n 2 x n k x 2 k k 1...knitvitüacmuvijbipbelak - 3 - !@> ekdwgfa cosa , cosb , coscmn p np pm m n...
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knkk
nk k
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i
KN³kmμkarniBnæ nig eroberog
lwm plÁún nig Esn Bisidæ
KN³kmμkarRtYtBinitübec©keTs
elak lwm qun elak Gwug sMNag elak nn; suxNa elakRsI Tuy rINa elak RBwm sunitü elak pl b‘unqay elak Titü em:g
KN³kmμkarRtYtBinitüGkçraviruTæ elak lwm miKÁsir
karIkMuBüÚTr½ kBaØa lI KuNÑaka elak Gwug sMNag
ii
Garmáfa esoePAKNitviTüaCMuvijBiPBelakPaKTI7 EdlGñksikSakMBugkan; enAkñúgédenH EckecjCabIEpñkEdlEpñkTI1 CakRmglMhat;eRCIserIs EpñkTI2 CaEpñkdMeNaHRsay nig EpñkTI3 CalMhat;Gnuvtþn_ . ral;bRbFanlMhat;nImYy²enAkñúgesovePAenH eyIgxJúM)aneRCIserIsykEt lMhat; NaEdlmanlkçN³Bi)ak mkeFVIdMeNaHRsayKMrUy:agek,aHk,aybMput . eKalbMNgénkareroberogcgRkgKWedIm,ITukCaÉksarCMnYysRmab;Gñk sikSakñúgRKb;mCÄdæan nig müa:geTotedIm,IcUlrYmelIksÞÜyvisy½KNitviTüakñúg RbeTskm<úCaeyIg[kan;EtrIkceRmInqab;rhs½Rsbtamsm½yviTüasaRsþTMenIb. esovePAenHminl¥hYseKhYsÉgenaHeT . kMhusqÁgedayGectnaR)akd CamanTaMgbec©keTs nig GkçraviruTæ . ehtuenHeyIgxJMúCaGñkniBnæ rgcaMCanic©nUv mtiriHKn;EbbsßabnaBIGñksikSaRKb;mCÄdæanedaykþIrIkrayedIm,IEklMGesovenH [kan;EtmansuRkitüPaBRbesIreLIgEfmeTot . CaTIbBa©b; xJMú)aTGñkniBnæsUmeKarBCUnBrGñksikSaTaMgGs;mansuxPaBl¥ manR)aCJaQøasév nig TTYl)aneCaKC½yCanic©kñúgkarsikSa .
)at;dMbg éf¶TI 25 ExFñÚ qñaM2010 GñkniBnæ nig RsayRCav lwm plÁún Tel : 017 768 246 Email: lim_phalkun@ymail.com Website: www.mathtoday.wordpress.com
KNitviTüaCMuvijBiPBelak
- 1 -
kRmglMhat;
!> cMeBaHRKb; IRx cUrRsayBa¢ak;fa ³ 22|sincos| baxbxa @> cMeBaHRKb;cMnYnBit xcUrRsaybBa¢ak;fa
bab
xa
x
1cossin 44
)0,0(; ba #> eK[GnuKmn_ 9)cot(tan12)cot(tan4)( 22 xxxxxf Edl )
2,0(
x . cUrrktémøtUcbMputénGnuKmn_ )(xf . $> eK[GnuKmn_
x
xy
sin45
cos3
cUrrktémøtUcbMput nig témøFMbMputénGnuKmn_ y . %> eK[GnuKmn_ 89)2(1243),( 2222 yxxyyxyxyxf cUrkMNt; x nig y edIm,I[GnuKmn_ ),( yxf mantémøtUcbMput
KNitviTüaCMuvijBiPBelak
- 2 -
rYckMNt;témøtUcbMputenaH . ^> eKmansmIkar 032 xx manb¤stageday 1x nig 2x cUrKNnatémø 5
23
1 419 xxA &> eK[GnuKmn_ cbxaxxf 2)( Edl IRcbaa ,,,0 . cUrRsayfaebIsmIkar xxf )( Kμanb¤skñúgsMNMucMnYnBitenaH smIkar xxff )]([ k¾Kμanb¤skñúgsMNMucMnYnBitEdr . *> edaHRsaysmIkar ³ xxxxx 126324 232 (> edaHRsaysmIkar ³
33
3 23 2
1952
)13(4)13(10
x
xx
!0> eK[smIkar 099999 xx cUrbgðajfasmIkarenHmanb¤s 0x CacMnYnBitEtmYyKt; rYcRsayfa
0x CacMnYnGsniTan . !!> eKdwgfa b
tan xa
.
cUrRsayfa 4 4cos x sin x 1
a b a b
KNitviTüaCMuvijBiPBelak
- 3 -
!@> eKdwgfa m n pcosa , cosb , cosc
n p p m m n
cUrKNnakenSam ³ 2 2 2
2 2 2
sin a sin b sin cM
2 2cosa sin a 2 2cosb sin b 2 2cosc sin c
!#> eKmanRtIekaN ABC mYyEdl BC a , AC b , AB c . tag R nig S erogKñaCakaM nig épÞRkLaénRtIekaN ABC enH . cUrRsayfa ³
2 2 2cos A cosB cosC R(sin A sin B sin C)
a b c 4S
!$> kñúgRKb;RtIekaN ABCcUrRsayfa ³
2(sin A sinB sinC)(1 cos A)(1 cosB)(1 cosC)
2
!%> eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa
2a1 cos A
2bc
Edl a , b , c CaRCugRtIekaN ABC . x> TajbBa¢ak;fa 1
(1 cos A)(1 cosB)(1 cosC)8
!^> eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa 2 2 2sin A sin B sin C 2sinBsinCcos A x> TajbBa¢ak;fa
2 2 2sin A sin B sin Ccot A cot B cot C
2sin AsinBsinC
KNitviTüaCMuvijBiPBelak
- 4 -
!&> eK[RtIekaN ABC mYymanmMu A ,B,C CamMuRsYcEdlepÞógpÞat; smPaB 1 1 1 1
cot A cot B cot C2 sin A sinB sinC
.
cUrRsayfa ABC CaRtIekaNsmgS½ ? !*> tag R CakaMrgVg;carwkkñúg nig S CaépÞRkLaénRtIekaN ABC mYy . k> cUrRsayfa
22R(cot A cot B)(cot B cotC)(cotC cot A)
S
x> ebI ABC CamMuRsYcenaHcUrTaj[)anfa ³ 1cos AcosBcosC
8 .
!(> eK[RtIekaN ABC mYymanRCug BC a , AC b , AB c carwkkñúgrgVg;mYymanp©it O nig kaM R . tag S nig OBCS CaépÞRkLaén OBC nig ABC erogKña . snμtfa A ,B,C CamMuRsYc . k> cUrRsayfa 2
OBC
1S a cot A
4 .
x> cUrTajbgðajfa 2 2 2a cot A b cot B c cot C 4S . K> cUrTajbgðajfa
2
abcacos A bcosB ccosC
2R .
KNitviTüaCMuvijBiPBelak
- 5 -
@0> eK[RtIekaN ABC mYy . D CacMNucmYYénRCug [BC] Edl BAD nig DAC . cUrRsaybBa¢ak;fa BC sin sin
AD sinB sinC
?
@!> bgðajfaebI c,b,a tagerogKñaCatYTI r,q,p énsIVútnBVnþmYyenaHeK)ansmPaB 0c)qp(b)pr(a)rq( . @@> eK[ctuekaNe)a:g ABCD mYymanépÞRkLaesμI 1 . cUrRsaybBa¢ak;fa ³ AB BC CD DA AC BD 4 8 ? ¬Austria-Poland, 1985¦ @#> eK[RtIekaNBIrmanRCug a ,b ,c knøHbrimaRt a b c
p2
nigkaMrgVg;carwkeRkA R . cUrRsaybBa¢ak;fa 2 pR 1
5 2aR bc 2
KNitviTüaCMuvijBiPBelak
- 6 -
@$> eKyk I nig O Cap©itrgVg;carwkkñúg nig carwkeRkAén ABC . rgVg; A carwkkñúgmMu A vab:HeTAnwg AB , AC nig BC erogKña Rtg;cMnuc K ,M nig N erogKña . ebIsinCacMnuckNþal P énGgát; KM sßitenAelIrgVg;carwkeRkA énRtIekaN ABC enaHcUrbgðajfabIcMnuc O , I , N rt;Rtg;Kña ? @%> cUrRsayfakaerénRKb;cMnYnKt;RtUvmanTRmg; k4 b¤ 14 k RKb; *INk . @^> cUrRsayfaebI 22 ba Eckdac;nwg 3 enaH a Eckdac;nwg 3 nig b Eckdac;nwg 3 . @&> cUrbgðajfa 44 n CacMnYnbzmEtkñugkrNI 1n . @*> eKdwgfa 0160321002004008 manktþabzm 250000p . cUrkMNt;ktþabzmenaH .
@(> cUrkMNt;EpñkKt;én
610
1
1
k k ?
#0> cUrkMNt;KUmYyéncMnYnKt;viC¢man ba , edaydwgfa ³ )(:)( baabi Eckmindac;nwg 7
777)(:)( babaii Eckdac;nwg 77 (IMO 1984 )
KNitviTüaCMuvijBiPBelak
- 7 -
#!> eKGnuKmn_ xxf 2121
1)(
cUrKNna )1
(...)2
()1
(n
nf
nf
nfS
#@> cUrRsaybBa¢ak;facMnYn ³ n n n nA 113 168 141 427 Eckdac;nwg 7 Canic©RKb;cMnYnKt;FmμCati n. ##> eK[ m nig n CaBIrcMnYnKt;viC¢manEdl 1 m n . cUrkMNt;témøtUcbMputén m n edIm,I[cMnYn m22 nig n22 manelxBIrxÞg;cugeRkaydUcKña . #$> eK[ 2f (n) 4n 7n 34 Edl n IN * cUrbgðajfaKμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT. #%>eK[ 2f (a) 16a 50a 13 Edl a kMNt;RKb;témø a EdleFIV[cMnYn f (a) Eckdac;nwg 49 #^> kMNt;RKb;cMnYnKt;viC¢man x edIm,I[ 3 2x 10x 115x 237 Eckdac;nwg 343 . #&> cUrRsayfaRKb;cMnYnKt;viC¢man n eK)an
237
134)(
n
nnf
CaRbPaKsRmYlmin)an ?
KNitviTüaCMuvijBiPBelak
- 8 -
#*> eKdwgfa ab
tan3 . cUrRsaybBa¢ak;fa ³
3 23 23 2
4
3 2
4
ba
1
b
sin
a
cos
#(> eKmansIVútcMnYnBit )a( n kMNt;eday ³ )
2
11....()
4
11)(
2
11(a n2n
cUrKNnalImIt nnalim
.
$0> eK[plbUk
222222n )1n(
1
n
11....
4
1
3
11
3
1
2
11
n
1S
cUrKNna nnSlim
.
