢man nig xusBI CabIcMnYnBitviC¢man nig xusBI CabIcMnYnBitviC¢man nig xusBI 1 . . . . x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar 6867 72log2log ====++++ xx 22222222----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar (((( )))) (((( )))) 2loglogloglog 2442 ====++++ xx 23232323----eK[sIVút eK[sIVút eK[sIVút eK[sIVút )21(log 2

2

n

nu ++++==== Edl Edl Edl Edl INn ∈∈∈∈ KNna KNna KNna KNna (((( )))) n

n

kkn uuuuuS ++++++++++++++++======== ∑∑∑∑

====....210

0 . . . .

24242424---- eK[sIVút eK[sIVút eK[sIVút eK[sIVút

−−−−==== 12

cos2ln nn

xu Edl Edl Edl Edl INn ∈∈∈∈

KNna KNna KNna KNna (((( )))) n

n

kkn uuuuuS ++++++++++++++++======== ∑∑∑∑

====....210

0 . . . .

25252525---- edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar

====

====

9001

6.5

4001

5.4

yx

yx


- 8 -

26262626----edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³ ³ ³ ³ 0logloglog30 3

42

5,03 ====−−−−++++ xxx xxx

27272727---- edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar ³³³³

====

====++++

64

26)loglog(5

yx

yx xy

28282828----eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½ 06)54(log5)54(log 3

632

6 ====++++−−−−−−−−−−−− −−−−−−−− xx xx ¿¿¿¿

29292929----eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½ )9(log

31

)9(log1 323

322)3( 3 xxxx xx

−−−−====−−−−++++ −−−−−−−−¿¿¿¿

30303030----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_

++++++++

++++++++====

1344

log.1

13log)(

2

222 xx

x

xxf

k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar

++++++++====

++++++++

1344

log1

13log

2

222 xx

x

x

x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam

++++++++

1

13log

22 x

x nig nig nig nig

++++++++1344

log2

2 xx viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña .


- 9 -

K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ )(xf elIcenøa¼ elIcenøa¼ elIcenøa¼ elIcenøa¼ ]3;0[ . . . . 31313131----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ IRxxxxf ∈∈∈∈++++++++==== ,)1ln()( 2 ccccUrRsaybBa¢ak´fa UrRsaybBa¢ak´fa UrRsaybBa¢ak´fa UrRsaybBa¢ak´fa )

11()

1(

bb

aa

fba

baf

++++++++

++++<<<<

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

cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb 0,0 >>>>>>>> ba .... 32323232----eK[ eK[ eK[ eK[ 1≥≥≥≥a nig nig nig nig 1≥≥≥≥b . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )

2(log2loglog 222

baba

++++≤≤≤≤++++

33333333----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³ (((( ))))nn

nun ++++

++++==== 222 log.

11log

Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ++++++++++++++++==== ....321 CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n K>kMnt; K>kMnt; K>kMnt; K>kMnt; n edIm,I[ edIm,I[ edIm,I[ edIm,I[ 100====nS . . . . 34343434----eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xxxf 2log2)( ++++==== Edl Edl Edl Edl 0>>>>x eKtag eKtag eKtag eKtag (((( )))) (((( ))))(((( )))))(;)(;)( 321 xfffuxffuxfu ============ nig nig nig nig (((( ))))(((( ))))(((( ))))....)(.... xffffu nn ==== . . . . k>eKyk k>eKyk k>eKyk k>eKyk nn uV 2log1++++==== . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa 2

1 nn VV ====++++


- 10 -

x>eRbIGnumanrYmKNitviTüacUrRsayfa x>eRbIGnumanrYmKNitviTüacUrRsayfa x>eRbIGnumanrYmKNitviTüacUrRsayfa x>eRbIGnumanrYmKNitviTüacUrRsayfa n

VVn2

1==== . . . . K>KNna K>KNna K>KNna K>KNna nV nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig x . . . . 35353535----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

(((( ))))2log )1( ++++==== ++++ nu nn Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrk>cUrk>cUrk>cUrbgðajfa bgðajfa bgðajfa bgðajfa (((( ))))nu CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ln....lnlnln 321 ++++++++++++++++==== CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n 36363636----eK[ eK[ eK[ eK[ xxxxf 2424 sin4coscos4sin)( ++++−−−−++++==== k> sRmYl k> sRmYl k> sRmYl k> sRmYl )(xf x>kMnt; x>kMnt; x>kMnt; x>kMnt; x edIm,I[ edIm,I[ edIm,I[ edIm,I[

21

)( ====xf

37373737----cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; INn∈∈∈∈ eK[ eK[ eK[ eK[ 12

sin12

cosππππππππ nn

nS ++++==== k> KNnatémø k> KNnatémø k> KNnatémø k> KNnatémø

12cos

ππππ nig nig nig nig 12

sinππππ

x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 0624 12 ====++++−−−− ++++++++ nnn SSS 38383838----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa

21

73

cos7

2cos

7cos ====++++−−−− ππππππππππππ

(IMO 1963)


- 11 -

39393939----KNna KNna KNna KNna )5

3sin41()

5sin41( 22 ππππππππ −−−−−−−−====A

40404040----eK[ eK[ eK[ eK[ 144

)( 2

3

−−−−++++====

xx

xxf . KNnatémø . KNnatémø . KNnatémø . KNnatémø )

7(cos

ππππf . . . .

41414141----eK[ eK[ eK[ eK[ )cos(sin1

)( xxk

xf kkk ++++====

Edl Edl Edl Edl ...;3;2;1====k cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa

121

)()( 64 ====−−−− xfxf . . . . 42424242----cUrbgðajfacMeBaHRKb;cMnYnKt;FmcUrbgðajfacMeBaHRKb;cMnYnKt;FmcUrbgðajfacMeBaHRKb;cMnYnKt;FmcUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati µCati µCati µCati n nig cMeBaHRKb; nig cMeBaHRKb; nig cMeBaHRKb; nig cMeBaHRKb; cMnYnBit cMnYnBit cMnYnBit cMnYnBit t

kx

2

ππππ≠≠≠≠ Edl Edl Edl Edl nt ....,,2,1,0==== nig nig nig nig k CacMnYnKt;eKmansmPaB CacMnYnKt;eKmansmPaB CacMnYnKt;eKmansmPaB CacMnYnKt;eKmansmPaB ³³³³

xxxxx

nn 2cotcot

2sin

1.....

4sin1

2sin1 −−−−====++++++++++++

(IMO 1966)

43434343----eK[ eK[ eK[ eK[ dcba ,,, CacMnYnenAkñúgcenøa CacMnYnenAkñúgcenøa CacMnYnenAkñúgcenøa CacMnYnenAkñúgcenøaH H H H ];0[ ππππ edaydwgfaedaydwgfaedaydwgfaedaydwgfa

++++====++++++++====++++

)cos2(cos4cos7cos

)sin2(sin4sin7sin

dcba

dcba

cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )cos(7)cos(2 cbda −−−−====−−−− . . . .


- 12 -

44444444----cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam ³³³³ )sin()sin()sin( xzzyyx −−−−++++−−−−++++−−−− 45454545----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag

2cba

p++++++++==== CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN .

kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa bc

cpbpA ))((2

sin−−−−−−−−==== rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng

BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH . xxxx----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa

81

2sin

2sin

2sin ≤≤≤≤CBA

46464646---- eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaNABC ehIyeKtag ehIyeKtag ehIyeKtag ehIyeKtag r nig nig nig nig R erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg; carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa

Rr

CBA ++++====++++++++ 1coscoscos 47474747----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag

2cba

p++++++++==== CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy r nig nig nig nig R

erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrprcabcab 422 ++++++++====++++++++ xxxx----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrrpcba 822 22222 −−−−−−−−====++++++++


- 13 -

48484848----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag

2cba

p++++++++==== CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy r

CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa ³³³³ k> k> k> k> r

Ccp

Bbp

Aap ====−−−−====−−−−====−−−−

2tan)(

2tan)(

2tan)(

x> x> x> x> p

rRCBA ++++====++++++++ 42

tan2

tan2

tan

K> K> K> K> rpCBA ====

2tan

2tan

2tan

X> X> X> X> rpCBA ====++++++++

2cot

2cot

2cot

49494949----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYyman mYyman mYyman mYymanRCug RCug RCug RCug cba ;; . . . . tag tag tag tag S CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 22

coscoscosR

abcCcBbAa ====++++++++

x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa SCcBbAa 4cotcotcot 222 ====++++++++ 50505050----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag r CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )(2cotcotcot RrCcBbAa ++++====++++++++


- 14 -

51515151----bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman

2tan

2tan

CBAbaba −−−−====

++++−−−−

52525252----bgðbgðbgðbgðajfakñúgRtIekaN ajfakñúgRtIekaN ajfakñúgRtIekaN ajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman )

2sin

2sin

2(sin4 222

222 CBAabc

cab

bca ++++++++≥≥≥≥++++++++

53535353----bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman 3333 16

81coscoscos

pc

C

b

B

a

A ≥≥≥≥++++++++

54545454----bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyEdl mYyEdl mYyEdl mYyEdl CBA ≠≠≠≠≠≠≠≠ eKmaneKmaneKmaneKman k> k> k> k> 0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa x> x> x> x> abc

BAACCBBAcACbCBa

3)sin()sin()sin(

)(sin)(sin)(sin 333333

====−−−−−−−−−−−−

−−−−++++−−−−++++−−−−

55555555----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

4sin.2

ππππnU

nn ==== Edl Edl Edl Edl *INn ∈∈∈∈

k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 4

sin4

cos4

)1(cos.2

ππππππππππππ nnn−−−−====

++++ x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa

4)1(

cos)2(4

cos)2( 1 ππππππππ ++++−−−−==== ++++ nnU nn

n K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk nn UUUUS ++++++++++++++++==== .........321 CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . .


- 15 -

56565656----eK[ eK[ eK[ eK[ x CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl 0217160 2 <<<<++++−−−− xx . . . . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 0

13sin <<<<

−−−−xππππ . . . .

57575757----eK[ eK[ eK[ eK[ 2

0ππππ<<<<<<<< x . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa ³³³³

21cos

11

sin1

1 ++++≥≥≥≥

++++

++++xx

55558888----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC . . . . 1 1 1 1----cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; [,0], ππππ∈∈∈∈yx cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³ )

2(sin2sinsin

yxyx

++++≤≤≤≤++++ . . . .

2 2 2 2----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa 2

33sinsinsin ≤≤≤≤++++++++ CBA . . . .

3 3 3 3----Tajbgðajfa Tajbgðajfa Tajbgðajfa Tajbgðajfa ³³³³ k/ k/ k/ k/

833

sin.sin.sin ≤≤≤≤CBA

x/ x/ x/ x/ 8

332

cos2

cos2

cos ≤≤≤≤CBA 59595959----eKeKeKeK[sIVúténcMnYnkMupøic [sIVúténcMnYnkMupøic [sIVúténcMnYnkMupøic [sIVúténcMnYnkMupøic )( nZ kMntéday kMntéday kMntéday kMntéday ³³³³

(((( ))))

∈∈∈∈++++====

++++====

++++ INnZZZ

iZ

nnn ;||212

31

1

0


- 16 -

( ( ( ( || nZ CamUDulén CamUDulén CamUDulén CamUDulén nZ ) . ) . ) . ) . snµtfa snµtfa snµtfa snµtfa INniZ nnnn ∈∈∈∈∀∀∀∀++++==== ,)sin.(cos θθθθθθθθρρρρ Edl Edl Edl Edl IRnnn ∈∈∈∈>>>> θθθθρρρρρρρρ ;,0 . . . . kkkk----rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag nθθθθ nignignignig 1++++nθθθθ ehIy ehIy ehIy ehIy nρρρρ nignignignig 1++++nρρρρ . . . . xxxx----rkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVút )( nθθθθ rYcKNnarYcKNnarYcKNnarYcKNna nθθθθ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . KKKK----cUrbgHajfacUrbgHajfacUrbgHajfacUrbgHajfa

2cos....

2cos

2coscos 121

00−−−−==== n

nθθθθθθθθθθθθθθθθρρρρρρρρ

rYbBa¢ak rYbBa¢ak rYbBa¢ak rYbBa¢ak nρρρρ GnuKmn_én GnuKmn_én GnuKmn_én GnuKmn_én n . . . . 60606060----eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit )( nU kMntelI kMntelI kMntelI kMntelI n eday½eday½eday½eday½

10 ====U nig nig nig nig aaUUINn nn sincos: 1 ++++====∈∈∈∈∀∀∀∀ ++++ Edl Edl Edl Edl

20

ππππ<<<<<<<< a . . . . k¿ tag k¿ tag k¿ tag k¿ tag

2cot

aUV nn −−−−==== . . . .

cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )( nV CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy . x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt )....(lim 10 n

nVVV ++++++++++++

+∞+∞+∞+∞→→→→ nig nig nig nig n

nU

+∞+∞+∞+∞→→→→lim


- 17 -

61616161----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMntéday ½kMntéday ½kMntéday ½kMntéday ½ 1;0 10 ======== UU nig nig nig nig nnn UaUUINn −−−−====∈∈∈∈∀∀∀∀ ++++++++ cos2: 12

Edl Edl Edl Edl IRa ∈∈∈∈ .... k¿ tag k¿ tag k¿ tag k¿ tag INnUaiaUZ nnn ∈∈∈∈∀∀∀∀−−−−−−−−==== ++++ ,)sin(cos1 . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa nn ZaiaZ )sin(cos1 ++++====++++ rYcTajrk rYcTajrk rYcTajrk rYcTajrk

nZ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ n nig nig nig nig a . . . . x¿ Tajrk x¿ Tajrk x¿ Tajrk x¿ Tajrk nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n rYcTajrk rYcTajrk rYcTajrk rYcTajrk n

aU

0lim

→→→→ . . . .

66662222----eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½ 223)

12cos()

4sin(4 ++++====++++++++ ππππππππ

xx 63636363----eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;elI kMnt;elI kMnt;elI kMnt;elI IN eday eday eday eday ³³³³

22

0 ====U nig nig nig nig INnU

U nn ∈∈∈∈∀∀∀∀

−−−−−−−−====++++ ,

2

11 2

1

KNna KNna KNna KNna nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . .


- 18 -

64646464---- eK[ eK[ eK[ eK[

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

====++++++++

====++++

====

4

3

2

2cos2222

2cos222

2cos22

ππππ

ππππ

ππππ

BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg BI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg 65656565----ykykykyk a; b ;c CaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nig A,B,C CargVas;mMuTaMgbICargVas;mMuTaMgbICargVas;mMuTaMgbICargVas;mMuTaMgbI énRtIekaN énRtIekaN énRtIekaN énRtIekaN ABCmYyEdl mYyEdl mYyEdl mYyEdl S CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH.

Rsayfa Rsayfa Rsayfa Rsayfa S

cbaCotCCotBCotA

4

222 ++++++++====++++++++ 66666666---- eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cABbACaBC ============ ,, . cUrRsaybBa¢k;fa . cUrRsaybBa¢k;fa . cUrRsaybBa¢k;fa . cUrRsaybBa¢k;fa ³³³³

abccba

cC

bB

aA

2coscoscos 222 ++++++++====++++++++

67676767----edaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkar k> k> k> k> (((( )))) 0)sin2(logsinlog2 3

22

2 ====++++ xx

x> x> x> x> 02coslog3coslog 2

2

22 ====++++++++

xx


- 19 -

68686868----edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar

(((( )))) (((( ))))(((( ))))

====

====yx

yx

sinlnsinln

2ln2ln

22

sin2sin2

Edl Edl Edl Edl 2

0ππππ<<<<<<<< x nig nig nig nig

20

ππππ<<<<<<<< y . . . . 69696969----edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar

++++====++++

====++++

2222

23

sinsin

sinsin yx

yx

70707070----eK[ eK[ eK[ eK[ 2

a0ππππ<<<<<<<< nig nig nig nig

2b0

ππππ<<<<<<<< . . . .

cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa 1bcosacos

bsinasin

2222

====

++++

lu¼RtaEt lu¼RtaEt lu¼RtaEt lu¼RtaEt ba ==== . . . . 71717171----eK[ eK[ eK[ eK[ ABC CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞatlkç&x&NÐ AcosCsinBsin1CsinBsin 22 ++++====++++ . . . . bgHajfa bgHajfa bgHajfa bgHajfa ABC CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg .


- 20 -

72727272----eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCúg mYymanRCúg mYymanRCúg mYymanRCúg c,b,a . . . . kMntRbePTénRtIekaN kMntRbePTénRtIekaN kMntRbePTénRtIekaN kMntRbePTénRtIekaN ABC ebIeKdwgfa ½ ebIeKdwgfa ½ ebIeKdwgfa ½ ebIeKdwgfa ½ )

c1

b1

a1

(21

cCcos

bBcos

aAcos ++++++++====++++++++

73737373----kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN ABC mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½

rR

4

2C

sin

1

2B

sin

1

2A

sin

1 ≥≥≥≥++++++++

Edl Edl Edl Edl r nig nig nig nig R CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . 74747474----k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa

aaa

a

2sin

1

sin2

1

2sin

2cos222 −−−−====

x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk ∑∑∑∑====

====n

kk

k

kn x

x

S0 2

2sin

2cos

.2

1 . . . .

��


- 21 -


- 22 -

lMhat´TI1lMhat´TI1lMhat´TI1lMhat´TI1 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ 22 3243 22)( xxxxxf −−−−++++++++−−−− ++++==== k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit k> cUrbgðajfaRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf . . . . x> rktémø x> rktémø x> rktémø x> rktémø x edIm, edIm, edIm, edIm,I[ I[ I[ I[ 20)( ====xf . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> bgðajfaRKb;cMnYnBit k> bgðajfaRKb;cMnYnBit k> bgðajfaRKb;cMnYnBit k> bgðajfaRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf cMeBaHRKb;cMnYnBitviC¢man cMeBaHRKb;cMnYnBitviC¢man cMeBaHRKb;cMnYnBitviC¢man cMeBaHRKb;cMnYnBitviC¢man a nig nig nig nig b eKman eKman eKman eKman (((( )))) 0

2 ≥≥≥≥−−−− ba eK)an eK)an eK)an eK)an 02 ≥≥≥≥++++−−−− baba b¤ b¤ b¤ b¤ abba 2≥≥≥≥++++ edayyk edayyk edayyk edayyk 432

2 ++++−−−−==== xxa nig nig nig nig 2322 xxb −−−−++++==== eK)an eK)an eK)an eK)an 22 3243 2.22)( xxxxxf −−−−++++++++−−−−≥≥≥≥ 1622)( 6 ====≥≥≥≥xf dUcenHRKb;cMnYnBit dUcenHRKb;cMnYnBit dUcenHRKb;cMnYnBit dUcenHRKb;cMnYnBit x eK)an eK)an eK)an eK)an 16)( ≥≥≥≥xf . . . . x> rktémø x> rktémø x> rktémø x> rktémø x edIm,I[ edIm,I[ edIm,I[ edIm,I[ 20)( ====xf eK)an eK)an eK)an eK)an 2022

22 3243 ====++++ −−−−++++++++−−−− xxxx

(((( ))))i

xx

xx

xxxx

0202

642

02022

43

43

)43(643

2

2

22

====−−−−++++

====−−−−++++

++++−−−−++++−−−−

++++−−−−−−−−++++−−−−


- 23 -

tag tag tag tag 02 432>>>>==== ++++−−−− xxt smIkar smIkar smIkar smIkar (((( ))))i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³

02064 ====−−−−++++t

t b¤ b¤ b¤ b¤ 064202 ====++++−−−− tt

03664100' >>>>====−−−−====∆∆∆∆ eKTajb¤seKTajb¤seKTajb¤seKTajb¤s 16610;4610 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 4====t eK)an eK)an eK)an eK)an 42 432

====++++−−−− xx naM[ naM[ naM[ naM[ 2432 ====++++−−−− xx b¤ b¤ b¤ b¤ 0232 ====++++−−−− xx eday eday eday eday 0====++++++++ cba enaH enaH enaH enaH 11 ====x b¤ b¤ b¤ b¤ 22 ====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 16====t eK)an eK)an eK)an eK)an 162 432

====++++−−−− xx naM[ naM[ naM[ naM[ 4432 ====++++−−−− xx b¤ b¤ b¤ b¤ 0)3( ====−−−−xx eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 01 ====x b¤ b¤ b¤ b¤ 32 ====x . . . . dUcenHeK)an dUcenHeK)an dUcenHeK)an dUcenHeK)an }3,2,1,0{∈∈∈∈x . . . .


- 24 -

lMhat´TI2lMhat´TI2lMhat´TI2lMhat´TI2

eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ 22

22)(

1

++++++++====

++++

x

x

xf

k> cMeBak> cMeBak> cMeBak> cMeBaHRKb;cMnYnBit HRKb;cMnYnBit HRKb;cMnYnBit HRKb;cMnYnBit m nig nig nig nig n ebI ebI ebI ebI 1====++++ nm enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa enaHcUrRsayfa ³³³³ 3)()( ====++++ nfmf . . . . x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk ∑∑∑∑

====

++++====

p

kp p

kfS

1 1 . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> k> k> k> Rsayfa Rsayfa Rsayfa Rsayfa 3)()( ====++++ nfmf ebI ebI ebI ebI 1====++++ nm eKman eKman eKman eKman (((( ))))

22

22 1

++++++++====

++++

m

m

mf nig nig nig nig (((( ))))22

22 1

++++++++====

++++

n

n

nf

22

22

22

22)()(

11

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

++++++++====++++

++++++++

n

n

m

m

nfmf

3

224

)224(3

224

2.32.312

2222

22222222

21

21

21

21

21

21

21

21

21

21

23

21

121

23

1

====

++++++++

++++++++====

++++++++

++++++++====

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

++++++++++++++++++++++++++++====

++++++++

++++++++

++++++++

++++++++

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

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

nm

nm

nm

nm

nmnm

nmnmnmnm

dUcenH ebI dUcenH ebI dUcenH ebI dUcenH ebI 1====++++ nm enaH enaH enaH enaH 3)()( ====++++ nfmf . . . .


- 25 -

x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk x> KNnaplbUk ∑∑∑∑====

++++====

p

kp p

kfS

1 1

eKman eKman eKman eKman (((( ))))ip

pf

pf

pfSn

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

++++++++

++++====

1......

12

11

b¤ b¤ b¤ b¤ (((( ))))iip

fpp

fp

pfSn

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

++++−−−−++++

++++====

11

....11

1

bUksmIkar bUksmIkar bUksmIkar bUksmIkar (((( ))))i nig nig nig nig (((( ))))ii eK)an eK)an eK)an eK)an ³³³³ pSn 33....332 ====++++++++++++==== naM[ naM[ naM[ naM[

23p

Sn ====

BIeRBaH BIeRBaH BIeRBaH BIeRBaH 311

1 ====

++++++++

++++ pp

fp

f

31

11

312

13

311

12

====

++++++++

++++

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

====

++++−−−−++++

++++

====

++++−−−−++++

++++

pf

pp

f

pp

fp

f

pp

fp

f

dUcenH dUcenH dUcenH dUcenH 2

311

pp

kfS

p

kp ====

++++==== ∑∑∑∑

==== . . . .


- 26 -

lMlMlMlMhat´TI3hat´TI3hat´TI3hat´TI3

eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xx

xx

ee

eexf −−−−

−−−−

++++−−−−====)(

k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit a nig nig nig nig b cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³ (((( ))))

)()(1)()(

bfafbfaf

baf++++

++++====++++

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> Rsayfak> Rsayfak> Rsayfak> Rsayfa )(xf CaG CaG CaG CaGnuKmn_essnuKmn_essnuKmn_essnuKmn_ess eKman eKman eKman eKman xx

xx

ee

eexf −−−−

−−−−

++++−−−−====)(

cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit cMeBaHRKb;cMnYnBit x eKman eKman eKman eKman 0>>>>++++ −−−− xx ee dUcenHGnuKmn_ dUcenHGnuKmn_ dUcenHGnuKmn_ dUcenHGnuKmn_ f manEdnkMnt; manEdnkMnt; manEdnkMnt; manEdnkMnt; IRD f ==== cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; fDx ∈∈∈∈ eK)an eK)an eK)an eK)an fDx ∈∈∈∈−−−− eK)aeK)aeK)aeK)an n n n (((( )))) 0)( ====

++++−−−−++++

++++−−−−====++++−−−− −−−−

−−−−

−−−−

−−−−

xx

xx

xx

xx

ee

ee

ee

eexfxf

eKTaj eKTaj eKTaj eKTaj )()( xfxf −−−−====−−−− dUcenH dUcenH dUcenH dUcenH )(xf CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess . CaGnuKmn_ess .