$!> eKmanes‘rIGnnþmYy ....2
n.....
2
3
2
2
2
1)
2
n( n
2
3
2
2
22
1nn
2
cUrRsayfaes‘rIxagelIenHCaes‘rIbRgYm . $@> cMeBaHRKb; *INn eKmanplbUk ³ 1n432n 2).1n2)(1n2(
3n2....
2.7.5
9
2.5.3
7
2.3.1
5S
k> kMNt;cMnYnBit a nig b edIm,I[eK)ansmPaB 1nn1n 2).1n2(
b
2).1n2(
a
2).1n2)(1n2(
3n2
x> KNna nS nig nnSlim
.
KNitviTüaCMuvijBiPBelak
- 9 -
$#> cMeBaHRKb; *INn eKmansIVútcMnYnBit )a( n kMNt;eday ³ 3a0 nig n2nn21n
3
1a
3
11a
k> KNna na CaGnuKmn_én n . x> KNnalImIt nn
alim
. $$> eK[ a nig b CaBIrcMnYnBitEdl
2ba0
cUrRsayfa bcos
abatanbtan
acos
ab22
$%> KNnaplbUk ³ n
n23
n2n
1nn Cn.....C9C4CS Edl
)!pn(!p
!nCp
n
$^> eK[GnuKmn_ 2x)x(f k> sikSaGefrPaB nig sg;Rkab )c( énGnuKmn_ f . x> eKBinitü n cMnuc n321 A...,,A,A,A sßitenAelIRkab )c( Edl )22,2(A1 nig )a,a(A 1nnn RKb; *INn . cUrkMNt;kUGredaenéncMnuc nA . K> cUrKNna nOA CaGnuKmn_én n rYcKNna n
nOAlim
.
KNitviTüaCMuvijBiPBelak
- 10 -
$&> eK[RtIekaN ABC mYy . emdüanénRtIekaNEdlKUsecj BIkMBUl C,B,A kat;rgVg;carwkeRkARtIekaNerogKñaRtg; P,N,M . tag cba ,, CaRCug nig cba mmm ,, Caemdüanén ABC . cUrRsayfa
9
16
222 224
4
224
4
224
4
bac
m
cab
m
cba
m cba $*> eK[ M CaTIRbCMuTm¶n;énRtIekaN ABC . ebIbnÞat; AB b:HeTAnwgrgVg;carwkeRkAénRtIekaN AMC enaHbgðajfa 32
sin
1
sin
1
CBMCMA
$(> eK[ 0,, cba . cUrRsaybBa¢ak;fa ³
2
9)(4)(4)(4 3 333 333 33
ba
bac
ac
acb
cb
cba %0> eK[ 1,1 ba nig 1x . cUrRsayfa xxx baba
2
log2loglog
KNitviTüaCMuvijBiPBelak
- 11 -
KNitviTüaCMuvijBiPBelak
- 12 -
lMhat;TI1 cMeBaHRKb; IRx cUrRsayBa¢ak;fa ³
22|sincos| baxbxa dMeNaHRsay RsayBa¢ak;fa ³
22|sincos| baxbxa
tagviucTr½ ),( bau
nig )sin,(cos xxv
yk CamMurvagviucTr½ u nig
v .
tamniymn½yplKuNsáaEl ³
cos||.||.
vuvu
eKTaj ||.||
.cos
vu
vu
eKman xbxavu sincos.
1||;|| 22 vbau
KNitviTüaCMuvijBiPBelak
- 13 -
eK)an 22
sincoscos
ba
xbxa
cMeBaHRKb;mMu 1cos1: IR b¤ 1|cos|
eK)an 1sincos22
ba
xbxa
dUcenH 22|sincos| baxbxa .
KNitviTüaCMuvijBiPBelak
- 14 -
lMhat;TI2 cMeBaHRKb;cMnYnBit xcUrRsaybBa¢ak;fa
bab
x
a
x
1cossin 44
)0,0(; ba dMeNaHRsay RsaybBa¢ak;fa
bab
x
a
x
1cossin 44
tamvismPaB SchwarzCauchy eK)an ³
baxx
bx
ax
bx
ax
bx
ax
22244
222244
)cos(sincossin
)(cos)(sincossin
eday 1cossin 22 xx
dUcenH bab
xa
x
1cossin 44
.
KNitviTüaCMuvijBiPBelak
- 15 -
lMhat;TI3 eK[GnuKmn_
9)cot(tan12)cot(tan4)( 22 xxxxxf Edl )
2,0(
x . cUrrktémøtUcbMputénGnuKmn_ )(xf . dMeNaHRsay rktémøtUcbMputénGnuKmn_ )(xf eKman
9)cot(tan12)cot(tan4)( 22 xxxxxf tag 2cottan2cottan xxxxt eK)an 2cottan)cot(tan 2222 xxxxt b¤ 2cottan 222 txx eK)an 8)32(912)2(4)( 22 tttxf eday 2t enaH 13432 t eK)an 781)( xf dUcenHtémøtUcbMputén )(xf esμInwg 7 .
KNitviTüaCMuvijBiPBelak
- 16 -
lMhat;TI4 eK[GnuKmn_
xx
ysin45
cos3
cUrrktémøtUcbMput nig témøFMbMputénGnuKmn_ y . dMeNaHRsay rktémøtUcbMput nig témøFMbMput eKman
xx
ysin45
cos3
eK)an xxyy cos3sin45 b¤ )1(5sin4cos3 yxyx tamvismPaB SchwarzCauchy eK)an ³
)2(169|sin4cos3| 2yxyx tam )1( nig )2( eK)an 2169|5| yy b¤ 22 16925 yy
1
992
2
y
y
eKTaj)an 11 y . dUcenH 1;1 maxmin yy .
KNitviTüaCMuvijBiPBelak
- 17 -
lMhat;TI5 eK[GnuKmn_
89)2(1243),( 2222 yxxyyxyxyxf cUrkMNt; x nig y edIm,I[GnuKmn_ ),( yxf mantémøtUcbMput rYckMNt;témøtUcbMputenaH . dMeNaHRsay kMNt; x nig y
89)2(1243),( 2222 yxxyyxyxyxf GnuKmn_enHGacsresr ³
5)3(4)2(3)6(),( 222 yxxyyxf edIm,I[GnuKmn_ ),( yxf mantémøtUcbMputluHRtaEt ³
03
02
06
y
x
xy
naM[ 3,2 yx
dUcenH 3,2 yx ehIytémøtUcbMputén ),( yxf KW 5 .
KNitviTüaCMuvijBiPBelak
- 18 -
lMhat;TI6 eKmansmIkar 032 xx manb¤stageday 1x nig 2x cUrKNnatémø 5
23
1 419 xxA dMeNaHRsay KNnatémø 5
23
1 419 xxA eKman 1x nig 2x Cab¤sénsmIkar 032 xx eK)an )1(031
21 xx nig )2(032
22 xx
tam )1( eK)an 312
1 xx
343)3(
3
1113
1
12
13
1
xxxx
xxx
tam )2( eK)an 322
2 xx
2119
12)3(7127
)127(
)963(
)96(
)3(
25
2
2222
25
2
225
2
2225
2
22
225
2
22
42
xx
xxxxx
xxx
xxxx
xxxx
xx
KNitviTüaCMuvijBiPBelak
- 19 -
yk 34 13
1 xx nig 2119 25
2 xx CMnYskñúgkenSam 5
23
1 419 xxA eK)an ³ )2119(4)34(19 21 xxA
141)(76
84765776
21
21
xx
xx
eday 121 xx eK)an 21714176 A dUcenH 217419 5
23
1 xxA .
KNitviTüaCMuvijBiPBelak
- 20 -
lMhat;TI7 eK[GnuKmn_ cbxaxxf 2)( Edl IRcbaa ,,,0 . cUrRsayfaebIsmIkar xxf )( Kμanb¤skñúgsMNMucMnYnBitenaH smIkar xxff )]([ k¾Kμanb¤skñúgsMNMucMnYnBitEdr . dMeNaHRsay karbgðaj eKman xxf )( smmUl xcbxax 2 b¤ )(0)1( 1
2 Ecxbax smIkar )( 1E Kμanb¤skñúgsMNMucMnYnBitluHRtaEt ³
)1(04)1( 21 acb
mü:ageToteKman ³ xcxbfxafxxff )()()]([ 2
]1)(][)([
)(])([])([ 22
baxxafxxf
xxfxxfbxxfa
smIkar xxff )]([ smmUl ³ 0]1)(][)([ baxxafxxf
KNitviTüaCMuvijBiPBelak
- 21 -
eday 0)( xxf CasmIkarKμanb¤s ¬ smμtikmμ ¦ eK)an 01)( baxxaf
)(01)1(
01)(
222
2
Ebacxbaxa
baxcbxaxa
DIsRKImINg;énsmIkar )( 2E KW ³ )1(4)1( 222
2 bacaba
)2(]44)1[(
]444)1[(22
22
acba
bacba
tam )1( nig )2( eKTaj)an 02 dUcenHebIsmIkar xxf )( Kμanb¤skñúgsMNMucMnYnBitenaH smIkar xxff )]([ k¾Kμanb¤skñúgsMNMucMnYnBitEdr .
KNitviTüaCMuvijBiPBelak
- 22 -
lMhat;TI8 edaHRsaysmIkar ³ xxxxx 126324 232 dMeNaHRsay edaHRsaysmIkar xxxxx 126324 232 smIkarmann½yluHRtaEt 01263 23 xxx smmUl 0)42(3 2 xxx smmUl 0]3)1([3 2 xx naM[ 0x . tag 42,3 2 xxbxa eK)an 42 xxba . smIkarGacsresr abba 2 elIkCakaer abba 4)( 2
ba
ba
abbaba
0)(
422
22
eKTaj 423 2 xxx b¤ 0452 xx eday 0 cba dUcenH 4;1 21 xx .
KNitviTüaCMuvijBiPBelak
- 23 -
lMhat;TI9 edaHRsaysmIkar ³
33
3 23 2
1952
)13(4)13(10
x
xx
dMeNaHRsay edaHRsaysmIkar ³
33
3 23 2
1952
)13(4)13(10
x
xx
smIkarGacsresr ³
0)13(419)52()13(10 3 2333 2 xxx EckGgÁTaMgBIrénsmIkarnwg 3 2)13( x eK)an ³
0413
13)52(
13
13.10 3333
2
3
x
x
x
x
tag 3
13
13
x
x
t eK)an 04)52(10 3323 tt
2333
333
3323
)52(25544
5825544
4.104)52(
KNitviTüaCMuvijBiPBelak
- 24 -
eKTajb¤s 333
33
1 54
102
1025252
t
33
3
3
33
2 21
105
1025252
t
-cMeBaH 3
54
t eK)an 33
54
13
13
x
x
b¤ )13(4)13(5 xx 93 x b¤ 2x .