- 27 -

x> Rsayfa x> Rsayfa x> Rsayfa x> Rsayfa (((( )))))()(1)()(

bfafbfaf

baf++++

++++====++++

eKman eKman eKman eKman 1

1)( 2

2

++++−−−−====

++++−−−−==== −−−−

−−−−

x

x

xx

xx

e

e

ee

eexf

eK)an eK)an eK)an eK)an (((( ))))1

1

1

122

22

)(2

)(2

++++−−−−====

++++−−−−====++++ ++++

++++

++++

++++

ba

ba

ba

ba

e

e

e

ebaf

(((( ))))1

12

2

++++−−−−==== a

a

e

eaf nig nig nig nig

1

1)( 2

2

++++−−−−==== b

b

e

ebf

eKmeKmeKmeKman an an an 1

1

1

1)()( 2

2

2

2

++++−−−−++++

++++−−−−====++++ b

b

a

a

e

e

e

ebfaf

(((( ))))iee

ebfaf ba

ba

)1)(1(

)1(2)()( 22

22

++++++++−−−−====++++

++++

ehIy ehIy ehIy ehIy )1)(1(

)1)(1(1)()(1 22

22

++++++++−−−−−−−−++++====++++ ba

ba

ee

eebfaf

(((( ))))iiee

ebfaf ba

ba

)1)(1(

)1(2)()(1 22

22

++++++++++++====++++

++++

EcksmPaB EcksmPaB EcksmPaB EcksmPaB (((( ))))i nig nig nig nig (((( ))))ii GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an ³³³³ )(

1

1)()(1)()(

22

22

bafe

ebfafbfaf

ba

ba

++++====++++−−−−====

++++++++

++++

++++

dUcenH dUcenH dUcenH dUcenH (((( )))))()(1)()(

bfafbfaf

baf++++

++++====++++ . . . .


- 28 -

lMhat´TI4lMhat´TI4lMhat´TI4lMhat´TI4 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ (((( )))) (((( ))))xx

xf 3232)( −−−−++++++++==== k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa k>cUrRsayfa )(xf CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 2)( ≥≥≥≥xf . . . . K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar 14)( ====xf . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>Rsayfa k>Rsayfa k>Rsayfa k>Rsayfa )(xf CaGnuKmn_KU CaGnuKmn_KU CaGnuKmn_KU CaGnuKmn_KU

(((( )))) (((( ))))xxxf 3232)( −−−−++++++++====

EdnkMnt; EdnkMnt; EdnkMnt; EdnkMnt; IRD f ==== cMeBaH cMeBaH cMeBaH cMeBaH fDx ∈∈∈∈ enaH enaH enaH enaH fDx ∈∈∈∈−−−− eK)an eK)an eK)an eK)an (((( )))) (((( )))) (((( )))) xx

xf−−−−−−−− −−−−++++++++====−−−− 3232

eday eday eday eday 1)32)(32( ====−−−−++++

enaH enaH enaH enaH xx

xf−−−−−−−−

++++++++

−−−−====−−−−

321

321

)(

(((( )))) (((( )))) (((( ))))xfxfxx ====−−−−++++++++====−−−− 3232)(

dUcenH dUcenH dUcenH dUcenH )(xf CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU . CaGnuKmn_KU .


- 29 -

x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x>cMeBaHRKb;cMnYn x bgðajfa bgðajfa bgðajfa bgðajfa 2)( ≥≥≥≥xf ³ ³ ³ ³ tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eK)an eK)an eK)an eK)an ³³³³ (((( )))) (((( )))) xxxx

)32()32(23232 −−−−++++≥≥≥≥−−−−++++++++ (((( )))) (((( )))) 23232 ≥≥≥≥−−−−++++++++ xx dUcenH dUcenH dUcenH dUcenH 2)( ≥≥≥≥xf cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; x . . . . K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar K>edaHRsaysmIkar 14)( ====xf eK)an eK)an eK)an eK)an (((( )))) (((( )))) 143232 ====−−−−++++++++ xx tag tag tag tag (((( )))) 032 >>>>++++==== x

t enaH enaH enaH enaH (((( ))))t

x 132 ====−−−−

smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr 141 ====++++t

t b¤ b¤ b¤ b¤ 01142 ====++++−−−− tt 2)34(48149' ========−−−−====∆∆∆∆

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s (((( ))))21 32347 ++++====++++====t

ehIy ehIy ehIy ehIy (((( )))) 222 32)32(347

−−−−++++====−−−−====−−−−====t ----cMeBaH cMeBaH cMeBaH cMeBaH (((( ))))2

1 32 ++++====t eK)an eK)an eK)an eK)an (((( )))) (((( ))))2

3232 ++++====++++ x naM[ naM[ naM[ naM[ 2====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH (((( )))) 2

1 32−−−−++++====t

eK)an eK)an eK)an eK)an (((( )))) (((( )))) 23232

−−−−++++====++++ x naM[ naM[ naM[ naM[ 2−−−−====x . . . .


- 30 -

lMhat´TI5lMhat´TI5lMhat´TI5lMhat´TI5 edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> 02433.49 2 ====++++−−−− ++++xx x> x> x> x> 01282.94 2 ====++++−−−− ++++xx K> K> K> K> 026.53 32132 ====++++−−−− ++++++++++++ xxx X> X> X> X> (((( )))) (((( )))) 102323 ====−−−−++++++++ xx g> g> g> g> (((( )))) (((( )))) 143232 ====++++++++−−−−

xx dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> 02433.49 2 ====++++−−−− ++++xx (((( )))) (((( )))) 02433363 2 ====++++−−−− xx tag tag tag tag (((( )))) 03 >>>>==== xt 81243324';0243362 ====−−−−====∆∆∆∆====++++−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 27918;9918 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 91 ====t eK)an eK)an eK)an eK)an 93 ====x naM[ naM[ naM[ naM[ 2====x ----cMeBaH cMeBaH cMeBaH cMeBaH 272 ====t eK)an eK)an eK)an eK)an 273 ====x naM[ naM[ naM[ naM[ 3====x dUcenH dUcenH dUcenH dUcenH 3;2 21 ======== xx . . . .


- 31 -

x> x> x> x> 01282.94 2 ====++++−−−− ++++xx (((( )))) (((( )))) 01282362 2 ====++++−−−− xx tag tag tag tag 02 >>>>==== xt 196128324';0128362 ====−−−−====∆∆∆∆====++++−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 321418;21418 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 21 ====t enaH enaH enaH enaH 22 ====x naM[ naM[ naM[ naM[ 1====x ----cMeBaH cMeBaH cMeBaH cMeBaH 322 ====t enaH enaH enaH enaH 322 ====x naM[ naM[ naM[ naM[ 5====x dUcenH dUcenH dUcenH dUcenH 5;1 21 ======== xx . . . . K> K> K> K> 026.53 32132 ====++++−−−− ++++++++++++ xxx (((( )))) (((( )))) (((( )))) 048630927 ====++++−−−− xxx EckGgÁTaMgBIEckGgÁTaMgBIEckGgÁTaMgBIEckGgÁTaMgBIrénsmIkarnwg rénsmIkarnwg rénsmIkarnwg rénsmIkarnwg 04 ≠≠≠≠x eK)an eK)an eK)an eK)an ³³³³

08

23

3023

27

0846

3049

27

2

====++++

−−−−

====++++

−−−−

xx

xx

tag tag tag tag 023 >>>>

====x

t enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr ³³³³

9216225';083027 2 ====−−−−====∆∆∆∆====++++−−−− tt eKTaj eKTaj eKTaj eKTaj

32

27315

;94

27315

21 ====++++========−−−−==== tt


- 32 -

----cMeBaH cMeBaH cMeBaH cMeBaH 94

1 ====t eK)an eK)an eK)an eK)an 94

23 ====

x

naM[ naM[ naM[ naM[ 2−−−−====x

----cMeBaH cMeBaH cMeBaH cMeBaH 32

2 ====t eK)an eK)an eK)an eK)an 32

23 ====

x

naM[ naM[ naM[ naM[ 1−−−−====x

dUcenH dUcenH dUcenH dUcenH 1;2 21 −−−−====−−−−==== xx . . . . X> X> X> X> (((( )))) (((( )))) 102323 ====−−−−++++++++ xx tag tag tag tag (((( )))) 023 >>>>++++==== x


x 123 ====−−−−


t b¤ b¤ b¤ b¤ 01102 ====++++−−−− tt

(((( )))) 06224125'2 >>>>========−−−−====∆∆∆∆

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s (((( ))))(((( )))) (((( ))))

++++====−−−−====−−−−====

++++====++++====−−−−22

2

21

2323625

23625

t

t

----cMeBaH cMeBaH cMeBaH cMeBaH (((( ))))223 ++++====t

eK)an eK)an eK)an eK)an (((( )))) (((( ))))22323 ++++====++++ x naM[ naM[ naM[ naM[ 2====x

----cMeBaH cMeBaH cMeBaH cMeBaH (((( )))) 223

−−−−++++====t eK)an eK)an eK)an eK)an (((( )))) (((( )))) 2

2323−−−−++++====++++ x naM[ naM[ naM[ naM[ 2−−−−====x

dUcenH dUcenH dUcenH dUcenH 2;2 21 −−−−======== xx . . . .


- 33 -

g> g> g> g> (((( )))) (((( )))) 143232 ====++++++++−−−−xx

tag tag tag tag (((( )))) 032 >>>>++++====x


x 132 ====−−−−


t b¤ b¤ b¤ b¤ 01142 ====++++−−−− tt 2)34(48149' ========−−−−====∆∆∆∆

eKTajb¤seKTajb¤seKTajb¤seKTajb¤s (((( ))))21 32347 ++++====++++====t

ehIy ehIy ehIy ehIy (((( )))) 222 32)32(347

−−−−++++====−−−−====−−−−====t ----cMeBaH cMeBaH cMeBaH cMeBaH (((( ))))2

1 32 ++++====t eK)an eK)an eK)an eK)an (((( )))) (((( ))))2

3232 ++++====++++x naM[ naM[ naM[ naM[ 4====x . . . .

----cMeBaH cMeBaH cMeBaH cMeBaH (((( )))) 21 32

−−−−++++====t eK)an eK)an eK)an eK)an (((( )))) (((( )))) 2

3232−−−−++++====++++

x naM[ naM[ naM[ naM[ 4−−−−====x . . . . dUcenH dUcenH dUcenH dUcenH 4;4 21 −−−−======== xx . . . .

��


- 34 -

lMhat´TI6lMhat´TI6lMhat´TI6lMhat´TI6 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³

x

xx

x

xx

x

xx

x

xx

6

6

5

5

4

4

3

32222 −−−−−−−−−−−−−−−−

====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³

x

xx

x

xx

x

xx

x

xx

6

6

5

5

4

4

3

32222 −−−−−−−−−−−−−−−−

====++++++++ xxxxxxxx 2222 2222

6543 −−−−−−−−−−−−−−−− ====++++++++ tag tag tag tag 11)1(2 22 −−−−≥≥≥≥−−−−−−−−====−−−−==== xxxt smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ (((( ))))itttt 6543 ====++++++++ ----ebI ebI ebI ebI 3====t enaHsmIkar enaHsmIkar enaHsmIkar enaHsmIkar (((( ))))i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³

2161256427

6543 3333

====++++++++====++++++++

216216==== epÞógpÞat; epÞógpÞat; epÞógpÞat; epÞógpÞat; dUcenH dUcenH dUcenH dUcenH 3====t Cab¤srbs;smIkar Cab¤srbs;smIkar Cab¤srbs;smIkar Cab¤srbs;smIkar )(i . . . .


- 35 -

----ebI ebI ebI ebI 31 <<<<≤≤≤≤−−−− t eKman eKman eKman eKman ³³³³

333

65

65

;64

64

;63

63

>>>>

>>>>

>>>>

ttt

eK)an eK)an eK)an eK)an 333

65

64

63

65

64

63

++++

++++

>>>>

++++

++++

ttt

16

543 >>>>++++++++t

ttt

naM[ naM[ naM[ naM[ tttt 6543 >>>>++++++++

dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar (((( ))))i Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH 31 <<<<≤≤≤≤−−−− t . . . .

----ebI ebI ebI ebI 3>>>>t eKman eKman eKman eKman ³³³³

333

65

65

;64

64

;63

63

<<<<

<<<<

<<<<

ttt


65

64

63

65

64

63

++++

++++

<<<<

++++

++++

ttt

16

543 <<<<++++++++t

ttt

naM[ naM[ naM[ naM[ tttt 6543 <<<<++++++++


- 36 -

dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar dUcenHsmIkar (((( ))))i Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH Kµanb¤scMeBaH 3>>>>t . . . .

srubmksmIkar srubmksmIkar srubmksmIkar srubmksmIkar (((( ))))i manb¤seTal manb¤seTal manb¤seTal manb¤seTal 3====t

cMeBaH cMeBaH cMeBaH cMeBaH 3====t eK)an eK)an eK)an eK)an 322 ====−−−− xx

b¤ b¤ b¤ b¤ 0322 ====−−−−−−−− xx naM[ naM[ naM[ naM[ 3;1 21 ====−−−−==== xx

dUcenH dUcenH dUcenH dUcenH 3;1 21 ====−−−−==== xx . . . .

��


- 37 -


8822642)1(2)1(2)1( ++++++++====++++ −−−−−−−−−−−− xxx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³³³³

8822642)1(2)1(2)1( ++++++++====++++ −−−−−−−−−−−− xxx

tag tag tag tag (((( )))) 0221 >>>>==== −−−−xX enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr enaHsmIkarGacsresr ³³³³

(((( ))))iXXXXX

XXX

)42)(2(6

82622

32

++++−−−−++++====++++−−−−

++++++++====++++

tag tag tag tag 2++++==== Xu nig nig nig nig 222 ++++−−−−==== xxv smIkar smIkar smIkar smIkar (((( ))))ii Gacsresr Gacsresr Gacsresr Gacsresr ³³³³

uvvu 2====++++ b¤ b¤ b¤ b¤ (((( )))) 02 ====−−−− vu

naM[ naM[ naM[ naM[ vu ==== b¤ b¤ b¤ b¤ 222 2 ++++−−−−====++++ XXX b¤ b¤ b¤ b¤ 0232 ====++++−−−− XX eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 2;1 21 ======== XX . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 11 ====X enaH enaH enaH enaH (((( )))) 12

21 ====−−−−x naM[ naM[ naM[ naM[ 1====x . . . . ----cMeBaHcMeBaHcMeBaHcMeBaH 22 ====X enaH enaH enaH enaH (((( )))) 22

21 ====−−−−x smmUl smmUl smmUl smmUl 1)1( 2 ====−−−−x eKTaj eKTaj eKTaj eKTaj 01 ====x b¤ b¤ b¤ b¤ 22 ====x . dUcenH . dUcenH . dUcenH . dUcenH }2;1;0{∈∈∈∈x . . . .


- 38 -


(((( )))) 222210114 6333 xxxxx −−−−====++++++++ −−−−−−−−−−−−

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIedaHRsaysmIedaHRsaysmIedaHRsaysmIkar kar kar kar ³³³³

(((( )))) 222210114 6333 xxxxx −−−−====++++++++ −−−−−−−−−−−− eKman eKman eKman eKman 222 )3(9)69(96 xxxxx −−−−−−−−====++++−−−−−−−−====−−−− eday eday eday eday 0)3( 2 ≥≥≥≥−−−− x enaHeK)an enaHeK)an enaHeK)an enaHeK)an (((( ))))ixx 96 2 ≤≤≤≤−−−− müa:geTottamvismPaB müa:geTottamvismPaB müa:geTottamvismPaB müa:geTottamvismPaB GMAM −−−− eKman eKman eKman eKman ³³³³

(((( )))) (((( ))))

(((( )))) (((( ))))iixxx

xxxxxx

xxxxxx

9333

33333

3.3.33333

22210114

3 2221011422210114

3 2)2(2101142)2(210114

≥≥≥≥++++++++

≥≥≥≥++++++++

≥≥≥≥++++++++

−−−−−−−−−−−−

−−−−++++−−−−++++−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−−−−−−−

tam tam tam tam (((( ))))i nig nig nig nig (((( ))))ii smIkarsmmUl smIkarsmmUl smIkarsmmUl smIkarsmmUl ³³³³ (((( ))))

(((( )))) (((( ))))

====++++++++

====−−−−−−−−−−−−−−−− 29333

19622210114

2

xxx

xx

bnÞab;BIedaHRsayeK)anb¤s bnÞab;BIedaHRsayeK)anb¤s bnÞab;BIedaHRsayeK)anb¤s bnÞab;BIedaHRsayeK)anb¤s 3====x . . . .


- 39 -


04882)14(4 ====++++−−−−−−−−++++ xx xx


04882)14(4 ====++++−−−−−−−−++++ xx xx tag tag tag tag 02 >>>>==== tx smIkarGacsrsr smIkarGacsrsr smIkarGacsrsr smIkarGacsrsr ³³³³

(((( ))))2

2

2

2

2

2

44

1923219628

)488)(1(4)14(

0488)14(

++++====

++++++++====

−−−−++++++++−−−−====

++++−−−−−−−−−−−−====∆∆∆∆

====++++−−−−−−−−++++

x

xx

xxx

xx

xtxt

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 82

)2()14( 2

1 ====++++++++−−−−−−−−

====xx

t

b¤ b¤ b¤ b¤ 62

)2()14( 2

2 ++++−−−−====++++−−−−−−−−−−−−

==== xxx

t ----cMeBaH cMeBaH cMeBaH cMeBaH 8====t eK)an eK)an eK)an eK)an 3282 ========x eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 3====x . . . .


- 40 -

----cMeBaH cMeBaH cMeBaH cMeBaH 6++++−−−−==== xt eK)an eK)an eK)an eK)an 62 ++++−−−−==== xx CasmIkarGCasmIkarGCasmIkarGCasmIkarGab;sIuscMnucRbsBVrvagExSekag ab;sIuscMnucRbsBVrvagExSekag ab;sIuscMnucRbsBVrvagExSekag ab;sIuscMnucRbsBVrvagExSekag (((( )))) xyc 2: ==== CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; (((( )))) 6: ++++−−−−==== xyd . . . . tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag (((( ))))c nigbnÞat; nigbnÞat; nigbnÞat; nigbnÞat; )(d kat;Kña kat;Kña kat;Kña kat;Kña Rtg;cMnucmanGab;sIus Rtg;cMnucmanGab;sIus Rtg;cMnucmanGab;sIus Rtg;cMnucmanGab;sIus 2====x EdlCab¤srbs;s EdlCab¤srbs;s EdlCab¤srbs;s EdlCab¤srbs;smIkar .mIkar .mIkar .mIkar . dUcenHsmIkarmanb¤sBIrKW dUcenHsmIkarmanb¤sBIrKW dUcenHsmIkarmanb¤sBIrKW dUcenHsmIkarmanb¤sBIrKW 2;3 21 ======== xx . . . .

2 3 4 5 6 7 8-1-2-3-4-5

2

3

4

5

6

7

-1

0 1

1

x

y

(((( )))) xyc 2: ====

(((( )))) 6: ++++−−−−==== xyd


- 41 -


04

21215

7443

2.34.9 ====−−−−++++

−−−−−−−− −−−−−−−− xxxx


04

21215

7443

2.34.9 11 ====−−−−++++

−−−−−−−− ++++−−−−++++−−−− xxxx

tag tag tag tag 02.3 1 >>>>==== ++++−−−− xt smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ 0

421

215

74432 ====−−−−++++

−−−−−−−− xtxt

−−−−−−−−

−−−−====∆∆∆∆4

21215

47443 2

xx

22

2

437

716

13692

25949

2130492

30116

1849

−−−−====++++−−−−====

++++−−−−++++−−−−====

xx

x

xxx

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s

++++−−−−====

====

10743

2

1

xt

t


- 42 -

----cMeBaH cMeBaH cMeBaH cMeBaH 43====t eK)an eK)an eK)an eK)an

43

2.3 1 ====++++−−−− x b¤ b¤ b¤ b¤ 21 22 −−−−++++−−−− ====x naM[ naM[ naM[ naM[ 3====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 107 ++++−−−−==== xt eK)an eK)an eK)an eK)an 1072.3 1 ++++−−−−====++++−−−− xx b¤ b¤ b¤ b¤

35

67

2 ++++−−−−====−−−− xx CasmIkarGab;sIuscMnucRbsBVrvag CasmIkarGab;sIuscMnucRbsBVrvag CasmIkarGab;sIuscMnucRbsBVrvag CasmIkarGab;sIuscMnucRbsBVrvag

ExSekag ExSekag ExSekag ExSekag (((( )))) xyc −−−−==== 2: nwgbnÞat; nwgbnÞat; nwgbnÞat; nwgbnÞat; (((( ))))35

67

: ++++−−−−==== xyd . . . .

FFFF tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag tamRkahVikeyIgeXIjfaExSekag (((( ))))c nigbnÞat; nigbnÞat; nigbnÞat; nigbnÞat; )(d kat;Kña kat;Kña kat;Kña kat;Kña Rtg;cMnucmanGRtg;cMnucmanGRtg;cMnucmanGRtg;cMnucmanGab;sIus ab;sIus ab;sIus ab;sIus 1;2 ====−−−−==== xx EdlCab¤srbs;smIkar EdlCab¤srbs;smIkar EdlCab¤srbs;smIkar EdlCab¤srbs;smIkar dUcenHsmIkarmanb¤sbI KW dUcenHsmIkarmanb¤sbI KW dUcenHsmIkarmanb¤sbI KW dUcenHsmIkarmanb¤sbI KW 3;1;2 321 ========−−−−==== xxx . . . .

2 3 4-1-2-3

2

3

4

5

0 1

1

x

y


- 43 -


32423233

31

3 −−−−++++−−−−−−−− ++++====++++ xxxxx


32423233

31

3 −−−−++++−−−−−−−− ++++====++++ xxxxx smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³

(((( )))) (((( ))))

(((( )))) 0271

393

093271

933

031

333

42

42

324232

====

−−−−−−−−

====−−−−−−−−−−−−

====++++−−−−−−−−

−−−−

−−−−

−−−−++++−−−−−−−−

xxx

xxxx

xxxxx

eK)an eK)an eK)an eK)an 093 ====−−−−x naM[ naM[ naM[ naM[ 2====x 0

271

3 22====−−−−−−−− xx naM[ naM[ naM[ naM[ 0342 ====++++−−−− xx

manb¤s manb¤s manb¤s manb¤s 3;1 21 ======== xx . . . . dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s ]3;2;1{∈∈∈∈x . . . .


- 44 -

lMhat´TI12lMhat´TI12lMhat´TI12lMhat´TI12 edaHRsayedaHRsayedaHRsayedaHRsayRbBnæ½RbBnæ½RbBnæ½RbBnæ½smIkar smIkar smIkar smIkar ³³³³

====

====

4321

43

721

32

yx

yx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar edaHRsayRbBnæ½smIkar ³³³³

(((( ))))

(((( ))))

====

====

ii

i

yx

yx

4321

43

721

32

smIkar smIkar smIkar smIkar (((( ))))i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³

(((( ))))(((( ))))13log233log

3.2log3log

721

log)3.2(log

22

2322

2

−−−−−−−−====++++====++++

====

−−−−−−−−

yx

yx

yx

smIkar smIkar smIkar smIkar (((( ))))ii Gacsresr Gacsresr Gacsresr Gacsresr ³³³³ (((( )))) (((( ))))

(((( ))))4333

33

2.3log4log

432log43log−−−−−−−−====++++

====

yx

yx


- 45 -

(((( ))))22log432log2 33 −−−−−−−−====++++ yx dksmIkar dksmIkar dksmIkar dksmIkar (((( ))))1 nwg nwg nwg nwg (((( ))))2 GgÁ nwg GgÁeK)an GgÁ nwg GgÁeK)an GgÁ nwg GgÁeK)an GgÁ nwg GgÁeK)an ³³³³

(((( )))) (((( ))))3log2log222log23log

3log22log42log23log

2332

2332

−−−−====−−−−−−−−====−−−−

y

yy

EckGgÁTaMgBIrnwg EckGgÁTaMgBIrnwg EckGgÁTaMgBIrnwg EckGgÁTaMgBIrnwg 2log23log 32 −−−− eKTajeKTajeKTajeKTaj)an )an )an )an 2−−−−====y . . . . yktémø yktémø yktémø yktémø 2−−−−====y CMnYskñúg CMnYskñúg CMnYskñúg CMnYskñúg (((( ))))2 eK)an eK)an eK)an eK)an ³³³³

2log432log4 33 −−−−−−−−====−−−−x naM[ naM[ naM[ naM[ 2−−−−====x dUcenH dUcenH dUcenH dUcenH 3;2 −−−−====−−−−==== yx . . . .

��


- 46 -


323.2 11

2 ====++++−−−−

−−−− xx

x


323.2 11

2 ====++++−−−−

−−−− xx

x smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa 01 ≠≠≠≠++++x b¤ b¤ b¤ b¤ 1−−−−≠≠≠≠x smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³

(((( ))))

0)3log1

11)(3(

03log21

11

)3(

3log21

13log11

2

32log3.2log

2

2

22

211

22

====++++

++++−−−−

====

−−−−++++−−−−++++−−−−

++++====++++−−−−++++−−−−

====

++++−−−−

−−−−

xx

xx

x

xx

x

xx

x

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ;31 ====x 3log1 22 −−−−−−−−====x . . . .