-cMeBaH 3
21
t eK)an 33
21
13
13
x
x
b¤ 13)13(2 xx 33 x b¤ 1x . dUcenH 2;1 xx .
KNitviTüaCMuvijBiPBelak
- 25 -
lMhat;TI10 eK[smIkar 099999 xx cUrbgðajfasmIkarenHmanb¤s 0x CacMnYnBitEtmYyKt; rYcRsayfa
0x CacMnYnGsniTan . dMeNaHRsay bgðajfasmIkarmanb¤sEtmYyKt; tagGnuKmn_ 9999)( 9 xxxf Edl IRx eK)an IRxxxxf 0)1(999)(' 88 naM[ )(xf CaGnuKmn_ekInCanic© . dUcenHRkab )(c tag f kat;GkS½ oxx' EtmYycMnucKt; . dUcenHsmIkar 099999 xx manb¤s 0x EtmYyKt;. bgðajfa 0x CacMnYnGsniTan ]bmafa 0x CacMnYnsniTanenaH
q
pxo Edl 1),( qpGCD
eK)an 0999)(9)( 9 qp
qp b¤ )111(9 89 pqqp
edayGgÁTIBIrEckdac;nwg q enaH p Eckdac;nwg q ¬pÞúyBIkarBit¦ dUcenH 0x CacMnYnGsniTan .
KNitviTüaCMuvijBiPBelak
- 26 -
lMhat;TI11 eKdwgfa b
tan xa
.
cUrRsayfa 4 4cos x sin x 1
a b a b
dMeNaHRsay Rsayfa
4 4cos x sin x 1
a b a b
eKman btan x
a naM[ 2 b
tan xa
eday cos xtan x
sin x
eK)an 2
2
sin x b
cos x a b¤
2 2 2 2cos x sin x cos x sin x 1
a b a b a b
eKTaj 2cos x 1
a a b
naM[
4
2
cos a(1)
a (a b)
ehIy 2sin x 1
b a b
naM[
4
2
sin b(2)
b (a b)
bUksmPaB (1) nig (2) GgÁ nig GgÁeK)an ³ 4 4
2 2 2
cos x sin x a b a b 1
a b (a b) (a b) (a b) a b
dUcenH 4 4cos x sin x 1
a b a b
.
KNitviTüaCMuvijBiPBelak
- 27 -
lMhat;TI12 eKdwgfa m n p
cosa , cosb , coscn p p m m n
cUrKNnakenSam ³ 2 2 2
2 2 2
sin a sin b sin cM
2 2cosa sin a 2 2cosb sin b 2 2cosc sin c
dMeNaHRsay KNnakenSam ³ eKman 2 2sin a 1 cos a (1 cosa)(1 cosa) nig 2 2 22 2cosa sin a 1 2cosa cos a (1 cosa) eK)an
2
2 2
sin a (1 cosa)(1 cosa) 1 cosa
2 2cosa sin a (1 cosa) 1 cosa
ehIy 2
2
sin b 1 cosb
2 2cosb sin b 1 cosb
nig 2
2
sin c 1 cosc
2 2cosc sin c 1 cosc
eK)an 1 cosa 1 cosb 1 coscE
1 cosa 1 cosb 1 cosc
m n p1 1 1n p p m m nm n p
1 1 1n p p m m n
n p m p m n m n p1
m n p
KNitviTüaCMuvijBiPBelak
- 28 -
lMhat;TI13 eKmanRtIekaN ABC mYyEdl BC a , AC b , AB c . tag R nig S erogKñaCakaM nig épÞRkLaénRtIekaN ABC enH . cUrRsayfa ³
2 2 2cos A cosB cosC R(sin A sin B sin C)
a b c 4S
dMeNaHRsay tamRTwsþIbTkUsIunUs nig sIunUsGnuvtþn_kñúgRtIekaN ABC eK)an ³
2 2 2a b c 2bccos A b¤ 2 2 2cos A b c a
a 2abc
a b c2R
sin A sinB sinC b¤
a 2R sin A
b 2R sinB
c 2R sinC
nig abcS
4R
eK)an 2 2 2 2cos A 4R (sin B sin C sin A)
a 8RS
b¤ 2 2 2cos A R(sin B sin C sin A)
(1)a 4S
dUcKñaEdr 2 2 2cosB R(sin A sin C sin B)
(2)b 4S
nig 2 2 2cosC R(sin A sin B sin C)
(3)c 4S
bUksmPaB (1) ,(2) &(3) eK)an ³ 2 2 2cos A cosB cosC R(sin A sin B sin C)
a b c 4S
Bit .
KNitviTüaCMuvijBiPBelak
- 29 -
lMhat;TI14 kñúgRKb;RtIekaN ABCcUrRsayfa ³
2(sin A sinB sinC)(1 cos A)(1 cosB)(1 cosC)
2
dMeNaHRsay RsaybBa¢ak;fa
2(sin A sinB sinC)(1 cos A)(1 cosB)(1 cosC)
2
tagRCug BC a , AC b , AB c nig a b cp
2
CaknøHbrimaRt
yk RCakaMrgVg;carwkeRkA nig S CaépÞRkLarbs; ABC . tamRTwsþIbTkUsuInUseKman 2 2 2a b c 2bccos A eK)an 2 2 2a (b 2bc c ) 2bc(1 cos A) eKTaj
2 2(b c) a (b c a)(b c a)1 cos A
2bc 2bc
eday a b cp
2
enaH a b c 2p nig b c a 2(p a)
eK)an 4p(p a) 2p(p a)1 cos A
2bc bc
dUcKñaEdr 2p(p b) 2p(p c)1 cosB , 1 cosC
ac ab
eK)an 2
2
8p .p(p a)(p b)(p c)(1 cos A)(1 cosB)(1 cosC) (1)
(abc)
tamrUbmnþehr:ug abcS p(p a)(p b)(p c)
4R
KNitviTüaCMuvijBiPBelak
- 30 -
eKTaj 2 2
p(p a)(p b)(p c) 1(2)
(abc) 16R
ykTMnak;TMng (2) CMnYskñúg (1) eK)an ³ 22
2
8p 1 p(1 cos A)(1 cosB)(1 cosC) . (3)
16R 2 R
tamRTwsþIbTsIunUs a b c2R
sin A sinB sinC
eKTaj a b c 2p psin A sinB sinC (4)
2R 2R R
tam (3) nig (4) eK)anTMnak;TMng ³ 2(sin A sinB sinC)
(1 cos A)(1 cosB)(1 cosC)2
Bit .
smÁal; ³ eKGacsYrbEnßmeToteday[RsaybBa¢ak;fa ³ 23
cos A cosB cosC sin A sinB sinC1
3 2
edayeRbIvismPaBmFümnBVnþ mFümFrNImaRteK)an ³ 3
1 cos A 1 cosB 1 cosC(1 cos A)(1 cosB)(1 cosC)
3
b¤ 3
cos A cosB cosC(1 cos A)(1 cosB)(1 cosC) 1
3
eday 2(sin A sinB sinC)
(1 cos A)(1 cosB)(1 cosC)2
dUcenH 23
cos A cosB cosC sin A sinB sinC1
3 2
KNitviTüaCMuvijBiPBelak
- 31 -
lMhat;TI15 eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa
2a1 cos A
2bc
Edl a , b , c CaRCugRtIekaN ABC . x> TajbBa¢ak;fa 1
(1 cos A)(1 cosB)(1 cosC)8
dMeNaHRsay k>RsaybBa¢ak;fa
2a1 cos A
2bc
tamRTwsþIbTsIunUseKman 2 2 2a b c 2bccos A tamvismPaBmFümnBVnþ mFümFrNImaRteKman 2 2b c 2bc eKTaj 2a 2bc 2bccos A 2bc(1 cos A) dUcenH
2a1 cos A
2bc .
x> TajbBa¢ak;fa 1(1 cos A)(1 cosB)(1 cosC)
8
tamsRmayxagelIeKman 2a
1 cos A (1)2bc
RsaydUcKñaEdr 2b
1 cosB (2)2ac
nig 2c
1 cosC (3)2ab
KuNvismPaB (1) , (2) , (3) GgÁ nig GgÁeKTTYl)an ³
1(1 cos A)(1 cosB)(1 cosC)
8 Bit .
KNitviTüaCMuvijBiPBelak
- 32 -
lMhat;TI16 eK[RtIekaN ABC mYy . k> cUrRsaybBa¢ak;fa 2 2 2sin A sin B sin C 2sinBsinCcos A x> TajbBa¢ak;fa ³
2 2 2sin A sin B sin Ccot A cot B cot C
2sin AsinBsinC
dMeNaHRsay k>RsaybBa¢ak;fa 2 2 2sin A sin B sin C 2sinBsinCcos A tag a , b , c CaRCugRtIekaN ABC nig R CakaMrgVg;carwkeRkARtIekaN. tamRTwsþIbTsIunUs a b c
2Rsin A sinB sinC
eKTaj a 2R sin A
b 2r sinB (1)
c 2R sinC
tamRTwsþIbTkUsIunUs 2 2 2a b c 2bccos A (2) yk (1) CMnYskñúg (2) eK)an ³
2 2 2 2 24R sin A 4R (sin B sin C 2sinBsinCcos A) sRmYl 24R kñúgGgÁTaMgBIrénsmPaBeK)an ³
2 2 2sin A sin B sin C 2sinBsinCcos A Bit
KNitviTüaCMuvijBiPBelak
- 33 -
dUcenH 2 2 2sin A sin B sin C 2sinBsinCcos A . x> TajbBa¢ak;fa ³
2 2 2sin A sin B sin Ccot A cot B cot C
2sin AsinBsinC
tamsRmayxagelIeKman 2 2 2sin A sin B sin C 2sinBsinCcos A
eKTaj 2 2 2sin B sin C sin A
cos A2sinBsinC
eday cos A
cot Asin A
eK)an 2 2 2sin B sin C sin A
cot A (i)2sin AsinBsinC
RsaydUcKñaEdreKTTYl)an 2 2 2sin A sin C sin B
cot B (ii)2sin AsinBsinC
nig 2 2 2sin A sin B sin C
cot C (iii)2sin AsinBsinC
eFIVplbUksmPaB (i) , (ii) & (iii) eK)an ³ 2 2 2sin A sin B sin C
cot A cot B cot C2sin AsinBsinC
Bit
dUcenH 2 2 2sin A sin B sin C
cot A cot B cot C2sin AsinBsinC
.