- 47 -

lMhat´TI14lMhat´TI14lMhat´TI14lMhat´TI14 eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

10 ====a nig TMnak;TMngkMeNIn nig TMnak;TMngkMeNIn nig TMnak;TMngkMeNIn nig TMnak;TMngkMeNIn (((( ))))nanana ++++

++++ ++++==== 121 24log

Edl Edl Edl Edl ...;2;1;0====n . . . . k>eKtag k>eKtag k>eKtag k>eKtag na

nb 21++++==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa 21 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n ....

x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa x>edayeRbIGnumanrYmKNitviTüacUrRsayfa n

nb 23==== . . . . K>TajrktY K>TajrktY K>TajrktY K>TajrktY na énsIVút énsIVút énsIVút énsIVút (((( ))))na CaGnuKmn_é CaGnuKmn_é CaGnuKmn_é CaGnuKmn_én n n n n . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> bgðajfa k> bgðajfa k> bgðajfa k> bgðajfa 2

1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n eKman eKman eKman eKman na

nb 21++++==== eK)an eK)an eK)an eK)an 1

1 21 ++++++++ ++++==== na

nb eday eday eday eday (((( ))))nanana ++++

++++ ++++==== 121 24log

++++++++

++++ ++++====nana

nb1242log

1 21

(((( )))) (((( ))))(((( )))) 22

2

1

12

1222

241

nna

nana

nana

b====++++====

++++++++====

++++++++==== ++++

dUcenH dUcenH dUcenH dUcenH 21 nn bb ====++++ . . . .


- 48 -

x>edayeRbIGnumanrYmKNitviTüax>edayeRbIGnumanrYmKNitviTüax>edayeRbIGnumanrYmKNitviTüax>edayeRbIGnumanrYmKNitviTüaRsayfa Rsayfa Rsayfa Rsayfa n

nb 23==== cMeBaH cMeBaH cMeBaH cMeBaH 0====n eK)an eK)an eK)an eK)an 32121 0

0 ====++++====++++==== ab Bit Bit Bit Bit ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH kn ==== KW KW KW KW k

kb 23==== Bit Bit Bit Bit eyIgnwgRsaeyIgnwgRsaeyIgnwgRsaeyIgnwgRsayfavaBitcMeBaH yfavaBitcMeBaH yfavaBitcMeBaH yfavaBitcMeBaH 1++++==== kn KW KW KW KW 12

1 3++++

++++ ====k

kb BitBitBitBit eyIgman eyIgman eyIgman eyIgman 2

1 kk bb ====++++ eday eday eday eday k

kb 23==== enaH enaH enaH enaH (((( )))) 12

22

1 33++++

++++ ========kk

kb Bit Bit Bit Bit dUcenH dUcenH dUcenH dUcenH n

nb 23==== . . . . K>TajrktY K>TajrktY K>TajrktY K>TajrktY na énsIVút énsIVút énsIVút énsIVút (((( ))))na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ³ ³ ³ ³ eyIgman eyIgman eyIgman eyIgman na

nb 21++++==== eKTaj eKTaj eKTaj eKTaj (((( ))))1log2 −−−−==== nn ba eday eday eday eday n

nb 23==== dUcenH dUcenH dUcenH dUcenH (((( ))))13log 2

2 −−−−====n

na . . . .

��


- 49 -

lMhat´TI15lMhat´TI15lMhat´TI15lMhat´TI15 eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit eK[sIVútcMnYnBit (((( ))))na kMnt;eday kMnt;eday kMnt;eday kMnt;eday 10 ====a nig nig nig nigTMnak;TMngkMeNIn TMnak;TMngkMeNIn TMnak;TMngkMeNIn TMnak;TMngkMeNIn (((( ))))nbnana

na ++++++++++++ ++++++++==== 121


nb 31++++==== . . . . bgðajfa bgðajfa bgðajfa bgðajfa 3

1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n .... x>x>x>x>KNna KNna KNna KNna nb nig nig nig nig na CaGCaGCaGCaGnuKmn_én nuKmn_én nuKmn_én nuKmn_én n . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> k> k> k> bgðajfa bgðajfa bgðajfa bgðajfa 3

1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n eyIgman eyIgman eyIgman eyIgman na

nb 31++++==== naM[ naM[ naM[ naM[ 1

1 31 ++++++++ ++++==== na

nb eday eday eday eday (((( ))))nbnana

na ++++++++++++ ++++++++==== 121

31 3327log eK)an eK)an eK)an eK)an nbnana

nb ++++++++++++ ++++++++++++==== 121

1 33271

(((( )))) (((( )))) (((( ))))(((( )))) 33

1

23

1

13

133333

nna

n

nananan

bb

b

====++++====

++++++++++++====

++++

++++

dUcenH dUcenH dUcenH dUcenH 31 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n ....


- 50 -

x>KNna x>KNna x>KNna x>KNna nb nig nig nig nig na CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ³ ³ ³ ³ eKman eKman eKman eKman 3

1 nn bb ====++++ RKb; RKb; RKb; RKb; 0≥≥≥≥n eK)an eK)an eK)an eK)an (((( )))) (((( ))))nn bb ln3ln 1 ====++++ tag tag tag tag (((( ))))nn bc ln==== enaH enaH enaH enaH nn cc 31 ====++++ naM[ naM[ naM[ naM[ (((( ))))nc CasIVútFrNImaRtmanersug CasIVútFrNImaRtmanersug CasIVútFrNImaRtmanersug CasIVútFrNImaRtmanersug 3====q nigtY nigtY nigtY nigtY (((( )))) (((( )))) 4ln31lnln 0

00 ====++++======== abc .... tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ 4ln30

nnn qcc ====××××====

eKTaj eKTaj eKTaj eKTaj (((( )))) 4ln3ln nnb ==== naM[ naM[ naM[ naM[ n

nb 34==== ehIyeday ehIyeday ehIyeday ehIyeday na

nb 31++++==== naM[ naM[ naM[ naM[ (((( )))) (((( ))))14log1log 3

33 −−−−====−−−−====n

nn ba dUcenH dUcenH dUcenH dUcenH n

nb 34==== nig nig nig nig (((( ))))14log 33 −−−−====

n

na . . . .

��


- 51 -

lMhat´TI16lMhat´TI16lMhat´TI16lMhat´TI16 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_

++++−−−−====

xx

xf11

lg)(

k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH .k>rkEdnkMnt;énGnuKmn_enH . x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit x>cMeBaHRKb;cMnYnBit 11 <<<<<<<<−−−− a nig nig nig nig 11 <<<<<<<<−−−− b cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa (((( )))) (((( ))))

++++++++====++++abba

fbfaf1

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>rkEdnkMnt;énGnuKmn_ k>rkEdnkMnt;énGnuKmn_ k>rkEdnkMnt;énGnuKmn_ k>rkEdnkMnt;énGnuKmn_ f

++++−−−−====

xx

xf11

lg)(

GnuKmn_mann½ykalNa GnuKmn_mann½ykalNa GnuKmn_mann½ykalNa GnuKmn_mann½ykalNa

−−−−≠≠≠≠

>>>>++++−−−−

1

011

xxx

dUcenH dUcenH dUcenH dUcenH [1;1] −−−−====fD . . . . x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa ³³³³ (((( )))) (((( ))))

++++++++====++++abba

fbfaf1


- 52 -

eyIgman eyIgman eyIgman eyIgman (((( ))))

++++−−−−====

aa

af11

lg nig nig nig nig (((( ))))

++++−−−−====

bb

bf11

lg

eyIg)an eyIg)an eyIg)an eyIg)an (((( )))) (((( ))))

++++−−−−++++

++++−−−−====++++

bb

aa

bfaf11

lg11

lg

++++++++++++++++++++−−−−====

++++++++−−−−−−−−====

abbaabba

baba

)(1)(1

lg

)1)(1()1)(1(

lg

(((( )))) (((( )))) (((( ))))(((( )))) (((( ))))i

abbaabba

bfaf

++++++++++++++++++++−−−−====++++

11

lg

müa:geTot müa:geTot müa:geTot müa:geTot

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

++++++++−−−−

====

++++++++

abba

abba

abba

f

11

11

lg1

(((( ))))iiabbaabba

abba

f

++++++++++++++++++++−−−−====

++++++++

)(1)(1

lg1

tam tam tam tam (((( ))))i nig nig nig nig (((( ))))ii eKTaj eKTaj eKTaj eKTaj (((( )))) (((( ))))

++++++++====++++abba

fbfaf1

. . . .


- 53 -

lMhat´TI17lMhat´TI17lMhat´TI17lMhat´TI17 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit (((( )))) 0≥≥≥≥nnu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

au ====0 nig nig nig nig (((( )))) nunn uu 2log

1 ====++++ Edl Edl Edl Edl 2>>>>a . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n . . . . x>eKtag x>eKtag x>eKtag x>eKtag nn uV 2log==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa 2

1 nn VV ====++++ . . . . K>eKtag K>eKtag K>eKtag K>eKtag nn VW 2log==== . bgðajfa . bgðajfa . bgðajfa . bgðajfa (((( ))))nW CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt X>KNna X>KNna X>KNna X>KNna nn VW ; nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n cMeBaH cMeBaH cMeBaH cMeBaH 0====n eK)an eK)an eK)an eK)an 20 >>>>==== au Bit Bit Bit Bit ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH kn ==== KW KW KW KW 2>>>>ku Bit Bit Bit Bit eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH 1++++==== kn KW KW KW KW 21 >>>>++++ku Bit Bit Bit Bit eyIgman eyIgman eyIgman eyIgman (((( )))) ku

kk uu 2log1 ====++++

eday eday eday eday 2>>>>ku enaH enaH enaH enaH 1log2 >>>>ku naM[ naM[ naM[ naM[ (((( )))) 22log

1 >>>>>>>>====++++ kku

kk uuu Bit Bit Bit Bit dUcenH dUcenH dUcenH dUcenH 2>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; ....;2;1;0====n . . . .


- 54 -

x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 21 nn VV ====++++

eyIgman eyIgman eyIgman eyIgman nn uV 2log==== naM[ naM[ naM[ naM[ 121 log ++++++++ ==== nn uV eday eday eday eday (((( )))) nu

nn uu 2log1 ====++++ eK)an eK)an eK)an eK)an ³³³³

(((( ))))[[[[ ]]]]2

22

2log21

.log.log

log

nnnnn

nunn

VVVuu

uV

============

====++++

dUcenH dUcenH dUcenH dUcenH 21 nn VV ====++++ . . . .

K> bgðajK> bgðajK> bgðajK> bgðajfa fa fa fa (((( ))))nW CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt CasIVútFrNImaRt ³ ³ ³ ³ eKman eKman eKman eKman nn VW 2log==== naM[ naM[ naM[ naM[ 121 log ++++++++ ==== nn VW eday eday eday eday 2

1 nn VV ====++++ eK)an eK)an eK)an eK)an ³³³³ (((( )))) nnn VVW 2

221 log2log ========++++

dUcenH dUcenH dUcenH dUcenH (((( ))))nW CasIVútFrNImaRtmanpleFobrYm CasIVútFrNImaRtmanpleFobrYm CasIVútFrNImaRtmanpleFobrYm CasIVútFrNImaRtmanpleFobrYm 2====q . . . . X>X>X>X>KNna KNna KNna KNna nn VW ; nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ³ ³ ³ ³ tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ n

n qWW ××××==== 0 eday eday eday eday (((( )))) (((( ))))auVW 22022020 logloglogloglog ============ dUcenH dUcenH dUcenH dUcenH (((( ))))aW n

n 22 loglog2==== . . . . ehIy ehIy ehIy ehIy (((( )))) n

nWn aV 2

2log2 ======== nig nig nig nig (((( ))))n

anVnu

22log22 ======== ....


- 55 -

lMhatlMhatlMhatlMhat´TI18´TI18´TI18´TI18 eK[ eK[ eK[ eK[ x nig nig nig nig y CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; CaBIrcMnYnBitepÞógpÞat; 12.544 −−−−++++====++++ yxyx . . . .

cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa

++++++++====++++3

22log21 2

yx

yx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay

Rsayfa Rsayfa Rsayfa Rsayfa

++++++++====++++3

22log1 2

yx

yx

eyIgman eyIgman eyIgman eyIgman 12.544 −−−−++++====++++ yxyx

(((( )))) (((( ))))(((( ))))(((( ))))(((( )))) (((( ))))(((( ))))

1

2

12

212

112

12

23

22

2.922

25222

22.522

2.522222

−−−−++++

−−−−++++

−−−−++++

++++++++−−−−++++

−−−−++++

====

++++

====++++

++++====++++

++++====++++

====−−−−++++

yxyx

yxyx

yxyx

yxyxyx

yxyxyx

eKTaj eKTaj eKTaj eKTaj

++++====

++++====−−−−++++3

22log2

322

log1 2

2

2

yxyx

yx

dUcenH dUcenH dUcenH dUcenH

++++++++====++++3

22log21 2

yx

yx . . . .


- 56 -

lMhat´TI19lMhat´TI19lMhat´TI19lMhat´TI19 edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) 0

4loglog 2

22 ====

++++ xx

x> x> x> x> (((( )))) 04

loglog3

5,02

2 ====

++++ x

x

K> K> K> K> 4log1log2 xx ++++==== X> X> X> X> (((( ))))(((( )))) (((( )))) 0212log312log 3

2

3 ====++++++++−−−−++++ xx g> g> g> g> 03437.87 2log12

2log ====++++−−−− ++++ xx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) 0

4loglog 2

22 ====

++++ xx lkçxNÐ lkçxNÐ lkçxNÐ lkçxNÐ 0>>>>x

smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³ ³ ³ ³ (((( )))) 04logloglog 222

2 ====−−−−++++ xx (((( )))) 02loglog 2

22 ====−−−−++++ xx

tag tag tag tag xt 2log==== eK)an eK)an eK)an eK)an 022 ====−−−−++++ tt eday eday eday eday 0====++++++++ cba eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 2;1 21 −−−−======== tt


- 57 -

----cMeBaH cMeBaH cMeBaH cMeBaH 1====t eK)an eK)an eK)an eK)an 1log2 ====x naM[ naM[ naM[ naM[ 2====x ----cMeBaH cMeBaH cMeBaH cMeBaH 2−−−−====t eK)an eK)an eK)an eK)an 2log2 −−−−====x naM[ naM[ naM[ naM[

41====x


;2 21 ======== xx . . . .

x> x> x> x> (((( )))) 04

loglog3

5,02

2 ====

++++ x

x lkçxNÐ lkçxNÐ lkçxNÐ lkçxNÐ 0>>>>x

smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ (((( ))))

(((( )))) 02log3log

04logloglog

22

2

21

3

21

22

====++++−−−−

====−−−−++++

xx

xx

tag tag tag tag xt 2log==== smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ 0232 ====++++−−−− tt naM[ naM[ naM[ naM[ 11 ====t b¤ b¤ b¤ b¤ 22 ====t

----cMeBaH cMeBaH cMeBaH cMeBaH 1====t eK)an eK)an eK)an eK)an 1log2 ====x naM[ naM[ naM[ naM[ 2====x ----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 2log2 ====x naM[ naM[ naM[ naM[ 4====x dUcenH dUcenH dUcenH dUcenH 4;2 21 ======== xx . . . . K> K> K> K> 4log1log2 xx ++++==== lkçxNÐ lkçxNÐ lkçxNÐ lkçxNÐ 0>>>>x ni ni ni nig g g g 1≠≠≠≠x smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr 2log21log2 xx ++++==== KuNGgÁTaMgBIrénsmIkarnwg KuNGgÁTaMgBIrénsmIkarnwg KuNGgÁTaMgBIrénsmIkarnwg KuNGgÁTaMgBIrénsmIkarnwg 0log2 ≠≠≠≠x eK)an eK)an eK)an eK)an ³³³³


- 58 -

(((( ))))(((( )))) 02loglog

2loglog

22

2

22

2

====−−−−−−−−

++++====

xx

xx

tag tag tag tag xt 2log==== eK)an eK)an eK)an eK)an 022 ====−−−−−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 2;1 21 ====−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 1−−−−====t eK)an eK)an eK)an eK)an 1log2 −−−−====x naM[ naM[ naM[ naM[

21====x

----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 2log2 ====x naM[ naM[ naM[ naM[ 4====x dUcenH dUcenH dUcenH dUcenH 4;

21

21 ======== xx . . . . X> X> X> X> (((( ))))(((( )))) (((( )))) 0212log312log 3

2

3 ====++++++++−−−−++++ xx tag tag tag tag (((( ))))12log3 ++++==== xt smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³

0232 ====++++−−−− tt naM[ naM[ naM[ naM[ 11 ====t b¤ b¤ b¤ b¤ 22 ====t ----cMeBaH cMeBaH cMeBaH cMeBaH 11 ====t eK)an eK)an eK)an eK)an (((( )))) 112log3 ====++++x naM[ naM[ naM[ naM[ 312 ====++++x b¤ b¤ b¤ b¤ 22 ====x naM[ naM[ naM[ naM[ 1====x . . . . ----cMeBacMeBacMeBacMeBaH H H H 22 ====t eK)an eK)an eK)an eK)an (((( )))) 212log3 ====++++x naM[ naM[ naM[ naM[ 912 ====++++x b¤ b¤ b¤ b¤ 82 ====x naM[ naM[ naM[ naM[ 3====x . . . . dUcenH dUcenH dUcenH dUcenH 3;1 ======== xx . . . .


- 59 -

g> g> g> g> 03437.87 2log122log ====++++−−−− ++++ xx

smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa 0>>>>x . . . . smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³

03437.567 2log2log2 ====++++−−−− xx tag tag tag tag xt 2log==== smIkarkøayCa smIkarkøayCa smIkarkøayCa smIkarkøayCa ³³³³

441343784';0343562 ====−−−−====∆∆∆∆====++++−−−− tt eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 492128;72128 21 ====++++========−−−−==== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 7====t eK)an eK)an eK)an eK)an 77 2log ====x b¤ b¤ b¤ b¤ 1log2 ====x naM[ naM[ naM[ naM[ 1====x . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 49====t eK)an eK)an eK)an eK)an 497 2log ====x b¤ b¤ b¤ b¤ 2log2 ====x naM[ naM[ naM[ naM[ 4====x . . . . dUcenH dUcenH dUcenH dUcenH 4;1 21 ======== xx . . . .

��


- 60 -

lMhat´TI20lMhat´TI20lMhat´TI20lMhat´TI20 edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) (((( )))) 06log5log 2

22 ====−−−−++++−−−−++++ xxxx

x> x> x> x> (((( )))) 0216log)5(2log 32

3 ====++++−−−−−−−−++++ xxxx

K> K> K> K> (((( )))) 044log3log21

2

21 ====−−−−++++++++++++

xxxx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam edaHRsaysmIkarxageRkam ³³³³ k> k> k> k> (((( )))) (((( )))) 06log5log 2

22 ====−−−−++++−−−−++++ xxxx

tag tag tag tag xt 2log==== Edl Edl Edl Edl 0>>>>x . smIkarkøayCa . smIkarkøayCa . smIkarkøayCa . smIkarkøayCa ³³³³ (((( )))) 0652 ====−−−−++++−−−−++++ xtxt

KNna KNna KNna KNna (((( )))) )6(45 2 −−−−−−−−−−−−====∆∆∆∆ xx

(((( ))))22

2

74914

2442510

−−−−====++++−−−−====

++++−−−−++++−−−−====

xxx

xxx

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s

++++−−−−====++++−−−−++++−−−−====

−−−−====−−−−++++++++−−−−====

62

75

12

75

2

1

xxx

t

xxt


- 61 -

----cMeBaH cMeBaH cMeBaH cMeBaH 11 −−−−====t eK)an eK)an eK)an eK)an 1log2 −−−−====x naM[ naM[ naM[ naM[ 21====x

----cMeBaH cMeBaH cMeBaH cMeBaH 62 ++++−−−−==== xt eK)an eK)an eK)an eK)an 6log2 ++++−−−−==== xx CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag (((( )))) xyc 2log: ==== CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; (((( )))) 6: ++++−−−−==== xyd . . . .

2 3 4 5 6 7 8-1-2

2

3

4

5

6

7

-1

-2

-3

0 1

1

x

y

A

(d)

(c)

tamRkahViktamRkahViktamRkahViktamRkahVikeyIgeXIjfa eyIgeXIjfa eyIgeXIjfa eyIgeXIjfa (((( ))))c kat; kat; kat; kat; (((( ))))d Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc A manGab;sIus manGab;sIus manGab;sIus manGab;sIus 4====x EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . dUcenH dUcenH dUcenH dUcenH 4;

21

21 ======== xx . . . .


- 62 -

x> x> x> x> (((( )))) 0216log)5(2log 32

3 ====++++−−−−−−−−++++ xxxx tag tag tag tag xt 3log==== Edl Edl Edl Edl 0>>>>x . smIkarGacsresr . smIkarGacsresr . smIkarGacsresr . smIkarGacsresr ³³³³

(((( )))) 0216522 ====++++−−−−−−−−++++ xtxt (((( )))) (((( )))) (((( ))))22 22165' −−−−====++++−−−−−−−−−−−−====∆∆∆∆ xxx

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s 72;3 21 ++++−−−−======== xtt ----cMeBaH cMeBaH cMeBaH cMeBaH 31 ====t eK)an eK)an eK)an eK)an 3log3 ====x naM[ naM[ naM[ naM[ 27====x ----cMeBaH cMeBaH cMeBaH cMeBaH 722 ++++−−−−==== xt eK)an eK)an eK)an eK)an 72log3 ++++−−−−==== xx CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag CasmIkarGab;sIuscMnucRbsBVrvagExSekag (((( )))) xyc 3log: ==== CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; CamYynwgbnÞat; (((( )))) 72: ++++−−−−==== xyd . . . .

2 3 4 5 6 7 8-1-2

2

3

4

5

6

7

-1

-2

-3

0 1

1

x

y

A

(d)

(c)


- 63 -

tamRkahVikeyIgeXIjfa tamRkahVikeyIgeXIjfa tamRkahVikeyIgeXIjfa tamRkahVikeyIgeXIjfa (((( ))))c kat; kat; kat; kat; (((( ))))d Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc Rtg;cMnuc A manGab;sIus manGab;sIus manGab;sIus manGab;sIus 3====x EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar . EdlCab¤srbs;smIkar .

K> K> K> K> (((( )))) 044log3log21

2

21 ====−−−−++++++++++++

xxxx

tag tag tag tag xt21log==== enaH enaH enaH enaH 044)3(2 ====−−−−++++++++++++ xtxt

manb¤s manb¤s manb¤s manb¤s 1;4 21 ++++−−−−====−−−−==== xtt . . . . ----cMeBaH cMeBaH cMeBaH cMeBaH 41 −−−−====t enaH enaH enaH enaH 4log

21 −−−−====x naM[ naM[ naM[ naM[ 16====x

----cMeBaH cMeBaH cMeBaH cMeBaH 12 ++++−−−−==== xt enaH enaH enaH enaH 1log21 ++++−−−−==== xx

2 3-1

2

-1

0 1

1

x

y

A

B

tamRkahVikeKTajb¤s tamRkahVikeKTajb¤s tamRkahVikeKTajb¤s tamRkahVikeKTajb¤s 2;1 ======== xx . . . .


- 64 -

lMhat´TI21lMhat´TI21lMhat´TI21lMhat´TI21 k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa k>cUrRsaybBa¢ak;fa AaBa BA loglog ==== Edl Edl Edl Edl BAa ;; CabIcMnY CabIcMnY CabIcMnY CabIcMnYnBitviC¢man nig xusBI nBitviC¢man nig xusBI nBitviC¢man nig xusBI nBitviC¢man nig xusBI 1 . . . . x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar 6867 72log2log ====++++ xx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>RsaybBa¢ak;fa k>RsaybBa¢ak;fa k>RsaybBa¢ak;fa k>RsaybBa¢ak;fa AaBa BA loglog ==== tag tag tag tag (((( ))))iAt Balog==== eK)an eK)an eK)an eK)an Bt aA loglog ==== eday eday eday eday

At

tB

BA log

loglog ====

At

ABt

BAt

aB

BaB

aB

B

loglog

log.loglog

logloglog

========

====

eKTaj eKTaj eKTaj eKTaj (((( ))))iiBt Aalog==== tam tam tam tam (((( ))))i nig nig nig nig (((( ))))ii eK)an eK)an eK)an eK)an AaBa BA loglog ==== . . . . x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar x>edaHRsaysmIkar 6867 72log2log ====++++ xx tamrUbmnþxagelIeK)an tamrUbmnþxagelIeK)an tamrUbmnþxagelIeK)an tamrUbmnþxagelIeK)an xx 2log72log 7====


- 65 -

smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³

823log

77

3437

6867.2

68677

32

32log

2log

2log

2log2log

========⇒⇒⇒⇒====

====

====

====

====++++

xx

x

x

x

xx

dUcenH dUcenH dUcenH dUcenH 8====x Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar .

��


- 66 -

lMhat´TI22lMhat´TI22lMhat´TI22lMhat´TI22 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar (((( )))) (((( )))) 2loglogloglog 2442 ====++++ xx


smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa smIkarmann½ykalNa

>>>>>>>>>>>>

0

0log

0log

2

4

x

x

x

b¤ b¤ b¤ b¤ 1>>>>x

smIkarGacsresr smIkarGacsresr smIkarGacsresr smIkarGacsresr ³³³³ (((( )))) 2loglog

21

log21

log 2222 ====++++

xx

(((( )))) (((( ))))

(((( ))))(((( ))))(((( ))))

16

4log

2loglog

6loglog3

2loglog23

1

2loglog21

loglog21

log

2

22

22

22

22222

================

====++++−−−−

====++++++++

x

x

x

x

x

xx

dUcenH dUcenH dUcenH dUcenH 16====x Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar . Cab¤srbs;smIkar .


- 67 -

lMhat´TI23lMhat´TI23lMhat´TI23lMhat´TI23 eK[sIVút eK[sIVút eK[sIVút eK[sIVút )21(log 2

2

n

nu ++++==== Edl Edl Edl Edl INn ∈∈∈∈ KNna KNna KNna KNna (((( )))) n

n

kkn uuuuuS ++++++++++++++++======== ∑∑∑∑

====....210

0 . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNna KNna KNna KNna (((( )))) n

n

kkn uuuuuS ++++++++++++++++======== ∑∑∑∑

====....210

0

eKman eKman eKman eKman )21(log 22

n

nu ++++====

−−−−

−−−−====++++

12

12log

2

12

2 n

n

eK)an eK)an eK)an eK)an ∑∑∑∑====

++++

−−−−

−−−−====n

kk

k

nS0 2

12

212

12log

(((( ))))12log

12

12....

15127

315

13

log

12

12log

122

2

12

2

0 2

12

2

−−−−====

−−−−

−−−−××××××××××××××××====

−−−−

−−−−====

++++

++++

====

++++

∏∏∏∏

n

n

n

n

kk

k

dUcenH dUcenH dUcenH dUcenH )12(log12

2 −−−−====++++n

nS . . . .


- 68 -

lMhat´TI24lMhat´TI24lMhat´TI24lMhat´TI24 eK[sIVút eK[sIVút eK[sIVút eK[sIVút

−−−−==== 12

cos2ln nn

xu Edl Edl Edl Edl INn ∈∈∈∈

KNna KNna KNna KNna (((( )))) n

n

kkn uuuuuS ++++++++++++++++======== ∑∑∑∑

====....210

0 . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNnaKNnaKNnaKNna nS ³ ³ ³ ³ eKman eKman eKman eKman 1cos22cos 2 −−−−==== aa eK)an eK)an eK)an eK)an

(((( ))))(((( ))))1cos21cos21cos412cos2 2 ++++−−−−====−−−−====++++ aaaa eKTaj eKTaj eKTaj eKTaj

1cos21cos2

1cos2++++++++====−−−−

aa

a edayyk edayyk edayyk edayyk n

xa

2====

eK)an eK)an eK)an eK)an

++++

++++====

−−−−====1

2cos2

122

cos2ln1

2cos2ln

n

n

nn x

xx

u

eyIg)an eyIg)an eyIg)an eyIg)an ∑∑∑∑====

++++

++++====

++++

++++====

n

knk

k

n xx

x

x

S0 1

2cos2

12cos2ln

12

cos2

122

cos2ln

dUcenH dUcenH dUcenH dUcenH (((( ))))

++++−−−−++++==== 12

cos2ln12cos2ln nn

xxS ....


- 69 -

lMhat´TI25lMhat´TI25lMhat´TI25lMhat´TI25 edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar

====

====

9001

6.5

4001

5.4

yx

yx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar ³³³³

====

====

9001

6.5

4001

5.4

yx

yx

eyIgman eyIgman eyIgman eyIgman )4001

ln()5.4(ln ====yx b¤ b¤ b¤ b¤ )1(5ln24ln25ln4ln −−−−−−−−====++++ yx ehIy ehIy ehIy ehIy )

9001

ln()6.5ln( ====yx b¤ b¤ b¤ b¤ )2(5ln26ln26ln5ln −−−−−−−−====++++ yx tam tam tam tam )2(&)1( eK)anRbBnæ½ eK)anRbBnæ½ eK)anRbBnæ½ eK)anRbBnæ½ ³³³³

)5ln(

)6ln(

5ln26ln26ln5ln

5ln24ln25ln4ln −−−−

−−−−−−−−====++++−−−−−−−−====++++

yx

yx


- 70 -

−−−−−−−−====++++

++++====−−−−−−−−

)4(5ln25ln.6ln26ln.5ln5ln

)3(6ln.5ln26ln.4ln26ln.5ln6ln.4ln22 yx

yx

bUksmIkar bUksmIkar bUksmIkar bUksmIkar )4(&)3( eK)an eK)an eK)an eK)an ³³³³ )5ln6ln.4(ln2)6ln.4ln5(ln 22 −−−−====−−−− x naM[ naM[ naM[ naM[ 2−−−−====x

tamsmIkar tamsmIkar tamsmIkar tamsmIkar 4001

5.4 ====yx eKTaj eKTaj eKTaj eKTaj 4001

5.4 2 ====−−−− y naM[eKTaj naM[eKTaj naM[eKTaj naM[eKTaj 2−−−−====y . . . . dUcenHRbB½næsmIkarmanKUcemøIy dUcenHRbB½næsmIkarmanKUcemøIy dUcenHRbB½næsmIkarmanKUcemøIy dUcenHRbB½næsmIkarmanKUcemøIy 2,2 −−−−====−−−−==== yx . . . .


- 71 -

lMhat´TI26lMhat´TI26lMhat´TI26lMhat´TI26 edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³ ³ ³ ³ 0logloglog30 3

42

5,03 ====−−−−++++ xxx xxx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar edaHRsaysmIkar ³ ³ ³ ³

0logloglog30 34

25,0

3 ====−−−−++++ xxx xxx

lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð

≠≠≠≠≠≠≠≠

≠≠≠≠>>>>

14

15.0

1

0

x

x

x

x

naMeGay naMeGay naMeGay naMeGay

≠≠≠≠

≠≠≠≠≠≠≠≠>>>>

41

2

1

0

x

x

x

x

smIkarxagelIGacsresr smIkarxagelIGacsresr smIkarxagelIGacsresr smIkarxagelIGacsresr ³³³³

04lnln

ln32lnln

ln210

0)4ln(

ln)5,0ln(

lnln

ln30

323

====++++

−−−−−−−−

++++

====−−−−++++

xx

xx

xx

xx

xx

tag tag tag tag xt ln==== eyIg)an eyIg)an eyIg)an eyIg)an 0

4ln3

2ln2

10 ====++++

−−−−−−−−

++++t

tt

t


- 72 -

2ln.3612ln722ln289

02ln22ln179

02ln332ln422ln22ln1010

0)2ln(3)2ln2(2)2ln2)(2ln(10

222

22

2222

====++++====∆∆∆∆

====−−−−++++

====++++−−−−++++++++−−−−++++

====−−−−−−−−++++++++++++−−−−

tt

tttttt

tttttt

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ³³³³

2ln4

92ln192ln17

2ln92

92ln192ln17

2

1

−−−−====−−−−−−−−====

====++++−−−−====

t

t

eday eday eday eday xt ln==== eKTaj eKTaj eKTaj eKTaj

−−−−====

====

2ln4ln

2ln92

ln

x

x

naM[ naM[ naM[ naM[ 161

;49 ======== xx

dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s dUcenHsmIkarmanb¤s 161

,4 29

1 ======== xx . . . .


- 73 -

lMhat´TI27lMhat´TI27lMhat´TI27lMhat´TI27 edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar ³³³³

====

====++++

64

26)loglog(5

yx

yx xy

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar edaHRsayRbBn½æsmIkar

====

====++++

)2(64

)1(26)loglog(5

yx

yx xy

lkçx½NÐ lkçx½NÐ lkçx½NÐ lkçx½NÐ

≠≠≠≠≠≠≠≠>>>>>>>>1,1

0,0

yx

yx

smIkar smIkar smIkar smIkar )1( Gacsresr Gacsresr Gacsresr Gacsresr 526

log1

log ====++++x

xy

y

b¤ b¤ b¤ b¤ 01log.526

)(log 2 ====++++−−−− xx yy tag tag tag tag xt ylog====


1441

25169

',015262 ====−−−−====∆∆∆∆====++++−−−− tt


513

,51

512

513

21 ====++++========−−−−==== tt

---- cMeBaH cMeBaH cMeBaH cMeBaH 51====t eK)an eK)an eK)an eK)an

51

log ====xy

naM[ naM[ naM[ naM[ )3(51

yx ====


- 74 -

yk ¬#¦ CYskñúg ¬@¦ eK)an yk ¬#¦ CYskñúg ¬@¦ eK)an yk ¬#¦ CYskñúg ¬@¦ eK)an yk ¬#¦ CYskñúg ¬@¦ eK)an 64. 51

====yy

naM[ naM[ naM[ naM[ 32====y ehIytam ¬#¦ eK)an ehIytam ¬#¦ eK)an ehIytam ¬#¦ eK)an ehIytam ¬#¦ eK)an 2)32( 51

========x .... ---- cMeBaH cMeBaH cMeBaH cMeBaH 5====t eK)an eK)an eK)an eK)an 5log ====xy naM[ naM[ naM[ naM[ )4(5yx ==== yk ¬$¦ CYskñúg ¬@¦ eK)an yk ¬$¦ CYskñúg ¬@¦ eK)an yk ¬$¦ CYskñúg ¬@¦ eK)an yk ¬$¦ CYskñúg ¬@¦ eK)an 64. 5 ====yy naM[ naM[ naM[ naM[ 2====y ehIytam ¬$¦ eK)an ehIytam ¬$¦ eK)an ehIytam ¬$¦ eK)an ehIytam ¬$¦ eK)an 3225 ========x . . . . dUcenHRbB½næmanKUcemøIy dUcenHRbB½næmanKUcemøIy dUcenHRbB½næmanKUcemøIy dUcenHRbB½næmanKUcemøIy ³³³³ )32,2( ======== yx b¤ b¤ b¤ b¤ )2,32( ======== yx . . . .


- 75 -

lMhat´TI28lMhat´TI28lMhat´TI28lMhat´TI28 eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½

06)54(log5)54(log 36

326 ====++++−−−−−−−−−−−− −−−−−−−− xx xx ¿¿¿¿

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½eda¼RsaysmIkar ½

)1(06)54(log5)54(log 36

326 ====++++−−−−−−−−−−−− −−−−−−−− xx xx ¿¿¿¿

l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ

≠≠≠≠−−−−>>>>−−−−

>>>>−−−−

16

06

054 3

x

x

x

smmUl smmUl smmUl smmUl

≠≠≠≠<<<<<<<<

5

6

23 3

x

x

x

¦ ¦ ¦ ¦

≠≠≠≠<<<<

5

23 3

x

x

tag tag tag tag )54(log 36 xy x −−−−==== −−−− smIkar smIkar smIkar smIkar )1( Gacsresr ½Gacsresr ½Gacsresr ½Gacsresr ½

12425,0652 ====−−−−====∆∆∆∆====++++−−−− yy eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 3,2 21 ======== yy ----cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ 2====y eKán eKán eKán eKán 2)54(log 3

6 ====−−−−−−−− xx


- 76 -

0)62)(3(

01812

123654

)6(54

2

23

23

23

====−−−−−−−−++++

====−−−−−−−−++++

++++−−−−====−−−−

−−−−====−−−−

xxx

xxx

xxx

xx

eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 71,71,3 321 ++++====−−−−====−−−−==== xxx ----cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ 3====y eKán eKán eKán eKán 3)54(log 3

6 ====−−−−−−−− xx

0)3(18

016210818

)6()54(

2

2

33

====−−−−

====++++−−−−

−−−−====−−−−

x

xx

xx

eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 3====x . . . . dUcenHsMNMub¤srbs;smIkarKW dUcenHsMNMub¤srbs;smIkarKW dUcenHsMNMub¤srbs;smIkarKW dUcenHsMNMub¤srbs;smIkarKW ³³³³

}71;71;3;3{ ++++−−−−−−−−∈∈∈∈x . . . .


- 77 -


)9(log31

)9(log1 323

322)3( 3 xxxx xx

−−−−====−−−−++++ −−−−−−−−¿¿¿¿


)1()9(log31

)9(log1 323

322)3( 3 xxxx xx

−−−−====−−−−++++ −−−−−−−−¿¿¿¿

l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ l&kçx&NÐ

≠≠≠≠−−−−>>>>−−−−

>>>>−−−−

13

03

09 32

x

x

xx


≠≠≠≠<<<<

<<<<≠≠≠≠

2

3

9,0

x

x

xx

¦ ¦ ¦ ¦

≠≠≠≠≠≠≠≠<<<<

2

0

3

x

x

x

smIkar smIkar smIkar smIkar )1( Gacsresr ½Gacsresr ½Gacsresr ½Gacsresr ½

[[[[ ]]]] 03)9(log

09)9(log6)9(log

)9(log32

)9(log91

1

2323

323

3223

323

3223

====−−−−−−−−

====++++−−−−−−−−−−−−

−−−−====−−−−++++

−−−−

−−−−−−−−

−−−−−−−−

xx

xxxx

xxxx

x

xx

xx


- 78 -

12727

02727

927279

)3(9

3)9(log

3232

332

323

========

====−−−−−−−−++++−−−−====−−−−

−−−−====−−−−

====−−−−−−−−

x

x

xxxxx

xxx

xxx

dUcen¼smIkarman¦s dUcen¼smIkarman¦s dUcen¼smIkarman¦s dUcen¼smIkarman¦s 1====x . . . .


eK[GnuKmeK[GnuKmeK[GnuKmeK[GnuKmn_ n_ n_ n_

++++++++

++++++++====

1344

log.1

13log)(

2

222 xx

x

xxf

k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar k¿ eda¼RsaysmIkar

++++++++====

++++++++

1344

log1

13log

2

222 xx

x

x

x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam

++++++++

1

13log

22 x

x nig nig nig nig

++++++++1344

log2

2 xx viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña . viC¢manRBmKña .

K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ )(xf elIcenøa¼ elIcenøa¼ elIcenøa¼ elIcenøa¼ ]3;0[ . . . .


- 79 -

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k¿ eda¼RsaysmIkar ½k¿ eda¼RsaysmIkar ½k¿ eda¼RsaysmIkar ½k¿ eda¼RsaysmIkar ½

++++++++====

++++++++

1344

log1

13log

2

222 xx

x

x

lkç&x&NÐ lkç&x&NÐ lkç&x&NÐ lkç&x&NÐ 31−−−−≥≥≥≥x

eyIgán eyIgán eyIgán eyIgán 1344

1

13 2

2 ++++++++====

++++++++

xx

x

x

222 )1(4)13( ++++====++++ xx smmUl smmUl smmUl smmUl

−−−−−−−−====++++

++++====++++

2213

22132

2

xx

xx ¦ ¦ ¦ ¦

====++++++++

====++++−−−−

0332

01322

2

xx

xx

eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s 21

,1 21 ======== xx . . . . x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x¿ kMNt´témø x

edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam edIm,I[kenßam

++++++++

1

13log

22 x

x nig nig nig nig

++++++++1344

log2

2 xx

viC¢manRBmKñalu¼RtaEt viC¢manRBmKñalu¼RtaEt viC¢manRBmKñalu¼RtaEt viC¢manRBmKñalu¼RtaEt ³³³³

≥≥≥≥++++++++

≥≥≥≥++++++++

11344

11

13

2

2

xx

x

x


≥≥≥≥++++

++++−−−−

≥≥≥≥++++++++−−−−

013

334

01

3

2

2

2

xxx

x

xx


- 80 -

naM[ naM[ naM[ naM[

−−−−>>>>

≤≤≤≤≤≤≤≤

31

30

x

x ¦ ¦ ¦ ¦ 30 ≤≤≤≤≤≤≤≤ x . . . .

dUcen¼ dUcen¼ dUcen¼ dUcen¼ ]3;0[∈∈∈∈x . . . . K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ K¿ rktémøFMbMputénGnuKmn_ )(xf elIcenøa¼ elIcenøa¼ elIcenøa¼ elIcenøa¼ ]3;0[

cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ ]3;0[∈∈∈∈x eKman eKman eKman eKman

≥≥≥≥

++++++++

≥≥≥≥

++++++++

01344

log

01

13log

2

2

22

xx

x

x

tamvismPaBkUsIueKman tamvismPaBkUsIueKman tamvismPaBkUsIueKman tamvismPaBkUsIueKman 0;;2

.2

≥≥≥≥∀∀∀∀

++++≤≤≤≤ baba

ba

eyIgman eyIgman eyIgman eyIgman

++++++++

++++++++====

1344

log.1

13log)(

2

222 xx

x

xxf

eyIgán eyIgán eyIgán eyIgán ³³³³ 22

222 )1344

(log21

)1

13(log

21

)(

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

++++++++≤≤≤≤

xx

x

xxf

(((( )))) [[[[ ]]]]3;0;44log

41

)(

)1344

()1

13(log

41

)(

22

22

22

∈∈∈∈∀∀∀∀====≤≤≤≤

++++++++

++++++++≤≤≤≤

xxf

xx

x

xxf

dUcen¼témøFMbMputénGnuKmn_ dUcen¼témøFMbMputénGnuKmn_ dUcen¼témøFMbMputénGnuKmn_ dUcen¼témøFMbMputénGnuKmn_ f esµInwg esµInwg esµInwg esµInwg 4====M . . . .


- 81 -

lMhat´TI31lMhat´TI31lMhat´TI31lMhat´TI31 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ IRxxxxf ∈∈∈∈++++++++==== ,)1ln()( 2 cUrRsaybBa¢ak´fa cUrRsaybBa¢ak´fa cUrRsaybBa¢ak´fa cUrRsaybBa¢ak´fa )

11()

1(

bb

aa

fba

baf

++++++++

++++<<<<

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

cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb 0,0 >>>>>>>> ba .... dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay RsaybBa¢ak´fa RsaybBa¢ak´fa RsaybBa¢ak´fa RsaybBa¢ak´fa )

11()

1(

bb

aa

fba

baf

++++++++

++++<<<<

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

eyIgman eyIgman eyIgman eyIgman IRxxxxf ∈∈∈∈++++++++==== ,)1ln()( 2

eyIgán eyIgán eyIgán eyIgán 2

2

2

2

1

12

21

1

)'1()('

xx

x

x

xx

xxxf

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

++++====

++++++++

++++++++====

222

2

1

1

)1)(1(

1

xxxx

xx

++++====

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

++++++++====

edayRKb edayRKb edayRKb edayRKb 01: 2 >>>>++++∈∈∈∈ xIRx naM[ naM[ naM[ naM[ IRx

xxf ∈∈∈∈∀∀∀∀>>>>

++++==== ,0

1

1)('

2

dUcen¼GnuKmn_ dUcen¼GnuKmn_ dUcen¼GnuKmn_ dUcen¼GnuKmn_ )(xf CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI IR.... müa¨geTotcMeBa¼RKb müa¨geTotcMeBa¼RKb müa¨geTotcMeBa¼RKb müa¨geTotcMeBa¼RKb 0,0 >>>>>>>> ba eKman ½eKman ½eKman ½eKman ½


- 82 -

aa

baa

++++<<<<

++++++++ 11 nig nig nig nig

bb

bab

++++<<<<

++++++++ 11

eKTaj eKTaj eKTaj eKTaj b

ba

aba

bba

a++++

++++++++

<<<<++++++++

++++++++++++ 1111

¦ ¦ ¦ ¦

bb

aa

baba

++++++++

++++<<<<

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

111

edaysarEtGnuKmn_ edaysarEtGnuKmn_ edaysarEtGnuKmn_ edaysarEtGnuKmn_ )(xf CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI CaGnuKmn_ekInCanic©elI IR ehtuen¼tamlkçN¼GnuKmn_ekIneKán ehtuen¼tamlkçN¼GnuKmn_ekIneKán ehtuen¼tamlkçN¼GnuKmn_ekIneKán ehtuen¼tamlkçN¼GnuKmn_ekIneKán ³³³³

bb

aa

baba

++++++++

++++<<<<

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

111

naM[ naM[ naM[ naM[ )11

()1

(b

ba

af

baba

f++++

++++++++

<<<<++++++++

++++ dUcen¼ dUcen¼ dUcen¼ dUcen¼ )

11()

1(

bb

aa

fba

baf

++++++++

++++<<<<

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

cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb cMeBa¼RKb 0,0 >>>>>>>> ba . . . .


- 83 -

lMhat´TI32lMhat´TI32lMhat´TI32lMhat´TI32 eK[ eK[ eK[ eK[ 1≥≥≥≥a nig nig nig nig 1≥≥≥≥b . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )

2(log2loglog 222

baba

++++≤≤≤≤++++

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgHajfa bgHajfa bgHajfa bgHajfa )

2(log2loglog 222

baba

++++≤≤≤≤++++ cMeBa¼ cMeBa¼ cMeBa¼ cMeBa¼ 1≥≥≥≥a nig nig nig nig 1≥≥≥≥b eyIgman eyIgman eyIgman eyIgman baba .2≥≥≥≥++++ ( vismPaBkUsIu )( vismPaBkUsIu )( vismPaBkUsIu )( vismPaBkUsIu ) ¦ ¦ ¦ ¦ ba

ba.

2≥≥≥≥

++++

naM[ naM[ naM[ naM[ )loglog(21

)2

(log 222 baba ++++≥≥≥≥

++++

¦ ¦ ¦ ¦ )1()2

(log2loglog 222ba

ba++++≤≤≤≤++++

müa¨geToteKman müa¨geToteKman müa¨geToteKman müa¨geToteKman baba 2222 log.log2loglog ≥≥≥≥++++

)2()loglog(

21

loglog

)loglog()log(log2

loglog.log2log)log(log2

22222

22222

222222

baba

baba

bbaaba

++++≥≥≥≥++++

++++≥≥≥≥++++

++++++++≥≥≥≥++++

tamTMnak´TMng tamTMnak´TMng tamTMnak´TMng tamTMnak´TMng )1( nig nig nig nig )2( eyIgTaj½ eyIgTaj½ eyIgTaj½ eyIgTaj½


- 84 -

)2

(log2loglog

)2

(log4)loglog(

)2

(log2)loglog(21

222

22

22

22

22

baba

baba

baba

++++≤≤≤≤++++

++++≤≤≤≤++++

++++≤≤≤≤++++

dUcen¼ dUcen¼ dUcen¼ dUcen¼ )2

(log2loglog 222ba

ba++++≤≤≤≤++++ . . . .


- 85 -

lMhat´TI33lMhat´TI33lMhat´TI33lMhat´TI33 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

(((( ))))nnn

un ++++

++++==== 222 log.

11log

Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ++++++++++++++++==== ....321 CaGnuKm CaGnuKm CaGnuKm CaGnuKmn_én n_én n_én n_én n K>kMnt; K>kMnt; K>kMnt; K>kMnt; n edIm,I[ edIm,I[ edIm,I[ edIm,I[ 100====nS . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman (((( ))))nn

nun ++++

++++==== 222 log.

11log

eday eday eday eday (((( )))) (((( ))))nnn

nn 2222 log1log

1log

11log −−−−++++====

++++====

++++

nig nig nig nig (((( )))) [[[[ ]]]] (((( ))))1log)(log)1(loglog 2222

2 ++++++++====++++====++++ nnnnnn eK)an eK)an eK)an eK)an (((( ))))[[[[ ]]]] (((( ))))[[[[ ]]]] 2

2

2

2 log1log nnun −−−−++++==== cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman (((( ))))nn 22 log)1(log >>>>++++ dUcenH dUcenH dUcenH dUcenH 0>>>>nu cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n . . . . x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ++++++++++++++++==== ....321


- 86 -

eKman eKman eKman eKman (((( ))))[[[[ ]]]] (((( ))))[[[[ ]]]] 2

2

2

2 log1log nnun −−−−++++==== ----ebI ebI ebI ebI [[[[ ]]]] [[[[ ]]]]2

2

2

21 1log2log:1 −−−−======== un ----ebI ebI ebI ebI [[[[ ]]]] [[[[ ]]]]2

2

2

22 2log3log:2 −−−−======== un ----ebI ebI ebI ebI [[[[ ]]]] [[[[ ]]]]2

2

2

23 3log4log:3 −−−−======== un -------------------------------------------------------------------------------------------------------------------------------------------------------- ----ebI ebI ebI ebI [[[[ ]]]] (((( ))))[[[[ ]]]]2

2

2

2 log)1(log: nnunn n −−−−++++======== eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an eFIVviFIbUkGgÁ nig GgÁénsmPaBTaMgenHeK)an ³³³³ dUcenH dUcenH dUcenH dUcenH (((( ))))[[[[ ]]]] 2

2 1log ++++==== nSn . . . . K>kMnt; K>kMnt; K>kMnt; K>kMnt; n ed ed ed edIm,I[ Im,I[ Im,I[ Im,I[ 100====nS eyIg)an eyIg)an eyIg)an eyIg)an (((( ))))[[[[ ]]]] 1001log 2

2 ====++++n (((( )))) 101log2 ====++++n 102421 10 ========++++n dUcenH dUcenH dUcenH dUcenH 1023====n . . . .