KNitviTüaCMuvijBiPBelak
- 34 -
lMhat;TI17 eK[RtIekaN ABC mYymanmMu A ,B,C CamMuRsYcEdlepÞógpÞat;smPaB
1 1 1 1cot A cot B cot C
2 sin A sinB sinC
.
cUrRsayfa ABC CaRtIekaNsmgS½ ? dMeNaHRsay Rsayfa ABC CaRtIekaNsmgS½ tag a , b , c CaRCug nig S CaépÞRkLaénRtIekaN ABC eKman 1 1 1
S bcsin A acsinB absinC2 2 2
eKTaj)an 2S 2S 2Ssin A , sinB , sibC
bc ac ab
eK)an 2 2 2
2 2 2b c a
cos A b c a2bccot A2Ssin A 4Sbc
ehIy 2 2 2 2 2 2a c b a b c
cot B , cot C4S 4S
eK)an 2 2 2a b c
cot A cot B cot C (1)4S
mü:ageTot 1 1 1 bc ca ab(2)
sin A sinB sinC 2S
tamsm μtikmμ 1 1 1 1cot A cot B cot C (3)
2 sin A sinB sinC
KNitviTüaCMuvijBiPBelak
- 35 -
yksmIkar (1) & (2) CMnYskñúg (3) eK)an ³
2 2 2a b c ab bc ca
4S 4S
b¤ 2 2 2a b c ab bc ca TMnak;TMngenHsmmUl 2 2 2(a b) (b c) (c a) 0
eKTaj a b 0
b c 0
c a 0
naM[ a b c .
edayRtIekaN ABC manRCugbIes μIKñavaCaRtIekaNsmgS½ . smÁal; ³ eKGacRsayfa
2 2 2sin A sin B sin Ccot A cot B cot C
2sin AsinBsinC
ehIysm μtikm μ 1 1 1 1cot A cot B cot C
2 sin A sinB sinC
eKTaj)ansmIkar ³ 2 2 2sin A sin B sin C 1 1 1 1
2sin AsinBsinC 2 sin A sinB sinC
2 2 2
2 2 2
sin A sin B sin C sin AsinB sinBsinC sinCsin A
(sin A sinB) (sinB sinC) (sinC sin A) 0
eKTaj sin A sinB 0
sinB sinC 0
sinC sin A 0
naM[ sin A sinB sinC
b¤ A B C enaH ABC CaRtIekaNsmgS½ .
KNitviTüaCMuvijBiPBelak
- 36 -
lMhat;TI18 tag R CakaMrgVg;carwkkñúg nig S CaépÞRkLaénRtIekaN ABC mYy . k> cUrRsayfa
22R(cot A cot B)(cot B cotC)(cotC cot A)
S
x> ebI ABC CamMuRsYcenaHcUrTaj[)anfa ³ 1cos AcosBcosC
8 .
dMeNaHRsay Rsayfa
22R(cot A cot B)(cot B cot C)(cot C cot A)
S
KUskMBs; aAA' h én ABC . tag BC a , AC b , AB c . kñúgRtIekaNEkg ABA' & AA'C eKman
BA' A'Ccot B ; cot C
AA' AA'
eK)an 2
a
BA' A'C a acot B cot C
AA' h 2S
. S CaépÞRkLa ABC
dUcKñaEdr 2 2b c
cot C cot A , cot A cot B2S 2S
A
B C A'
KNitviTüaCMuvijBiPBelak
- 37 -
eK)an 2 2 2
3
a b c(cot A cot B)(cot B cot C)(cot C cot A)
8S
eday abcS
4R enaH abc 4RS
eK)an 2 2
3
16R S(cot A cot B)(cot B cot C)(cot C cot A)
8S
dUcenH 22R
(cot A cot B)(cot B cot C)(cot C cot A)S
.
x> Taj [)anfa ³ 1cos AcosBcosC
8
man 22R
(cot A cot B)(cot B cot C)(cotC cot A) (i)S
ebI ABC CamMuRsYcenaH cot A 0 , cot B 0 , cot C 0 tamvismPaB mFümnBVnþ mFümFrNImaRteKman ³ cot A cot B 2 cot Acot B , cot B cotC 2 cot Bcot C
cot C cot A 2 cotCcot A
KuNvismPaBxagelIenH GgÁ nig GgÁ eK)an (cot A cot B)(cot B cotC)(cotC cot A) 8cot Acot BcotC (ii) tam (i)&(ii) eKTaj)an
22R8cot Acot Bcot C
S
Et 3
2abc 8R sin AsinBsinCS 2R sin AsinBsinC
4R 4R
eK)an 2
2
2R8cot Acot Bcot C
2R sin AsinBsinC
dUcenH 1cos AcosBcosC
8
KNitviTüaCMuvijBiPBelak
- 38 -
lMhat;TI19 eK[RtIekaN ABC mYymanRCug BC a , AC b , AB c carwkkñúgrgVg;mYymanp©it O nig kaM R . tag S nig OBCS CaépÞRkLaén OBC nig ABC erogKña . snμtfa A ,B,C CamMuRsYc . k> cUrRsayfa 2
OBC
1S a cot A
4 .
x> cUrTajbgðajfa 2 2 2a cot A b cot B c cot C 4S . K> cUrTajbgðajfa
2
abcacos A bcosB ccosC
2R .
dMeNaHRsay k> Rsayfa 2
OBC
1S a cot A
4
tag K CacMNuckNþalénRCug [BC] enaH [OK] [BC] . eKman BOC
BOK BAC A2
.
A
B C
O
K
KNitviTüaCMuvijBiPBelak
- 39 -
kñúgRtIekaNEkg OBK eKman ³ OK 2OK 2OK
cot BOK cot ABK BC a
eKTaj 1OK acot A
2 .
RkLaépÞénRtIekaN OBC KW 2OBC
1 1S BC.OK a cot A
2 4 Bit
dUcenH 2OBC
1S a cot A
4 .
x> Tajbgðajfa 2 2 2a cot A b cot B c cot C 4S tamsRmayxagelIeKman 2
OBC
1S a cot A
4
dUcKñaEdr 2OCA
1S b cot B
4 nig 2
OAB
1S c cot C
4
eday OBC OCA OABS S S S eK)an 2 2 21
S (a cot A b cot B c cot C)4
dUcenH 2 2 2a cot A b cot B c cot C 4S . K> Tajbgðajfa
2
abcacos A bcosB ccosC
2R
eKman 2 2 2 abca cot A b cot B c cotC 4S
R eRBaH abc
S4R
eday 2 2aa cot A .acos A 2Racos A , b cot B 2RbcosB
sin A
enaHeK)an 2
abcacos A bcosB ccosC
2R .
KNitviTüaCMuvijBiPBelak
- 40 -
lMhat;TI20 eK[RtIekaN ABC mYy . D CacMNucmYYénRCug [BC] Edl
BAD nig DAC . cUrRsaybBa¢ak;fa BC sin sin
AD sinB sinC
?
dMeNaHRsay RsaybBa¢ak;fa BC sin sin
AD sinB sinC
tamRTwsþIbTsIunUsGnuvtþn_ kñúg ABD& ADC eKman BD AD
sin sinB
b¤ sinBD .AD (1)
sinB
ehIy DC AD
sin sinC
b¤ sinDC .AD (2)
sinC
bUkTMnak;TMng (1) & (2) eK)an sin sinBD DC .AD
sinB sinC
eday BD DC BC enaH sin sinBC .AD
sinB sinC
dUcenH BC sin sin
AD sinB sinC
.
A
DB C
KNitviTüaCMuvijBiPBelak
- 41 -
lMhat;TI21 bgðajfaebI c,b,a tagerogKñaCatYTI r,q,p énsIVútnBVnþmYyenaHeK)ansmPaB
0c)qp(b)pr(a)rq( . dMeNaHRsay karbgðaj ebI c,b,a tagerogKñaCatYTI r,q,p énsIVútnBVnþ )u( n enaHeK)an
d)1r(uuc,d)1q(uub,d)1p(uua 1r1q1p
eKman 0c)qp(b)pr(a)rq( b¤ 0cqcpbpbraraq
b¤ 0)ab(r)ca(q)bc(p eday
d)pq(ab
d)rp(ca
d)qr(bc
enaH 0)pq(rd)rp(qd)qr(pd b¤ 0)prqrqrpqpqpr(d 00 Bit dUcenH 0c)qp(b)pr(a)rq( .
KNitviTüaCMuvijBiPBelak
- 42 -
lMhat;TI22 eK[ctuekaNe)a:g ABCD mYymanépÞRkLaesμI 1 . cUrRsaybBa¢ak;fa ³ AB BC CD DA AC BD 4 8 ? ¬Austria-Poland, 1985¦ dMeNaHRsay RsaybBa¢ak;fa ³ AB BC CD DA AC BD 4 8 tag AB a , BC b , CD c , DA d nigGgát;RTUg AC m , BD n . tagmMu AIB CID .
A
B
C
D
I
KNitviTüaCMuvijBiPBelak
- 43 -
RkLaépÞénctuekaN ABCD kMNt;eday ³ ABCD ABC ACD
1 1S S S absinB cdsin D (1)
2 2
b¤ ABCD ABD BCD
1 1S S S adsin A bcsinC (2)
2 2
bUkTMnak;TMng (1) nig (2) eK)an ³ ABCD
12S (adsin A absinB bcsinC cdsin D)
2
edayeKman sin x 1 RKb;cMnYnBit x enaHeKTaj)an ABCD
12S (ad ab bc cd)
2
b¤ ABCD
1S (a c)(b d)
4 eday ABCDS 1
eKTaj)an (a c)(b d) 4 tavismPaB AM GM eKman ³ a b c d 2 (a c)(b d) b¤ a b c d 4 (i) müa:geTot ³
KNitviTüaCMuvijBiPBelak
- 44 -
ABCD IAB IBC ICD IADS S S S S
1(IA.IB IB.IC IC.ID ID.IA)sin
21
(AC.IB AC.ID)sin21 1
AC.BDsin AC.BD2 2
eK)an ABCD
1S mn
2 eday ABCDS 1
eK)an mn 2 EttavismPaB AM GM eKman ³ m n 2 mn 2 2 8 (ii) bUkvismPaB (i) nig (ii) eK)an ³ a b c d m n 4 8 Bit dUcenH AB BC CD DA AC BD 4 8 .