- 87 -

lMhat´TI34lMhat´TI34lMhat´TI34lMhat´TI34 eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ eK[GnuKmn_ xxxf 2log2)( ++++==== Edl Edl Edl Edl 0>>>>x eKtag eKtag eKtag eKtag (((( )))) (((( ))))(((( )))))(;)(;)( 321 xfffuxffuxfu ============ nig nig nig nig (((( ))))(((( ))))(((( ))))....)(.... xffffu nn ==== . . . . k>eKyk k>eKyk k>eKyk k>eKyk nn uV 2log1++++==== . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa . cUrbgðajfa 2

1 nn VV ====++++ xxxx>eRbIGnumanrYmKNitviTüacUrRsayfa >eRbIGnumanrYmKNitviTüacUrRsayfa >eRbIGnumanrYmKNitviTüacUrRsayfa >eRbIGnumanrYmKNitviTüacUrRsayfa n

VVn2

1==== . . . . KKKK>KNna >KNna >KNna >KNna nV nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig x . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----bgðajfa bgðajfa bgðajfa bgðajfa 2

1 nn VV ====++++ eKman eKman eKman eKman nn uV 2log1++++==== enaH enaH enaH enaH 121 log1 ++++++++ ++++==== nn uvV eday eday eday eday (((( ))))(((( ))))(((( )))) (((( ))))1....)(.... −−−−======== nnn ufxffffu eK)an eK)an eK)an eK)an (((( )))) nu

nnn uufu 2log21 )( ++++

++++ ======== ehtuenH ehtuenH ehtuenH ehtuenH (((( ))))[[[[ ]]]]nu

nn uV 2log221 log1 ++++

++++ ++++====

(((( ))))22

21

221

)log1(

loglog21

nnn

nnn

VuV

uuV

====++++====

++++++++====

++++

++++

dUcenH dUcenH dUcenH dUcenH 21 nn VV ====++++ . . . .


- 88 -

x>eRbIGnumanrYmKNitviTüaRsayfa x>eRbIGnumanrYmKNitviTüaRsayfa x>eRbIGnumanrYmKNitviTüaRsayfa x>eRbIGnumanrYmKNitviTüaRsayfa n

VVn2

1==== eKman eKman eKman eKman 2

1 nn VV ====++++ ¬ tamsRmayxagelI ¦ ¬ tamsRmayxagelI ¦ ¬ tamsRmayxagelI ¦ ¬ tamsRmayxagelI ¦ eK)an eK)an eK)an eK)an 12

12

12 VVV ======== Bit Bit Bit Bit 22

14

12

23 VVVV ============ Bit Bit Bit Bit ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH ]bmafavaBitcMeBaH kn ==== KW KW KW KW k

VVk2

1==== eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH eyIgnwgRsayfavaBitcMeBaH 1++++==== kn KW KW KW KW 12

11

++++====++++

k

VVk eKman eKman eKman eKman 2

1 kk VV ====++++ eday eday eday eday k

VVk2

1==== eK)an eK)an eK)an eK)an 12

1

22

11

++++====

====++++

kk

VVVk Bit Bit Bit Bit dUdUdUdUcenH cenH cenH cenH n

VVn2

1==== . . . . K>KNna K>KNna K>KNna K>KNna nV nig nig nig nig nu CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig x ³ ³ ³ ³ eKman eKman eKman eKman n

VVn2

1==== eday eday eday eday (((( )))) (((( ))))2

2log2

2121 log1log1log1 2 xxuV x ++++====++++====++++==== ++++ dUcenH dUcenH dUcenH dUcenH (((( )))) 12

2log1++++

++++====n

xVn ehIy ehIy ehIy ehIy nn uV 2log1++++==== naM[ naM[ naM[ naM[ 12 −−−−==== nV

nu

dUcenH dUcenH dUcenH dUcenH (((( ))))(((( )))) 1log112

22 −−−−++++++++

====n

xnu . . . .


- 89 -

lMhat´TI35lMhat´TI35lMhat´TI35lMhat´TI35 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nu kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

(((( ))))2log )1( ++++==== ++++ nu nn Edl Edl Edl Edl ....;3;2;1====n . . . . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa (((( ))))nu CasIV CasIV CasIV CasIVútcuHCanic©cMeBaHRKb; útcuHCanic©cMeBaHRKb; útcuHCanic©cMeBaHRKb; útcuHCanic©cMeBaHRKb; 1≥≥≥≥n x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ln....lnlnln 321 ++++++++++++++++==== CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa (((( ))))nu CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman (((( ))))2log )1( ++++==== ++++ nu nn nig nig nig nig (((( ))))3log )2(1 ++++==== ++++++++ nu nn cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; 1≥≥≥≥n eKman eKman eKman eKman 1)2()2( 22 −−−−++++>>>>++++ nn b¤ b¤ b¤ b¤ )3)(1()12)(12()2( 2 ++++++++====++++++++−−−−++++>>>>++++ nnnnn eK)an eK)an eK)an eK)an (((( )))) [[[[ ]]]])3)(1(log2log )2(

2

)2( ++++++++>>>>++++ ++++++++ nnn nn b¤ b¤ b¤ b¤ (((( )))) (((( )))) (((( ))))13log1log2 )2()2( ++++++++++++>>>> ++++++++ nn nn tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eKman eKman eKman eKman ³³³³

(((( )))) (((( )))) (((( )))) (((( ))))2log1log22log1log )1()2()1()2( ++++++++≥≥≥≥++++++++++++ ++++++++++++++++ nnnn nnnn

b¤ b¤ b¤ b¤ (((( )))) (((( )))) (((( ))))222log1log )1()2( ≥≥≥≥++++++++++++ ++++++++ nn nn


- 90 -

tam tam tam tam (((( ))))1 nig nig nig nig (((( ))))2 eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³ (((( )))) (((( )))) (((( )))) (((( ))))

(((( )))) (((( ))))1

)2()1(

)2()2()1()2(

3log2log

3log1log2log1log

++++

++++++++

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

>>>>

++++>>>>++++

++++++++++++>>>>++++++++++++

nn

nn

nnnn

uu

nn

nnnn

dUcenH dUcenH dUcenH dUcenH (((( ))))nu CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; CasIVútcuHCanic©cMeBaHRKb; 1≥≥≥≥n . . . . x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk x>KNnaplbUk nn uuuuS ln....lnlnln 321 ++++++++++++++++==== eKman eKman eKman eKman (((( ))))

)1ln()2ln(

2log )1( ++++++++====++++==== ++++ n

nnu nn

eK)an eK)an eK)an eK)an (((( )))) (((( ))))

======== ∏∏∏∏∑∑∑∑

========

n

kk

n

kkn uuS

11lnln

eday eday eday eday (((( )))) (((( ))))(((( ))))∏∏∏∏∏∏∏∏

========

++++++++====

n

k

n

kk n

nu

11 1ln2ln

)2(log

2ln)2ln(

)1ln()2ln(

.....4ln5ln

.3ln4ln

.2ln3ln

2 ++++====++++====

++++++++====

nn

nn

dUcenH dUcenH dUcenH dUcenH (((( ))))[[[[ ]]]]2logln 2 ++++==== nSn . . . .


- 91 -

lMhat´TI36lMhat´TI36lMhat´TI36lMhat´TI36 eK[ eK[ eK[ eK[ xxxxf 2424 sin4coscos4sin)( ++++−−−−++++==== k> sRmYl k> sRmYl k> sRmYl k> sRmYl )(xf x>kMnt; x>kMnt; x>kMnt; x>kMnt; x edIm,I[ edIm,I[ edIm,I[ edIm,I[

21

)( ====xf dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> sRmYl k> sRmYl k> sRmYl k> sRmYl )(xf eKman eKman eKman eKman xxxxf 2424 sin4coscos4sin)( ++++−−−−++++==== eday eday eday eday xx 22 sin1cos −−−−==== nig nig nig nig xx 22 cos1sin −−−−====

)cos1(4cos)sin1(4sin)( 2424 xxxxxf −−−−++++−−−−−−−−++++====

xxxx

xx

xxxx

2222

2222

4242

sincos)cos2()sin2(

)cos2()sin2(

coscos44sinsin44

−−−−====−−−−−−−−−−−−====

−−−−−−−−−−−−====

++++−−−−−−−−++++−−−−====

dUcenH dUcenH dUcenH dUcenH xxf 2cos)( ==== . . . . x>kMnt; x>kMnt; x>kMnt; x>kMnt; x edIm,I[ edIm,I[ edIm,I[ edIm,I[

21

)( ====xf


2cos ====x naM[ naM[ naM[ naM[ ππππππππkx 2

3++++±±±±==== Edl Edl Edl Edl Zk ∈∈∈∈ . . . .


- 92 -

lMhat´TI37lMhat´TI37lMhat´TI37lMhat´TI37 cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; INn∈∈∈∈ eK[ eK[ eK[ eK[

12sin

12cos

ππππππππ nnnS ++++====

k> KNnatémø k> KNnatémø k> KNnatémø k> KNnatémø 12

cosππππ nig nig nig nig

12sin

ππππ x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 0624 12 ====++++−−−− ++++++++ nnn SSS dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> KNnatémø k> KNnatémø k> KNnatémø k> KNnatémø

12cos

ππππ nig nig nig nig 12

sinππππ

426

22

.23

22

.21

4sin

3sin

4cos

3cos

)43

cos(12

cos

++++====++++====

++++====

−−−−====

ππππππππππππππππ

ππππππππππππ

426

21

.22

22

.23

3cos

4sin

4cos

3sin

)43

sin(12

sin

−−−−====−−−−====

−−−−====

−−−−====




2612

sin;12

2612

cos−−−−====++++==== ππππππππ . . . .


- 93 -

x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa 0624 12 ====++++−−−− ++++++++ nnn SSS eKman eKman eKman eKman

12sin

12cos

ππππππππ nnnS ++++====

tag tag tag tag 12

sin;12

cos 21ππππππππ ======== xx enaH enaH enaH enaH nn

n xxS 21 ++++====


426

426

21 ====−−−−++++++++====++++ xx

ehIy ehIy ehIy ehIy 41

1626

426

.4

26. 21 ====−−−−====−−−−++++====xx

eK)an eK)an eK)an eK)an 1x nig nig nig nig 2x Cab¤ssmIkar Cab¤ssmIkar Cab¤ssmIkar Cab¤ssmIkar 041

262 ====++++−−−− xx

b¤ b¤ b¤ b¤ 01624 2 ====++++−−−− xx


====++++−−−−

====++++−−−−

01624

01624

22

2

12

1

xx

xx

b¤ b¤ b¤ b¤

====++++−−−−

====++++−−−−++++++++

++++++++

)(0624

)(0624

21

22

2

11

12

1

iixxx

ixxxnnn

nnn

bUksmIkar bUksmIkar bUksmIkar bUksmIkar )(i nig nig nig nig )(ii GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an ³³³³ 0)()(62)(4 21

12

11

22

21 ====++++++++++++−−−−++++ ++++++++++++++++ nnnnnn xxxxxx

dUcenH dUcenH dUcenH dUcenH 0624 12 ====++++−−−− ++++++++ nnn SSS . . . .


- 94 -

lMhat´TI38lMhat´TI38lMhat´TI38lMhat´TI38 cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa

21

73

cos7

2cos

7cos ====++++−−−−

ππππππππππππ (IMO 1963)

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa

21

73

cos7

2cos

7cos ====++++−−−− ππππππππππππ

tag tag tag tag 7

3cos

72

cos7

cosππππππππππππ ++++−−−−====S

KuNGgÁTaMgBIrénvismPaBnwg KuNGgÁTaMgBIrénvismPaBnwg KuNGgÁTaMgBIrénvismPaBnwg KuNGgÁTaMgBIrénvismPaBnwg 7

sin2ππππ eK)an eK)an eK)an eK)an ³³³³

7sin

73

cos27

sin7

2cos2

72

sin7

sin2ππππππππππππππππππππππππ ++++−−−−====S

7sin

)7

2sin

73

(sin)7

sin7

3(sin

72

sin

)7

2sin

74

(sin)7

sin7

3(sin

72

sin

ππππ

ππππππππππππππππππππ


====

−−−−++++−−−−−−−−====

−−−−++++−−−−−−−−====

eKTaj eKTaj eKTaj eKTaj 21====S


73

cos7

2cos

7cos ====++++−−−− ππππππππππππ . . . .


- 95 -

lMhat´TI39lMhat´TI39lMhat´TI39lMhat´TI39 KNna KNna KNna KNna )

53

sin41()5

sin41( 22 ππππππππ −−−−−−−−====A dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNna KNna KNna KNna )

53

sin41()5

sin41( 22 ππππππππ −−−−−−−−====A eKman eKman eKman eKman xxx cos3cos43cos 3 −−−−==== eKTaj eKTaj eKTaj eKTaj

xx

xcos

3cos3cos4 2 ====−−−−

xx

x

xx

x

cos3cos

sin41

cos3cos

3)sin1(4

2

2

====−−−−

====−−−−−−−−


cos

59

cos

53

cos

59

cos

5cos

53

cos

ππππ

ππππ

ππππ

ππππ

ππππ

ππππ

====××××====A

eday eday eday eday 5

cos)5

2cos(5

9cos

ππππππππππππππππ ====−−−−==== eKTaj)an eKTaj)an eKTaj)an eKTaj)an 1====A . . . . dUcenH dUcenH dUcenH dUcenH 1)

53

sin41()5

sin41( 22 ====−−−−−−−− ππππππππ . . . .


- 96 -


eK[ eK[ eK[ eK[ 144

)( 2

3

−−−−++++====

xx

xxf . KNnatémø . KNnatémø . KNnatémø . KNnatémø )

7(cos

ππππf . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNnatémø KNnatémø KNnatémø KNnatémø )

7(cos

ππππf

eK)an eK)an eK)an eK)an )(1

7cos4

7cos4

7cos

)7

(cos2

3

if−−−−++++

==== ππππππππ

ππππππππ

eKman eKman eKman eKman )7

3sin(

74

sinππππππππππππ −−−−====

)()17

cos47

cos4(81

7cos

)7

cos1(437

cos47

cos8

7sin43)1

7cos2(

7cos4

7sin4

7sin3)1

7cos2(

7cos

7sin4

73

sin7

2cos

72

sin2

23

23

22

32

ii−−−−++++====

−−−−−−−−====−−−−

−−−−====−−−−

−−−−====−−−−

====






tam tam tam tam )(i nig nig nig nig )(ii eKTaj eKTaj eKTaj eKTaj 81

)7

(cos ====ππππf . . . .


- 97 -

lMhat´TI41lMhat´TI41lMhat´TI41lMhat´TI41 eK[ eK[ eK[ eK[ )cos(sin

1)( xx

kxf kk

k ++++==== Edl Edl Edl Edl ...;3;2;1====k cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa

121

)()( 64 ====−−−− xfxf . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bbbbgðajfa gðajfa gðajfa gðajfa

121

)()( 64 ====−−−− xfxf

eKman eKman eKman eKman )cos(sin41

)( 444 xxxf ++++====

(((( ))))xx

xxxx

22

22222

cossin2141

]cossin2)cos(sin[41

−−−−====

−−−−++++====

ehIy ehIy ehIy ehIy )cos(sin61

)( 666 xxxf ++++====

[[[[ ]]]])cos(sincossin3)cos(sin61

)( 22223226 xxxxxxxf ++++−−−−++++====

(((( ))))xx 22 cossin3161 −−−−====


61

41

)()( 64 ====−−−−====−−−− xfxf


)()( 64 ====−−−− xfxf . . . .


- 98 -

lMhat´TI42lMhat´TI42lMhat´TI42lMhat´TI42 cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati cUrbgðajfacMeBaHRKb;cMnYnKt;FmµCati n nig cMeBaHRKb;cMnYnBit nig cMeBaHRKb;cMnYnBit nig cMeBaHRKb;cMnYnBit nig cMeBaHRKb;cMnYnBit

tk

x2

ππππ≠≠≠≠ Edl Edl Edl Edl nt ....,,2,1,0==== nig nig nig nig k CacMnYnKt;eKman CacMnYnKt;eKman CacMnYnKt;eKman CacMnYnKt;eKman

xxxxx

nn 2cotcot

2sin

1.....

4sin1

2sin1 −−−−====++++++++++++

(IMO 1966)

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfabgðajfabgðajfabgðajfa

xxxxx

nn 2cotcot

2sin

1.....

4sin1

2sin1 −−−−====++++++++++++

tag tag tag tag ∑∑∑∑====

====++++++++++++====n

kknn

xxxxS

1 2sin

1

2sin

1.....

4sin1

2sin1

eKman eKman eKman eKman a

aaa

aaa 2sin

2coscos22sincossin

2sin1 222 −−−−====++++====

aaaa

aa

a2cotcot

2sin2cos

2sincos2

2sin1 2

−−−−====−−−−====

eK)an eK)an eK)an eK)an (((( ))))∑∑∑∑====

−−−− −−−−====−−−−====n

k

nkkn xxxxS

1

1 2cotcot2cot2cot

dUcenH dUcenH dUcenH dUcenH xxxxx

nn 2cotcot

2sin

1.....

4sin1

2sin1 −−−−====++++++++++++


- 99 -

lMhat´TI43lMhat´TI43lMhat´TI43lMhat´TI43 eK[ eK[ eK[ eK[ dcba ,,, CacMnYnenAkñúgcenøaH CacMnYnenAkñúgcenøaH CacMnYnenAkñúgcenøaH CacMnYnenAkñúgcenøaH ];0[ ππππ edaydwgfa edaydwgfa edaydwgfa edaydwgfa

++++====++++++++====++++

)cos2(cos4cos7cos

)sin2(sin4sin7sin

dcba

dcba

cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )cos(7)cos(2 cbda −−−−====−−−− . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa )cos(7)cos(2 cbda −−−−====−−−− eKman eKman eKman eKman

++++====++++++++====++++

)cos2(cos4cos7cos

)sin2(sin4sin7sin

dcba

dcba

b¤ b¤ b¤ b¤

−−−−====−−−−−−−−====−−−−

bcda

bcda

cos7cos4cos8cos

sin7sin4sin8sin

b¤ b¤ b¤ b¤

−−−−====−−−−

−−−−====−−−−

)()cos7cos4()cos8cos(

)()sin7sin4()sin8sin(22

22

iibcda

ibcda

bUksmIkar bUksmIkar bUksmIkar bUksmIkar )(i nig nig nig nig )(ii GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an GgÁnwgGgÁeK)an ³³³³

)cos(56)cos(16

)cos(5665)cos(1665

cbda

cbda

−−−−−−−−====−−−−−−−−−−−−−−−−====−−−−−−−−

dUcenH dUcenH dUcenH dUcenH )cos(7)cos(2 cbda −−−−====−−−− . . . .


- 100 -

lMhat´TI44lMhat´TI44lMhat´TI44lMhat´TI44 cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam cUrsresrCaplKuNktþaénkenSam ³³³³

)sin()sin()sin( xzzyyx −−−−++++−−−−++++−−−− dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay sresrCaplKuNk+tþa sresrCaplKuNk+tþa sresrCaplKuNk+tþa sresrCaplKuNk+tþa tag tag tag tag )sin()sin()sin( xzzyyxT −−−−++++−−−−++++−−−−==== man man man man

22

cos2

sin2)sin()sin(zyxzx

zyyx++++−−−−−−−−====−−−−++++−−−−


cos2

sin2)sin(xzxz

xz−−−−−−−−====−−−−

−−−−−−−−++++−−−−−−−−====2

cos22

cos2

sin2xzzyxzx

T

2

sin2

sin2

sin4

2sin

2sin

2sin4

xzzyyx

yxyzzx

−−−−−−−−−−−−−−−−====

−−−−−−−−−−−−−−−−====

dUcenH dUcenH dUcenH dUcenH

2sin

2sin

2sin4)sin()sin()sin(

xzzyyxxzzyyx

−−−−−−−−−−−−−−−−====−−−−++++−−−−++++−−−−


- 101 -

lMhat´TI45lMhat´TI45lMhat´TI45lMhat´TI45 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag

2cba

p++++++++==== CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN . CaknøHbrimaRténRtIekaN .

kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa bc

cpbpA ))((2

sin−−−−−−−−==== rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng rYcsresrTMnak;TMng

BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH .BIreTotEdlRsedogKñaenH . xxxx----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa

81

2sin

2sin


dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----bgðajfa bgðajfa bgðajfa bgðajfa

bccpbpA ))((

2sin

−−−−−−−−==== tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs Abccba cos2222 −−−−++++==== eKTaj eKTaj eKTaj eKTaj

bcacb

A2

cos222 −−−−++++====

eday eday eday eday 2cos1

2sin2 AA −−−−====

bc

cbacbabc

cbabc

acbbc

4))((

4)(

4)(2 22222

−−−−++++++++−−−−====

−−−−−−−−====−−−−++++−−−−====


cbap

++++++++==== enaH enaH enaH enaH

−−−−====−−−−++++−−−−====++++−−−−

)(2

)(2

cpcba

bpcba


- 102 -

eK)an eK)an eK)an eK)an bc

cpbpbc

cpbpA ))((4

))((42

sin2 −−−−−−−−====−−−−−−−−====

dUcenH dUcenH dUcenH dUcenH bc

cpbpA ))((2

sin−−−−−−−−==== . . . .

TMnak;TMngBIreTOtEdlRsedogKñaenHKW TMnak;TMngBIreTOtEdlRsedogKñaenHKW TMnak;TMngBIreTOtEdlRsedogKñaenHKW TMnak;TMngBIreTOtEdlRsedogKñaenHKW ³³³³

accpapB ))((

2sin

−−−−−−−−==== nig nig nig nig ab

bpapC ))((2

sin−−−−−−−−====

xxxx----Rsayfa Rsayfa Rsayfa Rsayfa 81

2sin

2sin


eKman eKman eKman eKman bc

cpbpA ))((2

sin−−−−−−−−====

accpapB ))((

2sin

−−−−−−−−==== nig nig nig nig ab

bpapC ))((2

sin−−−−−−−−====

eK)an eK)an eK)an eK)an )())()((

2sin

2sin

2sin i

abccpbpapCBA −−−−−−−−−−−−====

tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eKman eKman eKman eKman

))((2

))((2

))((22

))((2)()(

bpapc

bpapbacba

bpapbap

bpapbpap

−−−−−−−−≥≥≥≥

−−−−−−−−≥≥≥≥−−−−−−−−++++++++

−−−−−−−−≥≥≥≥−−−−−−−−

−−−−−−−−≥≥≥≥−−−−++++−−−−

eKTaj)an eKTaj)an eKTaj)an eKTaj)an )1(4

))((2c

bpap ≤≤≤≤−−−−−−−−

dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr )2(4

))((2a

cpbp ≤≤≤≤−−−−−−−−


- 103 -

nig nig nig nig )3(4

))((2b

apcp ≤≤≤≤−−−−−−−− eFIvplKuNTMnak;TMng eFIvplKuNTMnak;TMng eFIvplKuNTMnak;TMng eFIvplKuNTMnak;TMng )2(;)1( nig nig nig nig )3( eK)an eK)an eK)an eK)an ³³³³

64)()()(

222222 cba

cpbpap ≤≤≤≤−−−−−−−−−−−−

eKTaj eKTaj eKTaj eKTaj )(81))()((

iiabc

cpbpap ≤≤≤≤−−−−−−−−−−−−

tamTMnak;TMtamTMnak;TMtamTMnak;TMtamTMnak;TMng ng ng ng )(i nig nig nig nig )(ii eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³

81

2sin

2sin

2sin ≤≤≤≤CBA . . . . lMhat´TI46lMhat´TI46lMhat´TI46lMhat´TI46 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC ehIyeKtag ehIyeKtag ehIyeKtag ehIyeKtag r nig nig nig nig R erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg;erogKñaCakaMrgVg; carwkkñúg nig kaMrgVg;carwkeRkAénRcarwkkñúg nig kaMrgVg;carwkeRkAénRcarwkkñúg nig kaMrgVg;carwkeRkAénRcarwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .tIekaN .tIekaN .tIekaN . cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa

Rr

CBA ++++====++++++++ 1coscoscos dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay Rsayfa Rsayfa Rsayfa Rsayfa

Rr

CBA ++++====++++++++ 1coscoscos


cos2

cos2coscosBABA

BA−−−−++++====++++


- 104 -

2

cos2

sin2

2cos)

22cos(2

BAC

BAC

−−−−====

−−−−−−−−==== ππππ


sin21cos 2 CC −−−−====

)2

cos2

(sin2

sin21coscoscosBACC

CBA−−−−−−−−−−−−====++++++++


cos2

cos2

cos2

sinBABABAC −−−−−−−−++++====−−−−−−−−

2

sin2

sin2BA−−−−====

eK)an eK)an eK)an eK)an )(

2sin

2sin

2sin41coscoscos i

CBACBA ++++====++++++++

eKdwgfa eKdwgfa eKdwgfa eKdwgfa ³³³³

bccpbpA ))((

2sin

−−−−−−−−==== ; ac

cpapB ))((2

sin−−−−−−−−====

nig nig nig nig ab

bpapC ))((2

sin−−−−−−−−====

abccpbpap

CBA))()((

41coscoscos−−−−−−−−−−−−++++====++++++++

tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug ³³³³

Rabc

prcpbpappS4

))()(( ========−−−−−−−−−−−−====


- 105 -

eKTaj eKTaj eKTaj eKTaj Srprp

Scpbpap ============−−−−−−−−−−−− 2

2

))()((

ehIy ehIy ehIy ehIy prRabc 4==== eK)an eK)an eK)an eK)an

Rr

prRSr

CBA ++++====++++====++++++++ 14

.41coscoscos

dUcenH dUcenH dUcenH dUcenH Rr

CBA ++++====++++++++ 1coscoscos . . . . lMhat´TI47lMhat´TI47lMhat´TI47lMhat´TI47 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag

2cba

p++++++++==== Caknø Caknø Caknø CaknøHbrimaRténRtIekaNehIy HbrimaRténRtIekaNehIy HbrimaRténRtIekaNehIy HbrimaRténRtIekaNehIy r nig nig nig nig R

erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN .erogKñaCakaMrgVg;carwkkñúg nig kaMrgVg;carwkeRkAénRtIekaN . kkkk----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrprcabcab 422 ++++++++====++++++++ xxxx----cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa Rrrpcba 822 22222 −−−−−−−−====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----bgðajfa bgðajfa bgðajfa bgðajfa Rrprcabcab 422 ++++++++====++++++++ ttttamrUbmnþehrug amrUbmnþehrug amrUbmnþehrug amrUbmnþehrug ³³³³

prcpbpappS ====−−−−−−−−−−−−==== ))()(( eKTaj eKTaj eKTaj eKTaj

pcpbpap

r))()(( −−−−−−−−−−−−====


- 106 -

pabcpcabcabpp

r

pabcpcabcabpcbap

r

pcpbpap

r

−−−−++++++++++++−−−−====

−−−−++++++++++++++++++++−−−−====

−−−−−−−−−−−−====

)(2

)()(

))()((

332

232

2

pabc

prcabcab

pabcpcabcabp

r

++++++++====++++++++

−−−−++++++++++++−−−−====

22

32 )(

eday eday eday eday R

abcprS

4======== eKTaj eKTaj eKTaj eKTaj rR

pabc

4====

dUcenH dUcenH dUcenH dUcenH Rrprcabcab 422 ++++++++====++++++++ . . . . xxxx----bgðajfa bgðajfa bgðajfa bgðajfa Rrrpcba 822 22222 −−−−−−−−====++++++++ tamsmPaB tamsmPaB tamsmPaB tamsmPaB )(2)( 2222 cabcabcbacba ++++++++++++++++++++====++++++++ eKTaj eKTaj eKTaj eKTaj )(2)( 2222 cabcabcbacba ++++++++−−−−++++++++====++++++++ eday eday eday eday pcba 2====++++++++ nig nig nig nig Rrprcabcab 422 ++++++++====++++++++ eK)an eK)an eK)an eK)an )4(24 222222 Rrprpcba ++++++++−−−−====++++++++ dUcenH dUcenH dUcenH dUcenH Rrrpcba 822 22222 −−−−−−−−====++++++++ . . . .