KNitviTüaCMuvijBiPBelak
- 45 -
lMhat;TI23 eK[RtIekaNBIrmanRCug a ,b ,c knøHbrimaRt a b c
p2
nigkaMrgVg;carwkeRkA R . cUrRsaybBa¢ak;fa 2 pR 1
5 2aR bc 2
dMeNaHRsay RsaybBa¢ak;fa 2 pR 1
5 2aR bc 2
tamvismPaBRtIekaNeKman ³ AB AC (AH BH) (AH HC) 2AH BC eKTaj)an AB AC BC b c a
AH p a2 2
H B C
A
a
b c
KNitviTüaCMuvijBiPBelak
- 46 -
tag S CaépÞRkLaénRtIekaN ABC enaHeK)an ³ 1 1
S BC.AH a.AH2 2
b¤ 2SAH
a
eKTaj)an 2Sp a
a b¤ 2S a(p a) Et abc
S4R
ehtuenH abc
2. a(p a)4R
b¤ bc 2pR 2aR b¤ bc 2aR 2pR naM[ pR 1
(i)bc 2aR 2
mü:ageTot]bmafa pR 2
2aR bc 5
Bit
eK)an 4aR 2bc 5pR (1) eday abc 1
S a.AH4R 2
b¤ bc 2R.AH tam (1) eK)an 4aR 4R.AH 5pR b¤ 5p
a AH (2)4
kñúgRtIekaNEkg ABH nig AHC eKman ³ 2 2 2AB AH BH nig 2 2 2AC AH HC b¤ 2 2 2c AH BH nig 2 2 2b AH HC enaH 2 2 2 2b c AH HC AH BH
KNitviTüaCMuvijBiPBelak
- 47 -
tamvismPaB Minkowsky eKman ³ 2 2 2 2 2 2
1 1 2 2 1 2 1 2x y x y (x x ) (y y ) ehtuenH 2 2 2 2b c 4.AH (BH HC) 4.AH a b¤ 2 2a b c a 4AH a b¤
2 2a 4AH ap (3)
2
tam (2) nig (3) eyIgnwgRsay[)anfa ³ 2 25 a 4AH a
a AH ( )4 2
Bit
b¤ 2 23a 8.AH 5 4.AH a b¤ 2 2 2 29a 48a.AH 64.AH 100.AH 25a b¤ 2 216a 48a.AH 36.AH 0 b¤ 2(4a 6AH) 0 Bit ehtuenH pR 2
2aR bc 5
(ii)
tamTMnak;TMng (i) nig (ii) eK)an ³ 2 pR 1
5 2aR bc 2
Bit
KNitviTüaCMuvijBiPBelak
- 48 -
lMhat;TI24 eKyk I nig O Cap©itrgVg;carwkkñúg nig carwkeRkAén ABC . rgVg; A carwkkñúgmMu A vab:HeTAnwg AB , AC nig BC erogKña Rtg;cMnuc K ,M nig N erogKña . ebIsinCacMnuckNþal P énGgát; KM sßitenAelIrgVg;carwkeRkA énRtIekaN ABC enaHcUrbgðajfabIcMnuc O , I , N rt;Rtg;Kña ? dMeNaHRsay RsaybBa¢ak;fabIcMnuc O , I , N rt;Rtg;Kña
A
B
C
PQ
K
MN
I
O
8
KNitviTüaCMuvijBiPBelak
- 49 -
yk Q CacMnucRbsBVTIBIrrvagbnÞat; PM CamYyrgVg; (ABC) tag KP PM x nig PQ y . eyIgman KB p c nig MC p b . tamRTwsþIbTsV½yKuNéncMnuc K nig M eFobnwgrgVg; (ABC) eKman KP.KQ KA.KB nig MP.MQ MC.MA b¤ x(x y) p(p c) nig x(x y) p(p b) bUksmIkarBIrenHeK)an 22x p(2p b a) pa b¤ 2 1
x pa2
ehtuenH 2 2MK 4x 2pa . tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN AKM eKman ³
2 2 2MK MA KA 2MA.KAcos A eday MA KA p nig 2MK 2pa enaHeK)an ³
2 2 2 2 A2pa 2p 2p cos A 4p sin
2 b¤ 2 A
a 2psin2
Et A (p b)(p c)sin
2 bc
eK)an 2p(p b)(p c)a
bc
b¤ abc 2p(p b)(p c) (1)
KNitviTüaCMuvijBiPBelak
- 50 -
yk D CacMeNalEkgén I elI BCehIysnμtfa N' CaRbsBV rvag (OI) CamYy [BC] nig tag E CacMnuckNþalén BC . ]bmafa N' N enaHeK)an N'C NC p b eday BD p b enaHeK)an ED EN' b¤ E CacMnuckNþal énGgát; [DN'] . RtIekaNEkg N'DI nigRtIekaNEkg N'EO manmMu DN'I rYmvaCaRtIekaNdUcKña . ehtuenHeK)an DI DN' 2DE
2EO EN' EN'
b¤ DI 2.EO eKman BOC
BAC EOC2
.
kñúgRtIekaNEkg EOC eKman OEcos EOC cos A
OC
eKTaj OE OC.cos A naM[ DI 2.OC cos A yk DI r nig OC R ¬kaMrgVg;carwkkñúg nig eRkA ABC ¦ eK)an r 2R cos A b¤ r
cos A2R
tamrUbmnþépÞRkLaén ABC eKman abc
S pr4R
nigRTwsþIbTkUsIunUs 2 2 2b c a
cos A2bc
KNitviTüaCMuvijBiPBelak
- 51 -
eK)an 2 2 2 2
Sb c a 2Sp
abc2bc abcp2S
b¤ 2 2 2 24S ap(b c a ) Et S p(p a)(p b)(p c) eK)an 2 2 24p(p a)(p b)(p c) ap(b c a ) b¤ 2 2 24(p a)(p b)(p c) a(b c a ) eday 2 2 2 2 2b c a (b c) a 2bc 2bc (b c a)(b c a)
2bc 4(p b)(p c)
enaH 4(p a)(p b)(p c) 2abc 4a(p b)(p c) b¤ abc 2(p a)(p b)(p c) 2a(p b)(p c) b¤ abc 2p(p b)(p c) (2) tamTMnak;TMng (1) nig (2) bBa¢ak;fakar]bma N' N Bit dUcenHbIcMnuc O , I , N rt;Rtg;Kña .
KNitviTüaCMuvijBiPBelak
- 52 -
lMhat;TI25 cUrRsayfakaerénRKb;cMnYnKt;RtUvmanTRmg; k4 b¤ 14 k RKb; *INk . dMeNaHRsay karbgðaj eyIgdwgfaRKb;cMnYnKt;viC¢man manrag n2 b¤ 1,12 nn eK)an 22 4)2( nn manrag k4 Edl 2nk ehIy 1)1(4144)12( 22 nnnnn manrag 14 k Edl )1( nnk . dUcenHkaerénRKb;cMnYnKt;RtUvmanTRmg; k4 b¤ 14 k RKb; *INk .
KNitviTüaCMuvijBiPBelak
- 53 -
lMhat;TI26 cUrRsayfaebI 22 ba Eckdac;nwg 3 enaH a Eckdac;nwg 3 nig b Eckdac;nwg 3 . dMeNaHRsay karbgðaj snμtfa a nig b CacMnYnEckmindac;nwg 3 enaHcMnYnBIrenHRtUvman rag 23,13,23,13 qbqbpapa Edl ...,2,1,0...;,2,1,0 qp . eK)an 1)23(3)13( 22 pppa ehIy 1]1)23([3)23( 22 pppa eKTaj)anfa 0,132 xxa nig 0,132 yyb eK)an 2)(322 yxba Eckmindac;nwg 3 EdlpÞúyBI smμtikmμ . dUcenHeKGacsnñidæanfaebI 22 ba Eckdac;nwg 3 enaH a Eckdac; nwg 3 nig b Eckdac;nwg 3 .
KNitviTüaCMuvijBiPBelak
- 54 -
lMhat;TI27 cUrbgðajfa 44 n CacMnYnbzmEtkñugkrNI 1n . dMeNaHRsay karbgðaj eKman 2224 4)2(4 nnn
]1)1[(]1)1[(
)22)(22(22
22
nn
nnnn
cMeBaHRKb; 1n eKman 11)1( 2 n nig 11)1( 2 n enaH 44 n minEmnCacMnYnbzm . ebI 1n enaH 544 n CacMnYnbzm . dUcenH 44 n CacMnYnbzmEtkñugkrNI 1n .
KNitviTüaCMuvijBiPBelak
- 55 -
lMhat;TI28 eKdwgfa 0160321002004008 manktþabzm 250000p cUrkMNt;ktþabzmenaH . dMeNaHRsay kMNt;ktþabzm edayeRCIserIs 2,1000 ba eK)an ³
543223450160321002004008 babbababaa
k
bababababa
ba
ba
.250501.1002.4
)998004)(1002004)(1002(
))()((
2
2222
66
Edl 250000k enaH 250501p .
KNitviTüaCMuvijBiPBelak
- 56 -
lMhat;TI29
cUrkMNt;EpñkKt;én
610
1
1
k k ?
dMeNaHRsay kMNt;EpñkKt;
tag 0,1
)( 2
1
xxx
xf
eK)an 0,021
)(' 2
3
xxxf enaH )(xf CaGnuKmn_cuH . cMeBaH 0k eK)an
kxdx
k
k
k
11
1 1
eKTaj
610
1
1610
1
11610
1
1x
dxx
dxk k
k
kk
ehIy
610
1
610
2
1610
1
11
1x
dxkk kk
eKTaj
1610
1
610
1
610
2
11
kk kxdx
k
eday
610
1
610
2
11
1
kk kk
KNitviTüaCMuvijBiPBelak
- 57 -
ehIy
610
1
1610
1
1
1000
11
kk kk
nig 1998)110(22 3610
1
610
1
xx
dx
eK)an
610
1
610
1
1
1000
11998
11
kk kk
b¤ 19991
1000
11998
610
1
k k
dUcenHEpñkKt;én
610
1
1
k k KW 1998 .
KNitviTüaCMuvijBiPBelak
- 58 -
lMhat;TI30 cUrkMNt;KUmYyéncMnYnKt;viC¢man ba , edaydwgfa ³
)(:)( baabi Eckmindac;nwg 7 777)(:)( babaii Eckdac;nwg 77 .
(IMO 1984 ) dMeNaHRsay kMNt;KUmYyéncMnYnKt;viC¢man ba , ³ tamrUbmnþeTVFajÚtuneK)an ³
222777 ))((7)( bababaabbaba tambRmab;eKman )(:)( baabi Eckmindac;nwg 7 dUcenHedIm,I[ 777)(:)( babaii Eckdac;nwg 77 luHRtaEt 22 baba Eckdac;nwg 37 . eKman 3222 7)( bababa enaH 19 ba edayeFIVkarsakl,gCMnYs 18,1 ba enaHeK)an ³
32222 7343181811 baba dUcenHKU 18,1 ba CacemøIy.
KNitviTüaCMuvijBiPBelak
- 59 -
mü:ageTottamRTwsþIbTGWEl ³ ebI 1),( naGCD enaH )mod(1)( na n . eKman ))(( 2233 babababa ehtuenHedIm,I[ 22 baba Eckdac;nwg 37 luHRtaEt
)7(mod 333 ba nig ba Eckmindac;nwg 7 . eKman 9837677)7( 2233 . ebI c CacMnYnKt;viC¢manEckmindac;nwg 7 enaHeK)an )7(mod1)( 3398 c . dUcenHedIm,I[ )7(mod1 33 a luHRtaEt 98ca ]TahrN_ ³ -ebIeKyk 2c enaH )7(mod182 398 ehtuenH )7(mod1182 33398 dUcenH 1,18 ba CacemøIymYy . -ebIeKyk 3c enaH )7(mod3243 398 ehtuenH )7(mod13243 33398 dUcenH 1,324 ba CacemøIyepSgmYyeTot .