- 107 -

lMhat´TI48lMhat´TI48lMhat´TI48lMhat´TI48 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag

2cba

p++++++++==== CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy CaknøHbrimaRténRtIekaNehIy r

CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa CakaMrgVg;carwkkñúgénRtIekaN . cUrRsayfa ³³³³ k> k> k> k> r

Ccp

Bbp

Aap ====−−−−====−−−−====−−−−

2tan)(

2tan)(

2tan)(

x> x> x> x> p

rRCBA ++++====++++++++4

2tan

2tan

2tan

K> K> K> K> rpCBA ====

2tan

2tan

2tan

X> X> X> X> rpCBA ====++++++++

2cot

2cot

2cot

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> k> k> k> r

Ccp

Bbp

Aap ====−−−−====−−−−====−−−−

2tan)(

2tan)(

2tan)(

A

'A B C

'B 'C

I


- 108 -

eKman eKman eKman eKman pCABCAB 2====++++++++ eday eday eday eday '''' BAACBCACAB ++++====++++==== ¬ eRBaH ¬ eRBaH ¬ eRBaH ¬ eRBaH '' BABC ==== ¦ ¦ ¦ ¦ ehIy ehIy ehIy ehIy '''' ACCAABCBCA ++++====++++==== ¬ eRBaH ¬ eRBaH ¬ eRBaH ¬ eRBaH CACB '' ==== nig nig nig nig '' ACAB ==== ¦ ¦ ¦ ¦ eK)an eK)an eK)an eK)an pACCABCBAAC 2'''' ====++++++++++++++++

paAC

pBCAC

pBCBCAC

PBCCABAAC

22'2

22'2

2'2

2)''('2

====++++====++++

====++++++++====++++++++++++

eKTaj eKTaj eKTaj eKTaj apABAC −−−−======== '' kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg AIC ' eKman eKman eKman eKman

apr

ACICA

−−−−========

''

2tan

eKTaj eKTaj eKTaj eKTaj rA

ap ====−−−−2

tan)( . . . . dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr

bprB−−−−

====2

tan b¤ b¤ b¤ b¤ rB

bp ====−−−−2

tan)(

nig nig nig nig cp

rC−−−−

====2

tan b¤ b¤ b¤ b¤ rC

cp ====−−−−2

tan)(

dUcenH dUcenH dUcenH dUcenH rC

cpB

bpA

ap ====−−−−====−−−−====−−−−2

tan)(2

tan)(2

tan)(


- 109 -

x> x> x> x> p

rRCBA ++++====++++++++ 42

tan2

tan2

tan

tag tag tag tag 2

tan2

tan2

tanCBA

T ++++++++====

eday eday eday eday cp

rCbp

rBap

rA−−−−

====−−−−

====−−−−

====2

tan;2

tan;2

tan

eK)an eK)an eK)an eK)an ³³³³

cpr

bpr

apr

T−−−−

++++−−−−

++++−−−−

====

[[[[ ]]]]

(((( ))))

(((( ))))

(((( ))))

prcabcabp

Scabcabpp

S

cabcabpcbapS

cpbpapabacbcpbapcapcbppr

cpbpapbpapcpapcpbpr

++++++++++++−−−−====

++++++++++++−−−−====

++++++++++++++++++++−−−−====

−−−−−−−−−−−−++++++++++++++++−−−−++++−−−−++++−−−−====

−−−−−−−−−−−−−−−−−−−−++++−−−−−−−−++++−−−−−−−−====

2

22

2

2

2

43

)(23.

))()(()()()(3

))()(())(())(())((

tamrUbmnþtamrUbmnþtamrUbmnþtamrUbmnþehrug ehrug ehrug ehrug ³³³³

prcpbpappS ====−−−−−−−−−−−−==== ))()(( eKTaj eKTaj eKTaj eKTaj

pcpbpap

r))()(( −−−−−−−−−−−−====


- 110 -

pabcpcabcabpp

r

pabcpcabcabpcbap

r

pcpbpap

r

−−−−++++++++++++−−−−====

−−−−++++++++++++++++++++−−−−====

−−−−−−−−−−−−====

)(2

)()(

))()((

332

232

2

pabc

prcabcab

pabc

cabcabpr

pabcpcabcabp

r

++++++++====++++++++

−−−−++++++++++++−−−−====

−−−−++++++++++++−−−−====

22

22

32 )(

eday eday eday eday R

abcprS

4======== eKTaj eKTaj eKTaj eKTaj rR

pabc

4====

eKTaj eKTaj eKTaj eKTaj Rrprcabcab 422 ++++++++====++++++++ ehtuenH ehtuenH ehtuenH ehtuenH

pRr

prrRprp

T44222 ++++====++++++++++++−−−−====

dUcenH dUcenH dUcenH dUcenH p

rRCBA ++++====++++++++4

2tan

2tan

2tan . . . .

K> K> K> K> prCBA ====

2tan

2tan

2tan

eKman eKman eKman eKman cp

rCbp

rBap

rA−−−−

====−−−−

====−−−−

====2

tan;2

tan;2

tan


- 111 -

eK)an eK)an eK)an eK)an ))()((2

tan2

tan2

tan3

cpbpaprCBA

−−−−−−−−−−−−====

pr

prr

Sr

S

rS

cpbpapppr

================

−−−−−−−−−−−−====

22

2

2

3

))()((

dUcenH dUcenH dUcenH dUcenH prCBA ====

2tan

2tan

2tan . . . .

X> X> X> X> rpCBA ====++++++++

2cot

2cot

2cot

eK)an eK)an eK)an eK)an r

cpr

bpr

apCBA −−−−++++−−−−++++−−−−====++++++++2

cot2

cot2

cot

rp

rppr

cbap

====−−−−====

++++++++−−−−====

23

)(3

dUcenH dUcenH dUcenH dUcenH rpCBA ====++++++++

2cot

2cot

2cot . . . .


- 112 -

lMhat´TI49lMhat´TI49lMhat´TI49lMhat´TI49 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag S CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig CaépÞRkla nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 22

coscoscosR


x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa x> cUrbgðajfa SCcBbAa 4cotcotcot 222 ====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>bgðajfa k>bgðajfa k>bgðajfa k>bgðajfa 22

coscoscosR


tag tag tag tag CcBbAaT coscoscos ++++++++==== tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs R

Cc

Bb

Aa

2sinsinsin

============


============

CRc

BRb

ARa

sin2

sin2

sin2

eK)an eK)an eK)an eK)an )2sin2sin2(sin CBART ++++++++====

[[[[ ]]]][[[[ ]]]]

[[[[ ]]]]CBAR

BABACR

BACBACR

CCBABAR

sinsinsin4

)cos()cos(sin2

)cos(sin2)cos(sin2

cossin2)cos()sin(2

====++++−−−−−−−−====

++++−−−−−−−−====++++−−−−++++====


- 113 -

eday eday eday eday Rc

CRb

BRa

A2

sin;2

sin;2

sin ============

eK)an eK)an eK)an eK)an 23 28.4

R

abc

R

abcRT ========

dUcenH dUcenH dUcenH dUcenH 22coscoscos

R

abcCcBbAa ====++++++++ . . . .

x> bgðajfa x> bgðajfa x> bgðajfa x> bgðajfa SCcBbAa 4cotcotcot 222 ====++++++++

tag tag tag tag CcBbAa cotcotcot 222 ++++++++====∑∑∑∑

SR

abcR

abc

R

abcR

CcBbAaR

CRcBRbARa

CcC

cBb

Bb

AaA

a

44

.42

.2

)coscoscos(2

cos2cos2cos2

cossin

cossin

cossin

2 ================

++++++++====++++++++====

++++++++====

dUcenH dUcenH dUcenH dUcenH SCcBbAa 4cotcotcot 222 ====++++++++ ....


- 114 -

lMhat´TI50lMhat´TI50lMhat´TI50lMhat´TI50 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cba ;; . . . . tag tag tag tag r CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig CakaMrgVg;carwkkñúg nig R CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . CakaMrgVg;carwkeRkARtIekaN . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa )(2cotcotcot RrCcBbAa ++++====++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa )(2cotcotcot RrCcBbAa ++++====++++++++ tag tag tag tag CcBbAaT cotcotcot ++++++++====

)()coscos(cos2

cos2cos2cos2

cossin

cossin

cossin

iCBAR

CRBRAR

CC

cB

Bb

AA

a

++++++++====++++++++====

++++++++====


cos2

cos2coscosBABA

BA−−−−++++====++++

2

cos2

sin2

2cos)

22cos(2

BAC

BAC

−−−−====

−−−−−−−−==== ππππ


sin21cos 2 CC −−−−====

)2

cos2

(sin2

sin21coscoscosBACC

CBA−−−−−−−−−−−−====++++++++


- 115 -


cos2

cos2

cos2

sinBABABAC −−−−−−−−

++++====−−−−−−−−

2

sin2

sin2BA−−−−====


sin2

sin2

sin41coscoscos iCBA

CBA ++++====++++++++ eKdwgfa eKdwgfa eKdwgfa eKdwgfa ³³³³

bccpbpA ))((

2sin

−−−−−−−−==== ; ac

cpapB ))((2

sin−−−−−−−−====

nig nig nig nig ab

bpapC ))((2

sin−−−−−−−−====

abccpbpap

CBA))()((

41coscoscos−−−−−−−−−−−−++++====++++++++

tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug tamrUbmnþehrug ³³³³

Rabc

prcpbpappS4

))()(( ========−−−−−−−−−−−−==== eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³

Srprp

Scpbpap ============−−−−−−−−−−−− 2

2

))()(( nig nig nig nig prRabc 4====


.41coscoscos iiRr

prRSr

CBA ++++====++++====++++++++

tam tam tam tam )(i nig nig nig nig )(ii eK)an eK)an eK)an eK)an )(2)1(2 RrRr

RT ++++====++++==== dUcenH dUcenH dUcenH dUcenH )(2cotcotcot RrCcBbAa ++++====++++++++ . . . .


- 116 -

lMhat´TI51lMhat´TI51lMhat´TI51lMhat´TI51 bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyeKman mYyeKman mYyeKman mYyeKman

2tan

2tan


++++−−−−

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa

2tan

2tan


++++−−−−

eKman eKman eKman eKman ARa sin2==== nig nig nig nig BRb sin2==== eK)an eK)an eK)an eK)an )sin(sin2 BARba −−−−====−−−−

)(

2sin

2sin4

2cos

2sin4

iCBA

Rba

BABARba

−−−−====−−−−

++++−−−−====−−−−

ehIy ehIy ehIy ehIy )sin(sin2 BARba ++++====++++

)(

2cos

2cos4

2cos

2sin4

iiBAC

Rba

BABARba

−−−−====++++

−−−−++++====++++

eFIviFIEck eFIviFIEck eFIviFIEck eFIviFIEck )(i nig nig nig nig )(ii GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an ³³³³

2tan

2tan


++++−−−− . . . .


- 117 -


)2

sin2

sin2

(sin4 222222 CBA

abc

cab

bca ++++++++≥≥≥≥++++++++ dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay

bgðajfa bgðajfa bgðajfa bgðajfa )2

sin2

sin2

(sin4 222222 CBA

abc

cab

bca ++++++++≥≥≥≥++++++++

tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs Abccba cos2222 −−−−++++==== eKTaj eKTaj eKTaj eKTaj A

bc

cb

bca

cos22

−−−−++++==== eday eday eday eday 2≥≥≥≥++++bc

cb


sin4cos22 22 A

Abca ====−−−−≥≥≥≥

dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr 2

sin4 22 B

cab ≥≥≥≥ nig nig nig nig

2sin4 2

2 Cabc ≥≥≥≥


sin42

sin42

sin4 222222 CBA

abc

cab

bca ++++++++≥≥≥≥++++++++

dUcenH dUcenH dUcenH dUcenH )2

sin2

sin2

(sin4 222222 CBA

abc

cab

bca ++++++++≥≥≥≥++++++++


- 118 -


3333 16

81coscoscos

pc

C

b

B

a

A ≥≥≥≥++++++++

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa 3333 16

81coscoscos

pc

C

b

B

a

A ≥≥≥≥++++++++

tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs tamRTwsþIbTkUsIunUs Abccba cos2222 −−−−++++==== eKTaj eKTaj eKTaj eKTaj )1(1

cos22

2

2

2

2 −−−−++++====a

c

a

b

a

Abc

dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr )2(1cos2

2

2

2

2

2 −−−−++++====b

a

b

c

b

Bca

nig nig nig nig )3(1cos2

2

2

2

2

2 −−−−++++====c

b

c

a

c

Aab bUksmIkar bUksmIkar bUksmIkar bUksmIkar )2(;)1( nig nig nig nig )3( eK)an eK)an eK)an eK)an ³³³³

33222

3

cos2cos2cos2

2

2

2

2

2

2

2

2

2

2

2

2

222

====−−−−++++++++≥≥≥≥

−−−−++++++++++++++++++++====

++++++++====

b

c

c

b

c

a

a

c

b

a

a

b

c

Cab

b

Bca

a

AbcT

eKTaj eKTaj eKTaj eKTaj 23coscoscos

222 ≥≥≥≥++++++++c

Cab

b

Bca

a

Abc


- 119 -

b¤ b¤ b¤ b¤ )(2

3coscoscos333 i

abcc

C

b

B

a

A ≥≥≥≥++++++++ eKman eKman eKman eKman 332 abccbap ≥≥≥≥++++++++==== eKTaj eKTaj eKTaj eKTaj

278 3p

abc ≤≤≤≤ naM[ naM[ naM[ naM[ )(16

812

33 ii

pabc≥≥≥≥

tam tam tam tam )(i nig nig nig nig )(ii eKTaj)an eKTaj)an eKTaj)an eKTaj)an ³³³³

3333 16

81coscoscos

pc

C

b

B

a

A ≥≥≥≥++++++++ . . . .

lMhat´TI54lMhat´TI54lMhat´TI54lMhat´TI54 bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN bgðajfakñúgRtIekaN ABC mYyEdl mYyEdl mYyEdl mYyEdl CBA ≠≠≠≠≠≠≠≠ eKmaneKmaneKmaneKman k> k> k> k> 0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa x> x> x> x> abc

BAACCBBAcACbCBa

3)sin()sin()sin(

)(sin)(sin)(sin 333333

====−−−−−−−−−−−−

−−−−++++−−−−++++−−−−

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa 0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa

A

H B C


- 120 -

kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg ABH eKman eKman eKman eKman c

BHABBH

B ========cos eKTaj eKTaj eKTaj eKTaj BcBH cos==== kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg kñúgRtIekaNEkg ACH eKman eKman eKman eKman

bHC

ACHC

C ========cos eKTaj eKTaj eKTaj eKTaj CbHC cos==== eday eday eday eday HCBHBCa ++++======== naM[ naM[ naM[ naM[ BcCba coscos ++++==== dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr AcCab coscos ++++==== nig nig nig nig AbBac coscos ++++==== tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs tamRTwsþIbTsIunUs

Bb

Aa

sinsin====

eK)an eK)an eK)an eK)an B

AcCaA

BcCbsin

coscossin

coscos ++++====++++

b¤ b¤ b¤ b¤

CBAcCAaBb

BABAccAaBb

BAc

CAaBb

AAcACaBBcBCb

cos)sin(cos)sinsin(

)cos()sin(cos)sinsin(

)2sin2(sin2

cos)sinsin(

sincossincoscossinsincos

−−−−−−−−====−−−−++++−−−−====−−−−

−−−−====−−−−

++++====++++

eKTaj eKTaj eKTaj eKTaj )(sinsin)sin( iBbAaBAc −−−−====−−−− dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr )(sinsin)sin( iiCcBbCBa −−−−====−−−− )(sinsin)sin( iiiAaCcACb −−−−====−−−− bUkTMnak;TMng bUkTMnak;TMng bUkTMnak;TMng bUkTMnak;TMng )(&)(;)( iiiiii eK)an eK)an eK)an eK)an ³³³³

0)sin()sin()sin( ====−−−−++++−−−−++++−−−− BAcACbCBa . . . .


- 121 -

x> x> x> x> abcBAACCB

BAcACbCBa3

)sin()sin()sin()(sin)(sin)(sin 333333

====−−−−−−−−−−−−

−−−−++++−−−−++++−−−−

tag tag tag tag )sin(;)sin(;)sin( BAczACbyCBax −−−−====−−−−====−−−−==== eK)an eK)an eK)an eK)an 0====++++++++ zyx b¤ b¤ b¤ b¤ zyx −−−−====++++ eK)an eK)an eK)an eK)an 33)( zyx −−−−====++++

xyzzyx

zyxyzx

zyyxxyx

zyxyyxx

3

3

)(3

33

333

333

333

33223

====++++++++

−−−−====++++−−−−

−−−−====++++++++++++

−−−−====++++++++++++

CMnYs CMnYs CMnYs CMnYs )sin(;)sin(;)sin( BAczACbyCBax −−−−====−−−−====−−−−==== rYcEckGgÁTaMgBIrnwg rYcEckGgÁTaMgBIrnwg rYcEckGgÁTaMgBIrnwg rYcEckGgÁTaMgBIrnwg )sin()sin()sin( BAACCB −−−−−−−−−−−−

abcBAACCB

BAcACbCBa3

)sin()sin()sin()(sin)(sin)(sin 333333

====−−−−−−−−−−−−

−−−−++++−−−−++++−−−−

��


- 122 -

lMhlMhlMhlMhat´TI55at´TI55at´TI55at´TI55 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;eday kMnt;eday kMnt;eday kMnt;eday ³³³³

4sin.2

ππππnU

nn ==== Edl Edl Edl Edl *INn ∈∈∈∈

k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa k>cUrbgðajfa 4

sin4

cos4

)1(cos.2


++++ x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa

4)1(

cos)2(4

cos)2( 1 ππππππππ ++++−−−−==== ++++ nnU nn

n K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk nn UUUUS ++++++++++++++++==== .........321 CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k>k>k>k>bgðajfa bgðajfa bgðajfa bgðajfa

4sin

4cos

4)1(

cos.2ππππππππππππ nnn

−−−−====++++

tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ bababa sinsincoscos)cos( −−−−====++++ eyIg)an eyIg)an eyIg)an eyIg)an ³³³³

)4

sin4

sin4

cos4

(cos2

)44

cos(24

)1(cos2



ni

n

nn

−−−−====

++++====++++

4

sin4

cos

)4

sin22

4cos

22

(2

ππππππππ

ππππππππ

nn

nn

−−−−====

−−−−====


- 123 -


sin4

cos4

)1(cos.2


++++ . . . .

x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa x> Taj[)anfa 4

)1(cos)2(

4cos)2( 1 ππππππππ ++++−−−−==== ++++ nn

U nnn

eeeeyIgman yIgman yIgman yIgman 4

sin4

cos4

)1(cos.2


++++


)1(cos2

4cos

4sin

ππππππππππππ ++++−−−−====nnn

KuNGgÁTaMgBIrnwg KuNGgÁTaMgBIrnwg KuNGgÁTaMgBIrnwg KuNGgÁTaMgBIrnwg n)2( eK)an eK)an eK)an eK)an

4)1(

cos)2(4

cos)2(4

sin)2( 1 ππππππππππππ ++++−−−−==== ++++ nnn nnn


)1(cos)2(

4cos)2( 1 ππππππππ ++++−−−−==== ++++ nn

U nnn . . . .

K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk K> KNnaplbUk nn UUUUS ++++++++++++++++==== .........321 eyIg)an eyIg)an eyIg)an eyIg)an ∑∑∑∑

========++++++++++++++++====

n

kknn UUUUUS

1321 )(...........

4

)1(cos)2(

4cos2

4)1(

cos)2(4

cos)2(

1

1

1

ππππππππ

ππππππππ

++++−−−−====

++++−−−−====

++++

====

++++∑∑∑∑

n

kk

n

n

k

kk


)1(cos)2(1 1 ππππ++++

−−−−==== ++++ nS n

n . . . .


- 124 -

lMhat´TI56lMhat´TI56lMhat´TI56lMhat´TI56 eK[ eK[ eK[ eK[ x CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl 0217160 2 <<<<++++−−−− xx . . . . cUrbgðajfa cUrbgðajfa cUrbgðajfa cUrbgðajfa 0

13sin <<<<

−−−−xππππ . . . .

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgðajfa bgðajfa bgðajfa bgðajfa 0

13sin <<<<

−−−−xππππ

tag tag tag tag 217160)( 2 ++++−−−−==== xxxf eeeebI bI bI bI 02171600)( 2 ====++++−−−−⇔⇔⇔⇔==== xxxf

150405041)21)(60(4)71( 2 ====−−−−====−−−−−−−−====∆∆∆∆ eKTaj¦s eKTaj¦s eKTaj¦s eKTaj¦s

53

1203971

,127

120171

21 ====++++========

−−−−==== xx eyIg)an eyIg)an eyIg)an eyIg)an 0217160)( 2 <<<<++++−−−−==== xxxf naM[ naM[ naM[ naM[

53

127 <<<<<<<< x b¤ b¤ b¤ b¤

59

347 <<<<<<<< x

54

1343 <<<<−−−−<<<< x naM[ naM[ naM[ naM[

34

131

54 <<<<

−−−−<<<<

x

eKTaj eKTaj eKTaj eKTaj 3

4135

4 ππππππππππππ <<<<−−−−

<<<<x


sin <<<<

−−−−xππππ . . . .

dUcenH ebI dUcenH ebI dUcenH ebI dUcenH ebI x CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl CacMnYnBitEdl 0217160 2 <<<<++++−−−− xx eK)an eK)an eK)an eK)an 0

13sin <<<<

−−−−xππππ . . . .