KNitviTüaCMuvijBiPBelak
- 60 -
lMhat;TI31 eKGnuKmn_ xxf 2121
1)(
cUrKNna )1
(...)2
()1
(n
nf
nf
nfS
dMeNaHRsay KNna S CadMbUgeyIgRtUvRsayfa ebI 1 qp enaH 1)()( qfpf eKman qp qfpf 2121 21
1)(;
21
1)(
qpqfpf 2121 21
1
21
1)()(
qpqp
qp
qp
qp
2222121
2121
2121
2121
2221
222
)21)(21(
222
ebI 1 qp enaH 0)(22222 qpqp
1222
222)()( 2121
2121
qp
qp
qfpf
KNitviTüaCMuvijBiPBelak
- 61 -
eKman )1()1
(...)2
()1
(n
nf
nf
nfS
b¤ )2()1
(...)2
()1
(n
fn
nf
n
nfS
bUksmIkar )1( nig )2( eK)an ³
11.....1112
)]1
()1
([.....)]1
()1
([2
)1(
nSn
fn
nf
n
nf
nfS
n
dUcenH 2
1
nS .
KNitviTüaCMuvijBiPBelak
- 62 -
lMhat;TI32 cUrRsaybBa¢ak;facMnYn ³ n n n nA 113 168 141 427 Eckdac;nwg 7 Canic©RKb;cMnYnKt;FmμCati n. dMeNaHRsay Rsayfa A Eckdac;nwg 7 tamrUbmnþ n n n 1 n 2 n 1a b (a b)(a a b ... b ) eK)an n n n nA (427 168 ) (141 113 ) 1 2
1 2 1 2
(427 168)m (141 113)m
259m 28m 7(37m 4m )
dUcenH A Eckdac;nwg 7 RKb; 1 2m , m IN * .
KNitviTüaCMuvijBiPBelak
- 63 -
lMhat;TI33 eK[ m nig n CaBIrcMnYnKt;viC¢manEdl 1 m n . cUrkMNt;témøtUcbMputén m n edIm,I[cMnYn m22 nig n22 manelx BIrxÞg;cugeRkaydUcKña . dMeNaHRsay kMNt;témøtUcbMputén m n ³ edIm,I[cMnYn m22 nig n22 manelxBIrxÞg;cugeRkaydUcKñaluHRtaEt
n m22 22 Eckdac;nwg 100 . eKman n m m n m22 22 22 (22 1) eday m22 CacMnYnKU nig n m22 1 CacMnYness . ehtuenH m n m22 (22 1) Eckdac;nwg 100 4 25 luHRtaEt
m22 Eckdac;nwg 4 nig n m22 1 Eckdac;nwg 25 . cMnYn m22 Eckdac;nwg 4 luHRtaEt m 2 . müa:geToteKman 222 96 5 4 elIkCakaereK)an 4 222 (96 5 4)
4 2 2
2
2
22 (96 5) 2(96 5).4 4
(96 154) 25 6
25a 6 , a 96 154
KNitviTüaCMuvijBiPBelak
- 64 -
eK)an 4p p p22 (25a 6) 25q 6 , q IN eday 6 5 1 enaHeK)an ³
pp p k
k 0
pk
k 2
6 (5 1) C(p,k)5
1 5p C(p,k)5
eKTaj p
4p k
k 222 1 5p 25q C(p,k)5
b¤ p
4p k
k 222 1 5p 25q C(p,k)5
eday p
k
k 225q C(p,k)5
Eckdac;nwg 25 ehtuenH 4p22 1
Eckdac;nwg 25 luHRtaEt 5p Eckdac;nwg 25 eBalKWeKRtUv[ p 5k , k IN * . dUcenH n m22 1 Eckdac;nwg 25 luHRtaEt n m 4p 20k eday m 2 enaH m n (n m) 2m 20k 4 24 dUcenHtémøGb,brmaén m n KW 24 .
KNitviTüaCMuvijBiPBelak
- 65 -
lMhat;TI34 eK[ 2f (n) 4n 7n 34 Edl n IN * cUrbgðajfaKμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT. dMeNaHRsay karbgðaj eKman 2f (n) 4n 7n 34
2
2
(4n 4n 1) (11n 33)
(2n 1) 11(n 3)
eyIgeXIjfaedIm,I[ f (n) Eckdac;nwg 121 luHRtaEt n 3 nig 2n 1 Eckdac;nwg 11 naM[eKman 1 2q ,q IN * Edl
1n 3 11q nig 22n 1 11q . eK)an 1 22(n 3) (2n 1) 22q 11q b¤ 1 211(2q q ) 7 CasmIkarKμancemøIykñúgsMNMu IN * BIeRBaH GgÁTImYyénsmIkarEckdac;nwg 11 EtGgÁTIBIrminEckdac;nwg 11 . dUcenH Kμan n EdleFIV[cMnYn f (n) Eckdac;nwg 121 eT.
KNitviTüaCMuvijBiPBelak
- 66 -
lMhat;TI35 eK[ 2f (a) 16a 50a 13 Edl a kMNt;RKb;témø a EdleFIV[cMnYn f (a) Eckdac;nwg 49 dMeNaHRsay kMNt;RKb;témø a eKman 2f (a) 16a 50a 13
2
2
(16a 8a 1) (42a 14)
(4a 1) 7(6a 2)
edIm,I[ f (a) Eckdac;nwg 49 luHRtaEteKman p , q Edl 4a 1 7p nig 6a 2 7q . eK)an 3(4a 1) 2(6a 2) 21p 14q 7 7(3p 2q) b¤ 3p 2q 1 eKGacsresr 3(p 1) 2(q 1) eKTaj p 1 2k nig q 1 3k RKb; k b¤ p 2k 1 , q 3k 1 eKTaj)an 4a 1 7(2k 1) naM[ 7k 3
a2
KNitviTüaCMuvijBiPBelak
- 67 -
eday a enaH 7k 3
2
.
edIm,I[ 7k 3
2
luHRtaEt k CacMnYnKt;essKW ³
k 2 1 , enaHeK)an 7(2 1) 3a 7 2
2
dUcenH a 7 2 , .
KNitviTüaCMuvijBiPBelak
- 68 -
lMhat;TI36 kMNt;RKb;cMnYnKt;viC¢man x edIm,I[ 3 2x 10x 115x 237 Eckdac;nwg 343 . dMeNaHRsay kMNt;RKb;cMnYnKt;viC¢man x tag 3 2f (x) x 10x 115x 237
3 2
2
(x 1) 7(x 1) 49(2x 5)
(x 1) (x 8) 49(2x 5)
edIm,I[ f (x) Eckdac;nwg 343 luHRtaEt ³ x 1 nig 2x 5 Eckdac;nwg 7 b¤ x 8 Eckdac;nwg 343 nig 2x 5 Eckdac;nwg 7 . -krNI x 1 nig 2x 5 Eckdac;nwg 7 enaHeKman 1 2q , q IN * Edl 1
2
x 1 7q
2x 5 7q
eK)an 1 22(x 1) (2x 5) 14q 7q
1 2
1 2
1 2
7 7(2q q )
1 2q q
2(q 1) (q 1)
eKTaj 1
2
q 1 k
q 1 2k
b¤ 1
2
q k 1
q 2k 1 , k IN
KNitviTüaCMuvijBiPBelak
- 69 -
eKTaj)an x 1 7(k 1) b¤ x 7k 6 , k IN dUcenH x 7k 6 , k IN CacemøIy . -krNI x 8 Eckdac;nwg 343 nig 2x 5 Eckdac;nwg 7 eK)an 1
2 1 2
x 8 343m
2x 5 7m , m ,m IN *
eKman 1 22(x 8) (2x 5) 686m 7m 1 2
1 2
21 686m 7m
3 98m m
eKGacsresr 1 298(m 1) m 95 naM[ 1
2
m 1 k
m 95 98k
b¤ 1
2
m k 1
m 98k 95 , k IN
dUcenH x 343(k 1) 8 343k 335 RKb; k IN .
KNitviTüaCMuvijBiPBelak
- 70 -
lMhat;TI37 cUrRsayfaRKb;cMnYnKt;viC¢man n eK)an
237
134)(
n
nnf
CaRbPaKsRmYlmin)an ? dMeNaHRsay bgðajfa )(nf CaRbPaKsRmYlmin)an tag )237,134( nnGCDd eK)an dyndxn .237,.134 Edl 1),( yxGCD . eK)an xdydnn 74)134(7)237(4
dxy
dxynn
).74(1
).74(91289228
eKTaj)an 1d enaH 1)237,134( nnGCD )ann½yfa 134 n nig 237 n CacMnYnbzmrvagKñaRKb; 1n dUcenH
237
134)(
n
nnf CaRbPaKsRmYlmin)an .
KNitviTüaCMuvijBiPBelak
- 71 -
lMhat;TI38 eKdwgfa
ab
tan3 . cUrRsaybBa¢ak;fa ³
3 23 23 2
4
3 2
4
ba
1
b
sin
a
cos
dMeNaHRsay RsaybBa¢ak;fa ³
3 23 23 2
4
3 2
4
ba
1
b
sin
a
cos
eKman a
btan3 naM[ 3
sin btan
cos a
eKTaj 3 3
cos sin
a b
b¤ 2 2 2 2
3 3 3 3 3 32 2 2 2 2 2
cos sin cos sin 1
a b a b a b
eKTaj 2
3 3 32 2 2
cos 1
a a b
nig
2
3 3 32 2 2
sin 1
b a b
eKTaj 34 2
3 3 32 2 2 2
cos a
a ( a b )
nig
34 2
3 3 32 2 2 2
sin b
b ( a b )
bUksmIkarBIenHeKTTYl)an 3 23 23 2
4
3 2
4
ba
1
b
sin
a
cos
Bit .
KNitviTüaCMuvijBiPBelak
- 72 -
lMhat;TI39 eKmansIVútcMnYnBit )a( n kMNt;eday ³
)2
11....()
4
11)(
2
11(a n2n
cUrKNnalImIt nnalim
.
dMeNaHRsay KNnalImIt nn
alim
yk
2
1q eK)an ³
)q1)......(q1)(q1(an22
n edayeRbIsmPaB
yx
yxyx
22
eK)an ³
1n2
1n2
1n2
n
n2
1n2
2
42
n
2
112
2
11
2
11
q1
q1a
q1
q1....
q1
q1.
q1
q1a
kalNa n enaH 1n2
2
1
xitCitsUnü
dUcenH 2alim nn
.
KNitviTüaCMuvijBiPBelak
- 73 -
lMhat;TI40 eK[plbUk
222222n )1n(
1
n
11....
4
1
3
11
3
1
2
11
n
1S
cUrKNna nnSlim
.
dMeNaHRsay KNnalImIt nn
Slim
eKBinitü 22
2222
22 )1k(k
k)1k()1k(k
)1k(
1
k
11
2
2
22
2
22
22
22
2222
1k
1
k
11
)1k(k
11
)1k(k
1)1k(k
)1k(k
1)1k(k2)1k(k
)1k(k
k1k2k)1k(k
eK)an 1n
11)
1n
11n(
n
1
1k
1
k
11
n
1S
n
1kn
dUcenH 11n
11limSlim
nnn
.