- 125 -

lMhat´TI57lMhat´TI57lMhat´TI57lMhat´TI57 eK[ eK[ eK[ eK[

20

ππππ<<<<<<<< x . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa . cUrRsaybBa¢ak;fa ³³³³

21cos

11

sin1

1 ++++≥≥≥≥

++++

++++xx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay RsaybBa¢ak;RsaybBa¢ak;RsaybBa¢ak;RsaybBa¢ak;

21cos

11

sin1

1 ++++≥≥≥≥

++++

++++xx

eyIgman eyIgman eyIgman eyIgman ³³³³

xxxxxx cossin1

cos1

sin1

1cos

11

sin1

1 ++++++++++++====

++++

++++

eKman eKman eKman eKman xxxx cossin

12

cos1

sin1 ≥≥≥≥++++

eKTaj eKTaj eKTaj eKTaj xxxxxx cossin

1cossin1

21cos

11

sin1

1 ++++++++≥≥≥≥

++++

++++


2sin21

cossin ≤≤≤≤==== xxx enaH enaH enaH enaH 2cossin1 ≥≥≥≥

xx

eK)an eK)an eK)an eK)an 2)21(2221cos

11

sin1

1 ++++====++++++++≥≥≥≥

++++

++++xx

dUcenH dUcenH dUcenH dUcenH 21cos

11

sin1

1 ++++≥≥≥≥

++++

++++xx

. . . .


- 126 -

lMhat´TI58lMhat´TI58lMhat´TI58lMhat´TI58 eK[eK[eK[eK[RtIekaN RtIekaN RtIekaN RtIekaN ABC . . . . 1111----cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; cMeBaHRKb; [,0], ππππ∈∈∈∈yx cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa ³³³³

)2

(sin2sinsinyx

yx++++≤≤≤≤++++ . . . .

2222----cUrRsayfa cUrRsayfa cUrRsayfa cUrRsayfa 2

33sinsinsin ≤≤≤≤++++++++ CBA . . . .

3333----Tajbgðajfa Tajbgðajfa Tajbgðajfa Tajbgðajfa ³³³³ k/ k/ k/ k/

833


x/ x/ x/ x/ 8

332

cos2

cos2

cos ≤≤≤≤CBA

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay 1111----Rsayfa Rsayfa Rsayfa Rsayfa )

2(sin2sinsin

yxyx

++++≤≤≤≤++++

eyIgman eyIgman eyIgman eyIgman

−−−−

++++====++++2

cos2

sin2sinsinyxyx

yx

eday eday eday eday [,0], ππππ∈∈∈∈yx eKTaj eKTaj eKTaj eKTaj

222,

20

ππππππππππππ <<<<−−−−<<<<−−−−<<<<

++++<<<<yxyx


sin >>>>

++++ yx nig nig nig nig 12

cos ≤≤≤≤

−−−− yx


- 127 -


++++≤≤≤≤

−−−−

++++====++++2

sin22

cos2

sin2sinsinyxyxyx

yx

dUcenH dUcenH dUcenH dUcenH )2

(sin2sinsinyx

yx++++≤≤≤≤++++ RKb; RKb; RKb; RKb; [,0], ππππ∈∈∈∈yx . . . .

2222----RRRRsayfa sayfa sayfa sayfa 2

33sinsinsin ≤≤≤≤++++++++ CBA

tamsRmayxagelIeKman tamsRmayxagelIeKman tamsRmayxagelIeKman tamsRmayxagelIeKman )2

(sin2sinsinyx

yx++++≤≤≤≤++++

eday eday eday eday CBA ,, CamMukñúgénRtIekaN CamMukñúgénRtIekaN CamMukñúgénRtIekaN CamMukñúgénRtIekaN ABC enaHeKman enaHeKman enaHeKman enaHeKman [,0],, ππππ∈∈∈∈CBA .... eyIg)an eyIg)an eyIg)an eyIg)an )(

2sin2sinsin a

BABA

++++≤≤≤≤++++

)(6

4sin2

)2

3sin(23

sinsin

bCBA

CBACCBA

C

++++++++<<<<

++++++++++++≤≤≤≤

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

bUkvismIkar bUkvismIkar bUkvismIkar bUkvismIkar )(a nig nig nig nig )(b GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an GgÁ nig GgÁeK)an ³³³³ )()

64

sin()2

sin(23

sinsinsinsin cCBABACBA

CBA

++++++++++++++++≤≤≤≤

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

edayeKman edayeKman edayeKman edayeKman )2

64

2sin(2)6

4sin()

2sin(

CBABACBABA

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

≤≤≤≤++++++++++++

++++ )()

3sin(2)

64

sin()2

sin( dCBACBABA ++++++++≤≤≤≤

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


- 128 -

tamTMnak;TMng tamTMnak;TMng tamTMnak;TMng tamTMnak;TMng )(c nig nig nig nig )(d eKTaj)an eKTaj)an eKTaj)an eKTaj)an

233

3sin3sinsinsin

3sin4

3sinsinsinsin

====

++++++++≤≤≤≤++++++++

++++++++≤≤≤≤

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

CBACBA

CBACBACBA


33sinsinsin ≤≤≤≤++++++++ CBA . . . .

3333----TajbgðajTajbgðajTajbgðajTajbgðaj k/ k/ k/ k/

833


tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman 2


tamvismPaBkUsIu tamvismPaBkUsIu tamvismPaBkUsIu tamvismPaBkUsIu 3 sinsinsin3sinsinsin CBACBA ≥≥≥≥++++++++ eKTaj eKTaj eKTaj eKTaj

233

sinsinsin33 ≤≤≤≤CBA


33sinsinsin ≤≤≤≤CBA


33sin.sin.sin ≤≤≤≤CBA . . . .

x/ x/ x/ x/ 8

332

cos2

cos2

cos ≤≤≤≤CBA

eyIgman eyIgman eyIgman eyIgman 2

cos2

sin22

cos2

sin2sinsinsinCCBABA

CBA ++++−−−−++++====++++++++

eday eday eday eday ππππ====++++++++ CBA b¤ b¤ b¤ b¤ 222CBA −−−−====

++++ ππππ


- 129 -


sin)22

cos(2

cos,2

cos)22

sin(2

sinCCBACCBA ====−−−−====

++++====−−−−====++++ ππππππππ

eKeKeKeK)an)an)an)an2

cos2

cos22

cos2

cos2sinsinsinCBABAC

CBA++++++++

−−−−====++++++++ )

2cos

2(cos

2cos2sinsinsin

BABACCBA

++++++++−−−−====++++++++

2

cos2

cos2

cos4sinsinsinCBA

CBA ====++++++++

tamsRtamsRtamsRtamsRmayxagelIeyIgman mayxagelIeyIgman mayxagelIeyIgman mayxagelIeyIgman 2


eyIgTaj eyIgTaj eyIgTaj eyIgTaj 2

332

cos2

cos2

cos4 ≤≤≤≤CBA

dUcendUcendUcendUcenH H H H 8

332

cos2

cos2

cos ≤≤≤≤CBA . . . .

��


- 130 -

lMhat´TI59lMhat´TI59lMhat´TI59lMhat´TI59 eK[sIVúténcMnYnkMupøic eK[sIVúténcMnYnkMupøic eK[sIVúténcMnYnkMupøic eK[sIVúténcMnYnkMupøic )( nZ kMnteday kMnteday kMnteday kMnteday ³³³³

(((( ))))

∈∈∈∈++++====

++++====

++++ INnZZZ

iZ

nnn ;||212

31

1

0

( ( ( ( || nZ Cam Cam Cam CamÜDulén ÜDulén ÜDulén ÜDulén nZ ) . ) . ) . ) . snµtfa snµtfa snµtfa snµtfa INniZ nnnn ∈∈∈∈∀∀∀∀++++==== ,)sin.(cos θθθθθθθθρρρρ Edl Edl Edl Edl IRnnn ∈∈∈∈>>>> θθθθρρρρρρρρ ;,0 . . . . kkkk----rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag rkTMnak´TMngrvag nθθθθ nignignignig 1++++nθθθθ ehIy ehIy ehIy ehIy nρρρρ nignignignig 1++++nρρρρ . . . . xxxx----rkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVútrkRbePTénsIVút )( nθθθθ rYcKNnarYcKNnarYcKNnarYcKNna nθθθθ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . KKKK----cUrbgHajfacUrbgHajfacUrbgHajfacUrbgHajfa

2cos....

2cos

2coscos 121

00−−−−==== n

nθθθθθθθθθθθθθθθθρρρρρρρρ

rYbBa¢ak rYbBa¢ak rYbBa¢ak rYbBa¢ak nρρρρ GnuKmn_én GnuKmn_én GnuKmn_én GnuKmn_én n .... dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkk----TMnak´TMngrvag TMnak´TMngrvag TMnak´TMngrvag TMnak´TMngrvag nθθθθ nig nig nig nig 1++++nθθθθ ehIy ehIy ehIy ehIy nρρρρ nig nig nig nig 1++++nρρρρ eyIgman eyIgman eyIgman eyIgman )sin.(cos nnnn iZ θθθθθθθθρρρρ ++++==== naM[ naM[ naM[ naM[ )sin(cos 1111 ++++++++++++++++ ++++==== nnnn iZ θθθθθθθθρρρρ


- 131 -

eday eday eday eday )||(21

1 nnn ZZZ ++++====++++ ehIy ehIy ehIy ehIy nnZ ρρρρ====|| eKán eKán eKán eKán ³³³³

[[[[ ]]]]nnnnnnn ii ρρρρθθθθθθθθρρρρθθθθθθθθρρρρ ++++++++====++++ ++++++++++++ )sin.(cos21

)sin(cos 111

)2

sin.2

(cos2

cos)sin.(cos

)sin.cos1(21

)sin.(cos

111

111

nnnnnnn

nnnnnn

ii

ii

θθθθθθθθθθθθρρρρθθθθθθθθρρρρ

θθθθθθθθρρρρθθθθθθθθρρρρ

++++====++++

++++++++====++++

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

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

eKTaján eKTaján eKTaján eKTaján 2

cos1n

nnθθθθρρρρρρρρ ====++++ nig nig nig nig

21n

nθθθθθθθθ ====++++

dUcen¼ dUcen¼ dUcen¼ dUcen¼ 21n

nθθθθθθθθ ====++++ nignignignig

2cos1

nnn

θθθθρρρρρρρρ ====++++ . . . . xxxx----RbePTénsIVút RbePTénsIVút RbePTénsIVút RbePTénsIVút )( nθθθθ nigKNna nigKNna nigKNna nigKNna nθθθθ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ½ ½ ½ ½ tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman tamsRmayxagelIeyIgman nn θθθθθθθθ

21

1 ====++++ naM[ naM[ naM[ naM[ (((( ))))nθθθθ CasIVútFrNImaRtmanersugesµI CasIVútFrNImaRtmanersugesµI CasIVútFrNImaRtmanersugesµI CasIVútFrNImaRtmanersugesµI

21====q . . . .

tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ nn q××××==== 0θθθθθθθθ


sin.3

cos2

31)sin(cos 0000

ππππππππθθθθθθθθρρρρ ii

iZ ++++====++++====++++====

eKTaján eKTaján eKTaján eKTaján 3

;1 00ππππθθθθρρρρ ========

dUcen¼ dUcen¼ dUcen¼ dUcen¼ nn

2

1.

3ππππθθθθ ==== . . . .


- 132 -

KKKK----bgHajfa bgHajfa bgHajfa bgHajfa 2

cos....2

cos2

coscos 2100

nn

θθθθθθθθθθθθθθθθρρρρρρρρ ==== tamtamtamtamsRmayxagelIeKman sRmayxagelIeKman sRmayxagelIeKman sRmayxagelIeKman ³³³³

2cos1

nnn

θθθθρρρρρρρρ ====++++ ¦ ¦ ¦ ¦ 2

cos1 n

n

n θθθθρρρρ

ρρρρ====++++

eKán eKán eKán eKán ∏∏∏∏ ∏∏∏∏−−−−====

====

−−−−====

====

++++

====

1

0

1

0

1 )2

cos(nk

k

nk

k

k

k

k θθθθρρρρ

ρρρρ

2

cos.........2

cos.2

cos.cos 1210

0

−−−−==== nn θθθθθθθθθθθθθθθθρρρρρρρρ

dUcen¼ dUcen¼ dUcen¼ dUcen¼ 2

cos....2

cos2

coscos 12100

−−−−==== nn

θθθθθθθθθθθθθθθθρρρρρρρρ . . . .

müa¨geToteyIgman müa¨geToteyIgman müa¨geToteyIgman müa¨geToteyIgman ³³³³

2cossin2

2cos

2sin2sin 1

nn

nnn

θθθθθθθθθθθθθθθθθθθθ ++++======== ( eRBa¼( eRBa¼( eRBa¼( eRBa¼

21n

nθθθθθθθθ ====++++ ))))

eKTaj eKTaj eKTaj eKTaj 1sin

sin.

21

2cos

++++====

n

nn

θθθθθθθθθθθθ

ehtuen¼ ehtuen¼ ehtuen¼ ehtuen¼ n

nn

nnn θθθθ

θθθθθθθθ

θθθθθθθθθθθθ

θθθθθθθθρρρρ

sinsin

.2

1sin

sin.....

sinsin

.sinsin

.2

1 01

2

1

1

0 ======== −−−−

dUcen¼ dUcen¼ dUcen¼ dUcen¼ )

2

1.

3sin(

1.

2

3

)2

1.

3sin

3sin

2

11

n

n

n

nn ππππππππ

ππππ

ρρρρ ++++====((((

==== . . . .


- 133 -

lMhat´TI60lMhat´TI60lMhat´TI60lMhat´TI60 eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit eK[sVúIténcMnYnBit )( nU kMntelI kMntelI kMntelI kMntelI n eday½eday½eday½eday½

10 ====U nig nig nig nig aaUUINn nn sincos: 1 ++++====∈∈∈∈∀∀∀∀ ++++ Edl Edl Edl Edl

20

ππππ<<<<<<<< a .... k¿ tag k¿ tag k¿ tag k¿ tag

2cot

aUV nn −−−−==== . . . .

cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa )( nV CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy .CasIVútFrNImaRtmYYy . x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt x¿KNnalImIt )....(lim 10 n

nVVV ++++++++++++

+∞+∞+∞+∞→→→→ nig nig nig nig n

nU

+∞+∞+∞+∞→→→→lim

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k¿ bgHajfa k¿ bgHajfa k¿ bgHajfa k¿ bgHajfa )( nV CasIVútFrNImaRtmYYy ½CasIVútFrNImaRtmYYy ½CasIVútFrNImaRtmYYy ½CasIVútFrNImaRtmYYy ½ man man man man

2cot

aUV nn −−−−==== naM[ naM[ naM[ naM[

2cot11

aUV nn −−−−==== ++++++++

Et Et Et Et aaUU nn sincos1 ++++====++++ eKán eKán eKán eKán

2cotsincos1

aaaUV nn −−−−++++====++++

)1

2tan

2cos

2sin2(

2cotcos

2cot

2cos

2sin2cos

−−−−++++====

−−−−++++====

aaaaaU

aaaaU

n

n


- 134 -

aVV

aa

Uaa

aUV

aaaUV

a

aaaa

aUV

nn

nnn

nn

nn

cos

cos)2

cot(cos2

cotcos

)12

sin2(2

cotcos

)1

2cos

2sin

2cos

2sin2(

2cotcos

1

1

21

1

====

−−−−====−−−−====

−−−−++++====

−−−−++++====

++++

++++

++++

++++

eday eday eday eday aVV nn cos1 ====++++ naM[ naM[ naM[ naM[ )( nV CasIVútFrNImaRtCasIVútFrNImaRtCasIVútFrNImaRtCasIVútFrNImaRtmanersug manersug manersug manersug acos nig tY nig tY nig tY nig tY

2cot1

2cot00

aaUV −−−−====−−−−==== . . . .

x¿x¿x¿x¿KNnalImItKNnalImItKNnalImItKNnalImIt )....(lim 10 nn

VVV +++++++++++++∞+∞+∞+∞→→→→

nig nig nig nig nn

U+∞+∞+∞+∞→→→→

lim eyIgmaneyIgmaneyIgmaneyIgman³³³³

aaa

qq

VVVVnn

n cos1cos1

).2

cot1(1

1...

11

010 −−−−−−−−−−−−====

−−−−−−−−====++++++++++++

++++++++ eyIgán eyIgán eyIgán eyIgán

−−−−−−−−−−−−====++++++++++++

++++

+∞+∞+∞+∞→→→→+∞+∞+∞+∞→→→→ aaa

VVVn

nn

n cos1cos1

)2

cot1(lim)....(lim1

10

eday eday eday eday 0coslim 1 ====+++++∞+∞+∞+∞→→→→

an

n

dUcen¼ dUcen¼ dUcen¼ dUcen¼ a

a

VVV nn cos1

2cot1

)....(lim 10 −−−−

−−−−====++++++++++++

+∞+∞+∞+∞→→→→ . . . .


- 135 -

müa¨geTot müa¨geTot müa¨geTot müa¨geTot 2

cota

UV nn −−−−==== naM[ naM[ naM[ naM[ 2

cota

VU nn ++++==== eday eday eday eday a

aqVV nn

n cos)2

cot1(0 −−−−====××××==== eKán eKán eKán eKán

2cotcos)

2cot1(

aa

aU n

n ++++−−−−==== nig nig nig nig

2cot

2cotcos)

2cot1(limlim

aaa

aU n

nn

n====

++++−−−−====+∞+∞+∞+∞→→→→+∞+∞+∞+∞→→→→


cotlima

U nn

====+∞+∞+∞+∞→→→→

. . . . lMhat´TI61lMhat´TI61lMhat´TI61lMhat´TI61 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnteday ½kMnteday ½kMnteday ½kMnteday ½

1;0 10 ======== UU nig nig nig nig nnn UaUUINn −−−−====∈∈∈∈∀∀∀∀ ++++++++ cos2: 12 Edl Edl Edl Edl IRa ∈∈∈∈ .... k¿ tag k¿ tag k¿ tag k¿ tag INnUaiaUZ nnn ∈∈∈∈∀∀∀∀−−−−−−−−==== ++++ ,)sin(cos1 . . . . cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa nn ZaiaZ )sin(cos1 ++++====++++ rYcTajrk rYcTajrk rYcTajrk rYcTajrk

nZ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ CaGnuKmn_ n nig nig nig nig a . . . . x¿ Tajrk x¿ Tajrk x¿ Tajrk x¿ Tajrk nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n rYcTajrk rYcTajrk rYcTajrk rYcTajrk n

aU

0lim

→→→→ . . . .


- 136 -

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k¿bgHajfa k¿bgHajfa k¿bgHajfa k¿bgHajfa nn ZaiaZ )sin(cos1 ++++====++++ eyIgman eyIgman eyIgman eyIgman nnn UaiaUZ )sin(cos1 −−−−−−−−==== ++++ eyIgán eyIgán eyIgán eyIgán 121 )sin(cos ++++++++++++ −−−−−−−−==== nnn UaiaUZ

[[[[ ]]]]

n

nn

nn

nn

nnn

Uaia

UaiaUaiaaia

UUaia

UUaia

UaiaUaU

)sin(cos

)sin(cos)sin(cos

)sincos

)(sin(cos

)sin(cos

)sin(coscos2

1

1

1

11

++++====−−−−−−−−++++====++++

−−−−++++====

−−−−++++====−−−−−−−−−−−−====

++++

++++

++++

++++++++

dUcen¼ dUcen¼ dUcen¼ dUcen¼ nn ZaiaZ )sin(cos1 ++++====++++ . . . . KNna KNna KNna KNna nZ CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n nig nig nig nig a ½ ½ ½ ½ eday eday eday eday nn ZaiaZ )sin(cos1 ++++====++++ naM[ naM[ naM[ naM[ )( nZ CasIVútFrNImaRténcMnYnkMupøicEdlmanersugCasIVútFrNImaRténcMnYnkMupøicEdlmanersugCasIVútFrNImaRténcMnYnkMupøicEdlmanersugCasIVútFrNImaRténcMnYnkMupøicEdlmanersug aiaq sincos ++++==== nigtY nigtY nigtY nigtY 1)sin(cos 010 ====−−−−−−−−==== UaiaUZ tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ )sin()cos()sin(cos0 nainaaiaqZZ nn

n ++++====++++====××××==== dUcen¼ dUcen¼ dUcen¼ dUcen¼ )sin(.)cos( nainaZn ++++==== . . . .


- 137 -

x¿Tajrk x¿Tajrk x¿Tajrk x¿Tajrk nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n ½ ½ ½ ½ eyIgman eyIgman eyIgman eyIgman )1()sin(cos1 nnn UaiaUZ −−−−−−−−==== ++++ nig nig nig nig )2()sin(cos1 nnn UaiaUZ ++++−−−−==== ++++ dksmIkar dksmIkar dksmIkar dksmIkar )1( nig nig nig nig )2( Gg:nwgGg:eKán ½Gg:nwgGg:eKán ½Gg:nwgGg:eKán ½Gg:nwgGg:eKán ½

nnn UaiZZ sin2====−−−− naM[ naM[ naM[ naM[ ai

ZZU nn

n sin2−−−−

==== Edl Edl Edl Edl 0sin ≠≠≠≠a . . . . eday eday eday eday )sin()cos( nainaZn ++++==== nig nig nig nig )sin()cos( nainaZn −−−−==== eKán eKán eKán eKán

ana

ainainanaina

Un sin)sin(

sin2)sin()cos()sin()cos( ====

++++−−−−++++====

dUcen¼ dUcen¼ dUcen¼ dUcen¼ a

naUn sin

)sin(==== . . . .

ehIy ehIy ehIy ehIy na

ana

nan

ana

Uaa

na

====××××========→→→→→→→→→→→→ sin)(

)sin(lim

sin)sin(

limlim000

dUcen¼ dUcen¼ dUcen¼ dUcen¼ nUn

a====

→→→→0lim . . . .


- 138 -


223)12

cos()4

sin(4 ++++====++++++++ ππππππππxx


223)12

cos()4

sin(4 ++++====++++++++ ππππππππxx

tamrUbmnþ tamrUbmnþ tamrUbmnþ tamrUbmnþ )sin()sin(cossin2 bababa −−−−++++++++==== smIkarxagelIGacsresrCabnþbnÞabxageRkam ½smIkarxagelIGacsresrCabnþbnÞabxageRkam ½smIkarxagelIGacsresrCabnþbnÞabxageRkam ½smIkarxagelIGacsresrCabnþbnÞabxageRkam ½

22

)3

2sin(

211)3

2sin(2

)21(6

sin)3

2sin(2

223)124

sin()124

sin(2

2

====++++

++++====++++++++

++++====

++++++++

++++====

−−−−−−−−++++++++++++++++++++

ππππ

ππππ

ππππππππ


x

x

x

xxxx

eKTaj eKTaj eKTaj eKTaj ππππππππππππkx 2

432 ++++====++++ b¤b¤b¤b¤ ππππππππππππππππ

kx 243

2 ++++−−−−====++++

dUcen¼ dUcen¼ dUcen¼ dUcen¼ Zkkxkx ∈∈∈∈++++====++++−−−−==== ;245

;24



- 139 -

lMhat´TI63lMhat´TI63lMhat´TI63lMhat´TI63 eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit eK[sIVúténcMnYnBit )( nU kMnt;elI kMnt;elI kMnt;elI kMnt;elI IN eday eday eday eday ³³³³

22

0 ====U nig nig nig nig INnU

U nn ∈∈∈∈∀∀∀∀

−−−−−−−−====++++ ,

2

11 2

1 KNna KNna KNna KNna nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay KNna KNna KNna KNna nU CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én CaGnuKmn_én n eyIgman eyIgman eyIgman eyIgman

4sin

22

0ππππ========U Bit Bit Bit Bit

]bmafavaBitdl;tYTI ]bmafavaBitdl;tYTI ]bmafavaBitdl;tYTI ]bmafavaBitdl;tYTI p KW KW KW KW 22

sin ++++====ppUππππ

eyIgnwgRsayfavaBitdl;tYTI eyIgnwgRsayfavaBitdl;tYTI eyIgnwgRsayfavaBitdl;tYTI eyIgnwgRsayfavaBitdl;tYTI )1( ++++p KW KW KW KW 31

2sin ++++++++ ====

ppUππππ Bit Bit Bit Bit

eyIgman eyIgman eyIgman eyIgman 2

11 2

1p

p

UU

−−−−−−−−====++++

tamkar]bma tamkar]bma tamkar]bma tamkar]bma 22

sin ++++====ppUππππ

eyIg)an eyIg)an eyIg)an eyIg)an 2

2sin11

22

1

++++++++

−−−−−−−−====

p

pU

ππππ

3

32

2

2sin

22

sin2

22

cos1

++++

++++++++========

−−−−====

p

pp ππππππππππππ

Bit Bit Bit Bit


sin ++++====nnUππππ . . . .


- 140 -

lMhat´TI64lMhat´TI64lMhat´TI64lMhat´TI64 eK[eK[eK[eK[

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

====++++++++

====++++

====

4

3

2

2cos2222

2cos222

2cos22

ππππ

ππππ

ππππ

BI]TaBI]TaBI]TaBI]TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg hrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg hrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg hrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBa¢k;rUbmnþenaHpg dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay rkrUbmnþTUeTA ½rkrUbmnþTUeTA ½rkrUbmnþTUeTA ½rkrUbmnþTUeTA ½ eKman eKman eKman eKman

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

====++++++++

====++++

====

4

3

2

2cos2222

2cos222

2cos22

ππππ

ππππ

ππππ

tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½

1)(

2cos22.........222 ++++====++++++++++++++++

nn

ππππ44444 344444 21

. . . .