KNitviTüaCMuvijBiPBelak
- 74 -
lMhat;TI41 eKmanes‘rIGnnþmYy ....
2
n.....
2
3
2
2
2
1)
2
n( n
2
3
2
2
22
1nn
2
cUrRsayfaes‘rIxagelIenHCaes‘rIbRgYm . dMeNaHRsay plbUkedayEpñkrbs;es‘rIenHKW n
2
3
2
2
22
n 2
n...
2
3
2
2
2
1S
BinitüGnuKmn_ cbkak)k(f 2 EdlRKb; *INk eKman ³
1kkk
2
2
)1k(f
2
)k(f
2
k
b¤ 1k
2
k
2
k
2
2
c)1k(b)1k(a
2
cbkak
2
k
b¤ cbbkaak2akc2bk2ak2k2 222 back)a2b(akk2 22 eKTaj 2a ehIy 4b0a2b nig 6bac ehtuenH 1k
2
k
2
k
2
2
12k8k2
2
6k4k2
2
k
eK)an
n
1k1k
2
k
2
n )2
12k8k2
2
6k4k2(S
n
2
2
6n4n6
eday 62
6n4n6limSlim n
2
nnn
dUcenHes‘rIGnnþxagelICaes‘rIbRgYm .
KNitviTüaCMuvijBiPBelak
- 75 -
lMhat;TI42 cMeBaHRKb; *INn eKmanplbUk ³
1n432n 2).1n2)(1n2(
3n2....
2.7.5
9
2.5.3
7
2.3.1
5S
k> kMNt;cMnYnBit a nig b edIm,I[eK)ansmPaB 1nn1n 2).1n2(
b
2).1n2(
a
2).1n2)(1n2(
3n2
x> KNna nS nig nnSlim
.
dMeNaHRsay k> kMNt;cMnYnBit a nig b edIm,I[eK)ansmPaB 1nn1n 2).1n2(
b
2).1n2(
a
2).1n2)(1n2(
3n2
eK)an )1n2(b)1n2(a23n2 ba2n)b2a4(3n2 eKTaj
3ba2
2b2a4 naM[ 1b,1a
dUcenH 1a nig 1b . x> KNna nS nig nn
Slim
tamsRmayxagelIenHcMeBaH 1a nig 1b eK)an ³
1nn1n 2).1n2(
1
2).1n2(
1
2).1n2)(1n2(
3n2
KNitviTüaCMuvijBiPBelak
- 76 -
b¤ 1pp1p 2)1p2(
1
2)1p2(
1
2)1p2)(1p2(
3p2
eK)an
n
1p1pn 2).1p2)(1p2(
3p2S
1n
n
1p1pp
2).1n2(
1
2
1
2)1p2(
1
2)1p2(
1
dUcenH
nn 2).1n2(
11
2
1S .
ehIy 2
1
2).1n2(
11lim
2
1Slim nnnn
.
KNitviTüaCMuvijBiPBelak
- 77 -
lMhat;TI43 cMeBaHRKb; *INn eKmansIVútcMnYnBit )a( n kMNt;eday ³
3a0 nig n2nn21n3
1a
3
11a
k> KNna na CaGnuKmn_én n . x> KNnalImIt nn
alim
. dMeNaHRsay k> KNna na CaGnuKmn_én n eKman n2nn21n
3
1a)
3
11(a
n2n
1n
nn21n
n2nn21n
3
11
1a
1a
)1a()3
11(1a
)3
11(a)
3
11(1a
eK)an
1n
0kk2
1n
0k k
1k
3
11
1a
1a
tamsmPaB ))((22 b¤
22
edayyk 1 nig k23
KNitviTüaCMuvijBiPBelak
- 78 -
eK)an k2
1k2
k2
3
11
3
11
3
11
ehtuenH )
3
11
3
11
(3
11
1n
0kk2
1k21n
0kk2
n2
n2
3
11
2
3
3
11
3
11
ehIyeday 2
1a
1a
1a
1a
1a n
0
n1n
0k k
1k
eKTaj )3
11(
2
3
2
1an2
n b¤ n2n
3
34a
dUcenH n2n3
34a
x> KNnalImIt nnalim
eK)an 4)3
34(limalim n2nnn
eRBaH 03
3lim n2n
dUcenH 4alim nn
.
KNitviTüaCMuvijBiPBelak
- 79 -
lMhat;TI44 eK[ a nig b CaBIrcMnYnBitEdl
2ba0
cUrRsayfa bcos
abatanbtan
acos
ab22
dMeNaHRsay Rsayfa
bcos
abatanbtan
acos
ab22
tagGnuKmn_ xtan)x(f Edl )2
,0[x
eK)an xcos
1)x('f 2 .
eday )x(f CaGnuKmn_Cab; nigmanedrIevelIcenøaH )2
,0[x
enaHtamRTwsþIbTtémømFüm enaHman )b,a(c Edl ³
)1(ab
atanbtan
ab
)a(f)b(f)c('f
ehIycMeBaH ]b,a[x Edl 2
ba0
eKman acosxcosbcos enaHeKTaj bcos
1)x('f
acos
122
yk cx eK)an )2(bcos
1)c('f
acos
122
tam )1( nig )2( eKTaj bcos
abatanbtan
acos
ab22
.
KNitviTüaCMuvijBiPBelak
- 80 -
lMhat;TI45 KNnaplbUk ³
nn
23n
2n
1nn Cn.....C9C4CS Edl
)!pn(!p
!nCp
n
dMeNaHRsay KNnaplbUk ³
nn
23n
2n
1nn Cn.....C9C4CS
tameTVFajÚtun nnn
33n
22n
1n
0n
n xC...xCxCxCC)x1( eFVIedrIevelIGgÁTaMgBIrénsmIkarenHeK)an ³
1nnn
23n
2n
1n
1n xnC....xC3xC2C)x1(n KuNGgÁTaMgBIrnwg x eK)an ³
nnn
33n
22n
1n
1n xnC....xC3xC2xC)x1(nx eFVIedrIevelIGgÁTaMgBIrénsmIkarenHeK)an ³
1nnn
22n
1n
2n1n xCn...xC4C)x1(x)1n(n)x1(n yk 1x CYskñúgsmPaBenHeK)an ³
nn
23n
2n
1n
2n1n Cn...C9C4C2).1n(n2.n nn
23n
2n
1n
2n Cn...C9C4C2)]1n(nn2[ dUcenH 2n
n 2).1n(nS
KNitviTüaCMuvijBiPBelak
- 81 -
lMhat;TI46 eK[GnuKmn_ 2x)x(f k> sikSaGefrPaB nig sg;Rkab )c( énGnuKmn_ f . x> eKBinitü n cMnuc n321 A...,,A,A,A sßitenAelIRkab )c( Edl )22,2(A1 nig )a,a(A 1nnn RKb; *INn . cUrkMNt;kUGredaenéncMnuc nA . K> cUrKNna nOA CaGnuKmn_én n rYcKNna n
nOAlim
.
dMeNaHRsay k> sikSaGefrPaB nig sg;Rkab )c( énGnuKmn_ f eKman 2x)x(f mann½yluHRtaEt 2x > EdnkMNt; ),2[D > TisedAGefrPaB -edrIev
2x2
1)x('f
RKb; 2x eK)an 02x2
1)x('f
naM[ f CaGnuKmn_ekInRKb; ,2x . -lImIt
2xlim)x(flim
xx
KNitviTüaCMuvijBiPBelak
- 82 -
-taragGefrPaB x 2
)x('f )x(f
-sMNg;Rkab 2xy:)c( x> kMNt;kUGredaenéncMnuc nA eday 2xy:)c()a,a(A 1nnn
0
2 3 4 5-1-2
2
3
4
-1
0 1
1
X
Y
2xy:)c(
KNitviTüaCMuvijBiPBelak
- 83 -
enaHeK)an 2aa n1n eKman )22,2(A)a,a(A 211 eK)an 21 2
cos22a
322 2
cos22
cos2222a
]bmafavaBitcMeBaHtYTI n KW 1nn 2cos2a
eyIgnwgRsayfavaBitdl;tYTI 1n KW 2n1n 2cos2a
eKman 2aa n1n eday 1nn 2cos2a
enaH 2n1n1n 2cos22
2cos2a
Bit
dUcenH )2
cos2,2
cos2(A 2n1nn
K> cUrKNna nOA CaGnuKmn_én n rYcKNna nn
OAlim
eK)an 2
1n2
nn aaOA eday 1nn 2
cos2a
nig 2n1n 2cos2a
dUcenH 2n2
1n2
n 2cos
2cos2OA
nig 22OAlim n
n
KNitviTüaCMuvijBiPBelak
- 84 -
lMhat;TI47 eK[RtIekaN ABC mYy . emdüanénRtIekaNEdlKUsecj BIkMBUl C,B,A kat;rgVg;carwkeRkARtIekaNerogKñaRtg;
P,N,M . tag cba ,, CaRCug nig cba mmm ,, Caemdüanén ABC . cUrRsayfa
9
16
222 224
4
224
4
224
4
bac
m
cab
m
cba
m cba dMeNaHRsay Rsayfa ³
9
16
222 224
4
224
4
224
4
bac
m
cab
m
cba
m cba tamvismPaB SchwarzCauchy eK)an ³
2222
2222
224
4
224
4
224
4
)(
)(
222 cba
mmm
bac
m
cab
m
cba
m cbacba
eyIgnwgRsayfa 3
4222
222
cba
mmm cba
KNitviTüaCMuvijBiPBelak
- 85 -
eyIgman AMCABC ¬mMusáat;edayFñÚrrYm AC ¦ ehIy C'MAB'AA ¬mMuTl;kMBUl ¦ eKTaj)an B'AA nig C'MA CaRtIekaNdUcKña . eK)an
a
2
m4
a
'AA
'CA.B'AM'A
M'A
B'A
'CA
'AA
ehIy a
2
a m4
amM'A'AAAM
RsaydUcKñaEdr b
2
b m4
bmBN nig
c
2
c m4
cmCP .
tag 222 CPBNAMT
A
B C
M
'A
KNitviTüaCMuvijBiPBelak
- 86 -
2c
4
2b
4
2a
42222
c2
b2
a
2
c
2
c2
b
2
b2
a
2
a
m
c
m
b
m
a
16
1
2
cbammm
)m4
cm()
m4
bm()
m4
am(T
eday )cba(4
3mmm 2222
c2
b2
a
eK)an
2c
4
2b
4
2a
4222
m
c
m
b
m
a
16
1)cba(
4
5T
tamvismPaB SchwartCauchy eK)an )cba(
3
4
mmm
)cba(
m
c
m
b
m
a 2222
c2
b2
a
2222
2c
4
2b
4
2a
4
eKTaj)an ³ )cba(
3
4)cba(
12
1)cba(
4
5T 222222222
naM[ 3
4222
222
cba
mmm cba Bit
dUcenH 9
16
222 224
4
224
4
224
4
bac
m
cab
m
cba
m cba
KNitviTüaCMuvijBiPBelak
- 87 -
lMhat;TI48 eK[ M CaTIRbCMuTm¶n;énRtIekaN ABC . ebIbnÞat; AB b:HeTAnwgrgVg;carwkeRkAénRtIekaN AMC enaHbgðajfa 32
sin
1
sin
1
CBMCMA
dMeNaHRsay bgðajfa 32
sin
1
sin
1
CBMCMA
tamvismPaB SchwarzCauchy
CBMCMACBMCMA
sinsin
4
sin
1
sin
1
G
A
M
B
C
'M
KNitviTüaCMuvijBiPBelak
- 88 -
eyIgnwgRsayfa 3
2CBMsinCAMsin
yk G CacMNuckNþalénRCug ]AB[ . tag c,b,a CaRbEvgRCug AB,CA,BC ehIy cba m,m,m CargVas;emdüanKUsBIkMBUl C,B,A erogKña . eday A CacMnucb:HrvagbnÞat; )AB( CamYyrgVg;p©it 'M enaHeKman 22 GC
3
1GC.GMGA ¬ eRBaH GC
3
1GM ¦
eday cmGC,2
c
2
ABGA eK)an 2
c
2
m3
1
4
c
tamRTwsþIbTemdüaneKman 4
c
2
bam
2222
c
eKTaj )4
c
2
ba(
3
1
4
c 2222
b¤ 222 c2ba .
ehtuenH 4
c3
4
c
2
c2m
2222
c b¤ c2
3mc
ehIy 4
b3
4
ac2b2
4
a
2
cbm
22222222
a
naM[ b2
3ma ehIy a
2
3mb .
tag S CaRkLaépÞénRtIekaN ABC ehIy 'BB,'AA Caemdüan . eK)an ³
KNitviTüaCMuvijBiPBelak
- 89 -
CAMsinAC'.AA4
1CAMsinAC.AM
2
1SCAM b¤ S
4
1S
2
1CAMsinbm
4
1S C'AAaCAM
eKTajJ)an bm
SCAMsin
a
ehIy am
SCBMsin
b
eK)an am
S
bm
SCBMsinCAMsin
ba
Et Csinab2
1S ehIy b
2
3ma nig a
2
3mb
eK)an )1(Csinab3
baCBMsinCAMsin
22
tamRTwsþIbTkUsIunUs Ccosab2bac 222 eKTaj Ccosab2ba
2
ba 2222
eRBaH 222 c2ba
ehtuenH )2(Ccosbc4ba 22 ykTMnak;TMng )2( CMnYskñúg )1( eK)an ³
3
2C2sin
3
2
3
CcosCsin4CBMsinCAMsin
3
2CBMsinCAMsin Bit
dUcenH 32sin
1
sin
1
CBMCMA .
KNitviTüaCMuvijBiPBelak
- 90 -
lMhat;TI49 eK[ 0,, cba . cUrRsaybBa¢ak;fa ³
2
9)(4)(4)(4 3 333 333 33
ba
bac
ac
acb
cb
cba dMeNaHRsay RsaybBa¢ak;fa ³
2
9)(4)(4)(4 3 333 333 33
ba
bac
ac
acb
cb
cba eKman )cb(bc3)cb(cb 333 tamvismPaB GMAM eKman bc2cb eKTaj
2
2
cbbc
naM[ 3)cb(4
3)cb(bc3
eK)an 33333 )cb(4
1)cb(
4
3)cb(cb
eKTaj 3 33 )cb(4cb b¤ 3 33 )cb(4acba
naM[ )1(1)(43 33
cb
a
cb
cba
dUcKñaEdr )2(1)(43 33
ac
b
ac
acb
KNitviTüaCMuvijBiPBelak
- 91 -
nig )3(1)(43 33
ba
c
ba
bac
edaybUkTMnak´TMng )3(,)2(,)1( eK)an ³
ba
c
ac
b
cb
aT
3 Edl
ba
bac
ac
acb
cb
cbaT
3 333 333 33 )(4)(4)(4
eyIgnwgRsayfa 2
3
ba
c
ac
b
cb
a
tag
pba
nac
mcb
eK)an pnmbaaccb naM[
2
pnmcba
eKTaj
2
pnmc
2
pnmb
2
mpna
eK)an p2
pnm
n2
pnm
m2
mpn
ba
c
ac
b
cb
a
KNitviTüaCMuvijBiPBelak
- 92 -
3
n
p
p
n
p
m
m
p
n
m
m
n
2
1
ba
c
ac
b
cb
a
tamvismPaB GMAM eK)an 2
n
p
p
n;2
p
m
m
p;2
n
m
m
n
eK)an 2
33222
2
1
ba
c
ac
b
cb
a
dUcenH 2
9
2
33 T Bit .
KNitviTüaCMuvijBiPBelak
- 93 -
lMhat;TI50 eK[ 1,1 ba nig 1x . cUrRsayfa xxx baba
2
log2loglog
dMeNaHRsay Rsayfa xxx baba
2
log2loglog
eKman a
xx
a log
1log nig
bx
xb log
1log
tamvismPaB SchwarzCauchy eK)an ³
baxx
xxba loglog
4loglog
eyIgnwgRsayfa xba ba
xx 2
log2loglog
4
b¤ 2
log2loglogba
ba xxx
cMeBaH 1a nig 1b eyIgman b.a2ba ( vismPaB GMAM b¤ b.a
2
ba
KNitviTüaCMuvijBiPBelak
- 94 -
naM[ )loglog(2
1)
2(log ba
baxxx
b¤ )1()2
(log2loglogba
ba xxx
mü:ageToteKman ³ baba xxxx log.log2loglog
)2()loglog(2
1loglog
)loglog()log(log2
loglog.log2log)log(log2
2
2
baba
baba
bbaaba
xxxx
xxxx
xxxxxx
tamTMnak;TMng )1( nig )2( eyIgTaj ³
)2
(log2loglog
)2
(log4)loglog(
)2
(log2)loglog(2
1
2
2
baba
baba
baba
xxx
xxx
xxx
dUcenH xxx baba
2
log2loglog
KNitviTüaCMuvijBiPBelak
- 95 -
!> eK[
2,0
.
cUrbgðajfa 1cos
cos
sin
sin2
4
2
4
?
@> eK[GnuKmn_ x
xxxf
sin2
cossin1)(
RKb; IRx bgðajfa 1)(3
1 xf .
#> eK[ 1,, cba . cUrRsaybBa¢ak;fa ³
cbaac
c
cb
b
ba
a acb
9logloglog 222
$> edaHRsaysmIkar ³ 0)1(213)1( 3 223 243 22 xxxxxx %> eKmanGnuKmn_ ³ 9)cot(tan2cottan)( 22 xxxxxf Edl
20
x . cUrrktémøtUcbMputénGnuKmn_enH ?
KNitviTüaCMuvijBiPBelak
- 96 -
^> cUrbgðajfaebI k CacMnYnKUenaH 12 n
k Eckdac;nwg 22 n cMeBaHRKb;cMnYnKt;FmμCati n . &> cMnYnKt; n mYynwgehAfa Good ebIeyIgGacsresr ³ kaaan ....21 Edl kaaa ...,, 21 CacMnYnKt; viC¢man ¬mincaM)ac;xusKña ¦ EdlepÞógpÞat; ³ 1
1....
11
21
kaaa
.
bgðajfaebI n KW Good enaH 82 n nig 92 n k¾ Good .
*> RKb; 1n bgðajfa 2!
!2
1
4
n
n
n
n
.
(> cUrbgðajfaebI n CacMnYnKt;FmμCatimYyenaHeK)an ³
30325
345 nnnn k¾CacMnYnKt;FmμCatiEdr .
!0> bgðajfa 24
13
2
1....
2
1
1
1
nnn RKb; 1n .
!!> cUrbgðajfa nnn 45 35 Eckdac;nwg 120 RKb; INn !@> cUrbgðajfa 3579 496 nnnn Eckdac;nwg 8640 RKb; INn .
KNitviTüaCMuvijBiPBelak
- 97 -
!#> eK[ n CacMnYnKt;viC¢man . cUrbgðajfa ³ )2)....(2)(1( nnn Eckdac;nwg n2 ? !$> bgðajfa 124 22
nn Eckdac;nwg 7 Canic©RKb; INn !%> bgðajfaebI 22 ba Eckdac;nwg 7 enaH a Eckdac;nwg 7 nig b Eckdac;nwg 7. !^> cUrkMNt;RKb;cMnYnKt;viC¢man a nig b ebIeKdwgfa ³ 8511322 ba nig 1764),( baLCM !&> RKb; 1n cUrRsayfa ³ 1)2187,13( 233 nnnnnGCD !*> ebI 1),( nmGCD cUrRsayfa
!!.
)!1(
nm
nm CacMnYnKt; . !(> ebI 1n nig 1)6,( nGCD enaHbgðajfa
)!2(!
)!42(
nn
n
CacMnYnKt; .
@0> cUrRsayfa 504
39 nn CacMnYnKt;ebI INn .
KNitviTüaCMuvijBiPBelak
- 98 -
@!> RKb;cMnYnBit x nig y cUrRsayfa ³ yxyx
21
1
41
1
41
1 @@> RKb; IRx bgðajfa ³
3
4
cos1
1
sin1
122
xx
@#> RtIekaN ABC mYymanmMukñúgCamMuRsYc . cUrRsayfa 6
cos
1
cos
1
cos
1
CBA
@$> kñúgRKb;RtIekaN ABC cUrRsayfa ³ 4
sin
1
sin
1
sin
1222 CBA
@%> RKb;cMnYnBitviC¢man cba ,, cUrRsayfa ³
cbaabc
c
cab
b
bca
a
1
888 222 @*> eK[ cba ,, CaRCugrbs;RtIekaNmYy . cUrRsayfa ³
abccbacbacbabba
3
)(
1
)(
1
)(
1222
@(> bgðajfa 11
cos2sin)1(2
2
a
xaxa RKb; IRx .
KNitviTüaCMuvijBiPBelak
- 99 -
#0> cUrRsayfa 20
7
9sin
3
1
#!> RKb; )1,1(x bgðajfa 21
1
1
144
xx
#@> eK[ zyx ,, CacMnYnBitviC¢man .
cUrRsayfa 2
3
zx
z
zy
y
yx
x
##> eK[ cba ,, CaRCugrbs;RtIekaNmYy . cUrRsayfa ³
abcbacabcacbabccbaabc 3
1
)(8
1
)(8
1
)(8
1333
#$>ebI cba ,, CabIcMnYnBitviC¢manEdl 1 cabcab cUrRsayfa ³
abc
ac
cb
ba
16
16
16
1333
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