- 141 -

RsaybBaØak´rUbmnþen¼ ½RsaybBaØak´rUbmnþen¼ ½RsaybBaØak´rUbmnþen¼ ½RsaybBaØak´rUbmnþen¼ ½ eyIgtag eyIgtag eyIgtag eyIgtag

4444 34444 21)(

2......222n

nA ++++++++++++++++====

cMeBa¼RcMeBa¼RcMeBa¼RcMeBa¼RKb Kb Kb Kb *INn ∈∈∈∈ . . . . eyIgman eyIgman eyIgman eyIgman

212

cos22ππππ========A Bit Bit Bit Bit

eyIg«bmafavaBitdltYTI eyIg«bmafavaBitdltYTI eyIg«bmafavaBitdltYTI eyIg«bmafavaBitdltYTI p KW KW KW KW ³³³³

1)(

2cos22......222 ++++====++++++++++++++++====

pp

pAππππ

4444 34444 21 Bit Bit Bit Bit

eyIgnwgRsayfavaBitdltYTI eyIgnwgRsayfavaBitdltYTI eyIgnwgRsayfavaBitdltYTI eyIgnwgRsayfavaBitdltYTI 1++++p KW KW KW KW 21

2cos2 ++++++++ ====

ppAππππ Bit Bit Bit Bit

eyIgman eyIgman eyIgman eyIgman pp AA ++++====++++ 21 edaytamkar«bma edaytamkar«bma edaytamkar«bma edaytamkar«bma

12cos2 ++++====

ppAππππ

eyIgáneyIgáneyIgáneyIgán22

211

2cos2

2cos4

2cos22 ++++++++++++++++ ========++++====

ppppAππππππππππππ Bit Bit Bit Bit


)(2

cos22.........222 ++++====++++++++++++++++n

n

ππππ44444 344444 21

. . . .


- 142 -

lMhat´TI65lMhat´TI65lMhat´TI65lMhat´TI65 ykykykyk a; b ;c CaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nigCaRbEvgRCug nig A,B,C CargVas;mMuTaMgbIénRtIekaN CargVas;mMuTaMgbIénRtIekaN CargVas;mMuTaMgbIénRtIekaN CargVas;mMuTaMgbIénRtIekaN ABCmYyEdl mYyEdl mYyEdl mYyEdl S CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH. CaRklaépÞRtIekaNenaH.

Rsayfa Rsayfa Rsayfa Rsayfa S

cbaCotCCotBCotA

4

222 ++++++++====++++++++ dMdMdMdMeNa¼eNa¼eNa¼eNa¼RsRsRsRsayayayay tamRTWsþIbTsIunus tamRTWsþIbTsIunus tamRTWsþIbTsIunus tamRTWsþIbTsIunus

(((( ))))(((( ))))(((( ))))3cot4

2cot4

1cot4

sincos

)sin21

(4

cos2

222

222

222

22

222

gCSabc

gBScab

gAScba

AA

Abccb

Abcba

−−−−++++====

−−−−++++====

−−−−++++====

−−−−++++====

−−−−++++====

bUk bUk bUk bUk (1),(2),(3) eyIg)an eyIg)an eyIg)an eyIg)an

Scba

gCgBgA

gCgBgAScbacba

4cotcotcot

)cotcot(cot4)(2222

222222

++++++++====++++++++

++++++++−−−−++++++++====++++++++

dUcenH dUcenH dUcenH dUcenH S

cbagCggA

4cotBcotcot

222 ++++++++====++++++++


- 143 -

lMhat´TI66lMhat´TI66lMhat´TI66lMhat´TI66 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCug mYymanRCug mYymanRCug mYymanRCug cABbACaBC ============ ,, cUrRsaybBa¢k;facUrRsaybBa¢k;facUrRsaybBa¢k;facUrRsaybBa¢k;fa

abccba

cC

bB

aA

2coscoscos 222 ++++++++====++++++++

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman eKman eKman eKman ³³³³

Abccba cos2222 −−−−++++==== smmUl smmUl smmUl smmUl bc

acbA

2cos

222 −−−−++++====

Baccab cos2222 −−−−++++==== smmUl smmUl smmUl smmUl ac

bcaB

2cos

222 −−−−++++====

Cabbac cos2222 −−−−++++==== smmUl smmUl smmUl smmUl ab

cbaC

2cos

222 −−−−++++==== tag tag tag tag

cC

bB

aA

Mcoscoscos ++++++++====

abccba

abccbabcaacb

abccba

abcbca

abcacb

2

2

222

222

222222222

222222222

++++++++====

−−−−++++++++−−−−++++++++−−−−++++====

−−−−++++++++−−−−++++++++

−−−−++++====

dUcenH dUcenH dUcenH dUcenH abc

cbac

Cb

Ba

A2

coscoscos 222 ++++++++====++++++++ . . . .


- 144 -

lMhat´TI67lMhat´TI67lMhat´TI67lMhat´TI67 eeeedaHRsaysmIkardaHRsaysmIkardaHRsaysmIkardaHRsaysmIkar k> k> k> k> (((( )))) 0)sin2(logsinlog2 3

22

2 ====++++ xx


2

22 ====++++++++

xx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkaredaHRsaysmIkar k> k> k> k> (((( )))) 0)sin2(logsinlog2 3

22

2 ====++++ xx lkç½xNÐ½ lkç½xNÐ½ lkç½xNÐ½ lkç½xNÐ½ 0sin >>>>x smIkarGacsresmIkarGacsresmIkarGacsresmIkarGacsresr sr sr sr ³³³³

(((( ))))(((( )))) 01sinlog3sinlog2

02log)(sinlogsinlog2

22

2

23

22

2

====++++++++

====++++++++

xx

xx

tag tag tag tag xX sinlog2==== 0132 2 ====++++++++ XX manb¤s manb¤s manb¤s manb¤s

21

;1 21 −−−−====−−−−==== XX

----cMeBaH cMeBaH cMeBaH cMeBaH 1−−−−====X eK)an eK)an eK)an eK)an 1sinlog2 −−−−====x b¤ b¤ b¤ b¤ 21

sin ====x

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ππππππππkx 2

6++++==== b¤ b¤ b¤ b¤ )(2

65

Zkkx ∈∈∈∈++++==== ππππππππ


- 145 -

----cMeBaH cMeBaH cMeBaH cMeBaH 21−−−−====X eK)an eK)an eK)an eK)an

21

sinlog2 −−−−====x b¤ b¤ b¤ b¤ 22

sin ====x

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ππππππππkx 2

4++++==== b¤ b¤ b¤ b¤ )(2

43

Zkkx ∈∈∈∈++++==== ππππππππ


2

22 ====++++++++

xx

lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð lkç½xN½Ð 0cos >>>>x tag tag tag tag xt coslog

22==== enaH enaH enaH enaH tx −−−−====coslog 2

eK)an eK)an eK)an eK)an 0232 ====++++−−−− tt manb¤s manb¤s manb¤s manb¤s 2;1 21 ======== tt ----cMeBaH cMeBaH cMeBaH cMeBaH 11 ====t eK)an eK)an eK)an eK)an 1coslog

22 ====x b¤ b¤ b¤ b¤

22

cos ====x

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s )(24

Zkkx ∈∈∈∈++++±±±±==== ππππππππ

----cMeBaH cMeBaH cMeBaH cMeBaH 22 ====t eK)an eK)an eK)an eK)an 2coslog22 ====x b¤ b¤ b¤ b¤

21

cos ====x

eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s )(23

Zkkx ∈∈∈∈++++±±±±==== ππππππππ


- 146 -

lMhat´TI68lMhat´TI68lMhat´TI68lMhat´TI68 edaHRsayedaHRsayedaHRsayedaHRsayRbB½næRbB½næRbB½næRbB½næsmIkarsmIkarsmIkarsmIkar

(((( )))) (((( ))))(((( ))))

====

====yx

yx

sinlnsinln

2ln2ln

22

sin2sin2

Edl Edl Edl Edl 2

0ππππ<<<<<<<< x nig nig nig nig

20

ππππ<<<<<<<< y . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar

(((( )))) (((( ))))(((( ))))

====

====

)(22

)(sin2sin2

sinlnsinln

2ln2ln

ii

iyx

yx

smIkasmIkasmIkasmIkar r r r )(i Gacsresr Gacsresr Gacsresr Gacsresr ³³³³ (((( )))) (((( ))))

)(2ln43

sinln21

sinln

sinln21

2ln41

sinln2ln

)sinln2ln21

(2ln21

)sinln2(ln2ln

sin2lnsin2ln2ln2ln

ayx

yx

yx

yx

−−−−====−−−−

++++====++++

++++====++++

====


- 147 -

smIkar smIkar smIkar smIkar )(ii Gacsresr Gacsresr Gacsresr Gacsresr ³³³³ (((( )))) (((( ))))

)(sinln2sinln

2lnsinln2lnsinln21

2lnsinln2lnsinln

2ln2ln sinlnsinln

byx

yx

yx

yx

====

====

====

====

yksmIkar yksmIkar yksmIkar yksmIkar )(b CYkñúg CYkñúg CYkñúg CYkñúg )(a eK)an eK)an eK)an eK)an ³³³³

====

−−−−====

−−−−====

−−−−====−−−−

22

lnsinln

2ln21

sinln

2ln43

sinln23

2ln43

sinln21

sinln2

y

y

y

yy


sin ====y na na na naM[ M[ M[ M[ 4ππππ====y eRBaH eRBaH eRBaH eRBaH

20

ππππ<<<<<<<< y

tam tam tam tam )(b eKTaj eKTaj eKTaj eKTaj

====

22

ln2sinln x


sin ====x naM[ naM[ naM[ naM[ 6ππππ====x eRBaH eRBaH eRBaH eRBaH

20

ππππ<<<<<<<< x

dUcenH dUcenH dUcenH dUcenH 6ππππ====x nig nig nig nig

4ππππ====y . . . .


- 148 -

lMhat´TI69lMhat´TI69lMhat´TI69lMhat´TI69 edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar

++++====++++

====++++

2222

23

sinsin

sinsin yx

yx

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay edaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkaredaHRsayRbB½næsmIkar

++++====++++

====++++

)(2222

)(23

sinsin

sinsin ii

iyx

yx

tam tam tam tam )(i eKTaj eKTaj eKTaj eKTaj )(sin23

sin iiixy −−−−==== yk yk yk yk )(iii CYskñúg CYskñúg CYskñúg CYskñúg )(ii eK)an eK)an eK)an eK)an ³³³³

222.222

2222sinsin

sin23

sin

++++====++++

++++====++++−−−−

−−−−

xx

xx

tag tag tag tag 02sin >>>>==== xt eK)an eK)an eK)an eK)an 22

122 ++++====++++

tt

b¤ b¤ b¤ b¤ 022)22(2 ====++++++++−−−− tt manb¤s manb¤s manb¤s manb¤s 2;2 21 ======== tt


- 149 -

----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 22sin ====x naM[ naM[ naM[ naM[ 21

sin ====x eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s ππππππππ

kx 26

++++==== b¤ b¤ b¤ b¤ )(26

5Zkkx ∈∈∈∈++++==== ππππππππ

ehIytam ehIytam ehIytam ehIytam )(iii eK)an eK)an eK)an eK)an 121

23

sin ====−−−−====y

eKTaj eKTaj eKTaj eKTaj Zkky ∈∈∈∈++++==== ;22

ππππππππ ----cMeBaH cMeBaH cMeBaH cMeBaH 2====t eK)an eK)an eK)an eK)an 22sin ====x naM[ naM[ naM[ naM[ 1sin ====x eKTajb¤s eKTajb¤s eKTajb¤s eKTajb¤s )(2

2Zkkx ∈∈∈∈++++==== ππππππππ

ehIytam ehIytam ehIytam ehIytam )(iii eK)an eK)an eK)an eK)an 21

23

sin ====−−−−====y

eKTaj eKTaj eKTaj eKTaj Zkkyky ∈∈∈∈++++====++++==== ;26

5;2

6ππππππππππππππππ . . . .


- 150 -

lMhat´TI70lMhat´TI70lMhat´TI70lMhat´TI70 eK[ eK[ eK[ eK[

2a0

ππππ<<<<<<<< nig nig nig nig 2

b0ππππ<<<<<<<< . . . .

cUrbgHajfa cUrbgHajfa cUrbgHajfa cUrbgHajfa 1bcosacos

bsinasin

2222

====

++++

lu¼RtaEt lu¼RtaEt lu¼RtaEt lu¼RtaEt ba ==== . . . . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay kkkkarbgHajarbgHajarbgHajarbgHaj

eKman eKman eKman eKman 1bcosacos

bsinasin

2222

====

++++

smmUl smmUl smmUl smmUl 1bcosacos

bsinasin

)bcosb(sin 2

4

2

422 ====

++++++++

0)acosbcosbsin

asinbsinbcos

1acosbcosbsin

asinbsinbcos

acosasin21

1acosbcosbsin

asinbsinbcos

acosasin

222

42

24

2

222

42

24

2

244

====

−−−−

====++++++++−−−−

====++++++++++++

eKTaj eKTaj eKTaj eKTaj acosbcosbsin

asinbsinbcos 22 ====


- 151 -

smmUl smmUl smmUl smmUl bcosbsin

acosasin

2

2

2

2

==== smmUl smmUl smmUl smmUl btanatan 22 ==== eday eday eday eday

2a0

ππππ<<<<<<<< nig nig nig nig 2

b0ππππ<<<<<<<<

ena¼eKTena¼eKTena¼eKTena¼eKTaj aj aj aj ba ==== . . . .


- 152 -

lMhat´TI71lMhat´TI71lMhat´TI71lMhat´TI71 eK[ eK[ eK[ eK[ ABC CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ

AcosCsinBsin1CsinBsin 22 ++++====++++ . . . . bgHajfa bgHajfa bgHajfa bgHajfa ABC CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay bgHajfa bgHajfa bgHajfa bgHajfa ABC CaRtIekaNEkg CaRtIekaNEkg CaRtIekaNEkg CaRtIekaNEkg tamRTwsþIbTsIutamRTwsþIbTsIutamRTwsþIbTsIutamRTwsþIbTsIunUseKman nUseKman nUseKman nUseKman R2

Csinc

Bsinb

Asina ============

eKTaj eKTaj eKTaj eKTaj CsinR2c,BsinR2b,AsinR2a ============ eday eday eday eday Acosbc2cba 222 −−−−++++==== ( RTwsþIbTkUsIunUs ) ( RTwsþIbTkUsIunUs ) ( RTwsþIbTkUsIunUs ) ( RTwsþIbTkUsIunUs ) eKTaján ½eKTaján ½eKTaján ½eKTaján ½

)AcosCsinBsin2CsinB(sinR4AsinR4 22222 −−−−++++==== )1(AcosCsinBsin2CsinBsinAsin 222 −−−−++++==== Et Et Et Et )2(AcosCsinBsin1CsinBsin 22 ++++====++++ yksmIkar yksmIkar yksmIkar yksmIkar )2( CYsenAkñúg CYsenAkñúg CYsenAkñúg CYsenAkñúg )1( eKTaj eKTaj eKTaj eKTaj 1Asin2 ==== naM[ naM[ naM[ naM[ 090A ==== . dUcen¼ . dUcen¼ . dUcen¼ . dUcen¼ ABC CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg . CaRtIekaNEkg .


- 153 -

lMhat´TI72lMhat´TI72lMhat´TI72lMhat´TI72 eK[RtIekaN eK[RtIekaN eK[RtIekaN eK[RtIekaN ABC mYymanRCúg mYymanRCúg mYymanRCúg mYymanRCúg c,b,a . . . . kMnt´RbePTénRtIekaN kMnt´RbePTénRtIekaN kMnt´RbePTénRtIekaN kMnt´RbePTénRtIekaN ABC ebIeK ebIeK ebIeK ebIeKdwgfa ½dwgfa ½dwgfa ½dwgfa ½ )

c1

b1

a1

(21

cCcos

bBcos

aAcos ++++++++====++++++++

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay RbePTénRtIekaN RbePTénRtIekaN RbePTénRtIekaN RbePTénRtIekaN ABC tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman ½ eKman ½ eKman ½ eKman ½

bc2acb

Acos222 −−−−++++==== naM[ naM[ naM[ naM[ )1(

abc2acb

aAcos 222 −−−−++++====

dUcKñaEdreKTaj dUcKñaEdreKTaj dUcKñaEdreKTaj dUcKñaEdreKTaj )2(abc2

bcab

Bcos 222 −−−−++++====

nig nig nig nig )3(abc2

cbac

Ccos 222 −−−−++++==== bUkTMnak´TMng bUkTMnak´TMng bUkTMnak´TMng bUkTMnak´TMng )3(,)2(,)1( Gg:nwgGg:eKán ½ Gg:nwgGg:eKán ½ Gg:nwgGg:eKán ½ Gg:nwgGg:eKán ½

abc2cba

cCcos

bBcos

aAcos 222 ++++++++====++++++++

eday eday eday eday )c1

b1

a1

(21

cCcos

bBcos

aAcos ++++++++====++++++++

eKTaján ½eKTaján ½eKTaján ½eKTaján ½


- 154 -

0)ac()cb()ba(

0)aac2c()cbc2b()bab2a(

ac2bc2ab2c2b2a2

acbcabcba

abc2abcabc

abc2cba

)c1

b1

a1

(21

abc2cba

222

222222

222

222

222

222

====−−−−++++−−−−++++−−−−

====++++−−−−++++++++−−−−++++++++−−−−

++++++++====++++++++

++++++++====++++++++

++++++++====++++++++

++++++++====++++++++

eKTajánsmPaB eKTajánsmPaB eKTajánsmPaB eKTajánsmPaB cba ======== . . . . dUcen¼ dUcen¼ dUcen¼ dUcen¼ ABC CaRtIekaNsmg&ß . CaRtIekaNsmg&ß . CaRtIekaNsmg&ß . CaRtIekaNsmg&ß .


- 155 -

lMhat´TI73lMhat´TI73lMhat´TI73lMhat´TI73 kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN kñúgRtIekaN ABC mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½ mYycUrRsaybBa¢ak´fa ½

rR

4

2C

sin

1

2B

sin

1

2A

sin

1 ≥≥≥≥++++++++

Edl Edl Edl Edl r nig nig nig nig R CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . CakaMrgVgćarwkkñúg nig carwkeRkARtIekaN . dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay Rsayfa Rsayfa Rsayfa Rsayfa (((( ))))1

rR

4

2C

sin

1

2B

sin

1

2A

sin

1 ≥≥≥≥++++++++

tag tag tag tag cAB,bAC,aBC ============ tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman ½ eKman ½ eKman ½ eKman ½

Acos.bc2cba 222 −−−−++++==== eday eday eday eday 2A

sin21Acos 2−−−−==== ena¼ ena¼ ena¼ ena¼ )

2A

sin21(bc2cba 2222 −−−−−−−−++++====

eKTaj eKTaj eKTaj eKTaj bc4

)cba)(cba(bc4

)cb(a2A

sin22

2 ++++−−−−−−−−++++====−−−−−−−−====

tag tag tag tag 2

cbap

++++++++==== ( knø¼brimaRténRtIekaN ) ( knø¼brimaRténRtIekaN ) ( knø¼brimaRténRtIekaN ) ( knø¼brimaRténRtIekaN ) eKán eKán eKán eKán )cp(2cba −−−−====−−−−++++ nig nig nig nig )bp(2cba −−−−====++++−−−− eKán eKán eKán eKán

bc)cp)(bp(

2A

sin2 −−−−−−−−====


- 156 -

naM[ naM[ naM[ naM[ bc

)cp)(bp(2A

sin−−−−−−−−==== . dUcKñaEdreKTaj ½ . dUcKñaEdreKTaj ½ . dUcKñaEdreKTaj ½ . dUcKñaEdreKTaj ½

ab)bp)(ap(

2C

sin;ac

)cp)(ap(2B

sin−−−−−−−−====−−−−−−−−====

eKán eKán eKán eKán abc

)cp)(bp)(ap(2C

sin2B

sin2A

sin−−−−−−−−−−−−====

eKman eKman eKman eKman R4

abcpr)cp)(bp)(ap(pS ========−−−−−−−−−−−−====

eKTaj eKTaj eKTaj eKTaj S.R4abc ==== nig nig nig nig S.rpS

)cp)(bp)(ap(2

========−−−−−−−−−−−−

eKán eKán eKán eKán R4r

S.R4S.r

2C

sin2B

sin2A

sin ======== . . . . vismPaB vismPaB vismPaB vismPaB (((( ))))1 smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½

(((( ))))22

2C

sin

2B

sin2A

sin

2B

sin

2A

sin2C

sin

2A

sin

2C

sin2B

sin

2C

sin2B

sin2A

sin4

14

2C

sin

1

2B

sin

1

2A

sin

1

≥≥≥≥++++++++

≥≥≥≥++++++++

eday eday eday eday 2A

sina

apbca

)cp)(bp()ap(2C

sin2B

sin 2

2 −−−−====−−−−−−−−−−−−====


- 157 -

eKTaj eKTaj eKTaj eKTaj a

ap

2A

sin

2C

sin2B

sin −−−−==== . dUcKñaEdreKTaján ½ . dUcKñaEdreKTaján ½ . dUcKñaEdreKTaján ½ . dUcKñaEdreKTaján ½

bbp

2B

sin

2A

sin2C

sin −−−−==== nig nig nig nig c

cp

2C

sin

2B

sin2A

sin −−−−====

vismPaB vismPaB vismPaB vismPaB (((( ))))2 smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½ smmUleTAnwg ½ 2

ccp

bbp

aap ≥≥≥≥−−−−++++−−−−++++−−−−

tamvismPaB tamvismPaB tamvismPaB tamvismPaB GMAM −−−− eKán ½ eKán ½ eKán ½ eKán ½ a)ap(2a)ap(p −−−−≥≥≥≥++++−−−−==== naM[ naM[ naM[ naM[

p)ap(2

aap −−−−≥≥≥≥−−−−

dUcKñaEdr dUcKñaEdr dUcKñaEdr dUcKñaEdr p

)bp(2b

bp −−−−≥≥≥≥−−−− nig nig nig nig

p)cp(2

ccp −−−−≥≥≥≥

−−−− eKán eKán eKán eKán

Bit2c

cpb

bpa

ap

p)cp()bp()ap(

2c

cpb

bpa

ap

≥≥≥≥−−−−++++−−−−++++−−−−

−−−−++++−−−−++++−−−−≥≥≥≥−−−−++++−−−−++++−−−−

dUcen¼dUcen¼dUcen¼dUcen¼ rR

4

2C

sin

1

2B

sin

1

2A

sin

1 ≥≥≥≥++++++++ . . . .


- 158 -

lMhat´TI74lMhat´TI74lMhat´TI74lMhat´TI74 k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa k> cUrRsayfa

aaa

a

2sin

1

sin2

1

2sin

2cos222 −−−−====


====n

kk

k

kn x

x

S0 2

2sin

2cos

.2

1

dMeNa¼dMeNa¼dMeNa¼dMeNa¼RsRsRsRsayayayay k> Rsayfa k> Rsayfa k> Rsayfa k> Rsayfa

aaa

a

2sin

1

sin2

1

2sin

2cos222 −−−−====

eKman eKman eKman eKman 1cos22cos 2 −−−−==== aa nig nig nig nig aaa cossin22sin ==== eK)an eK)an eK)an eK)an

aa

a

a

a22

2

2 cossin4

1cos2

2sin

2cos −−−−====

aa

aaaa

a

2sin

1

sin2

1cossin4

1

cossin4

cos2

22

2222

2

−−−−====

−−−−====

dUcenH dUcenH dUcenH dUcenH aaa

a

2sin

1

sin2

1

2sin

2cos222 −−−−====


====n

kk

k

kn x

x

S0 2

2sin

2cos

.2

1


- 159 -

eKman eKman eKman eKman aaa

a

2sin

1

sin2

1

2sin

2cos222 −−−−====

yk yk yk yk 12 ++++==== kx

a eK)an eK)an eK)an eK)an ³³³³

kkk

k

xxx

x

2sin

1

2sin2

1

2sin

2cos

21

22−−−−====

++++

KuNGgÁTaMgBIr nwg KuNGgÁTaMgBIr nwg KuNGgÁTaMgBIr nwg KuNGgÁTaMgBIr nwg k2

1 eK)an eK)an eK)an eK)an ³³³³

kk

kk

k

k

k xxx

x

2sin2

1

2sin2

1

2sin

2cos

.2

12

1212

−−−−====

++++++++

xx

xxS

nn

n

kk

kk

kn

sin1

2sin2

12

sin2

1

2sin2

1

11

0 21

21

−−−−====

−−−−====

++++++++

====++++

++++∑∑∑∑

dUcenH dUcenH dUcenH dUcenH xx

S

nn

n sin1

2sin2

1

11

−−−−====

++++++++

. . . .

lwm pl:ún nig Esn Bisidæ - WordPress.com · Mathematical Olympiad Treasures ( Titu Andreescu and...

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