lwm pl:ún nig Esn Bisidæ aCMuviijBPBelak 43-eK[RtIekaN ABC...

lwm pl:ún nig Esn Bisidæ briBaØabR&tKNitviTüa nig BaNiC¢kmμ

2 2n

k nk 1

k 3 1 n 3n 3.

2 23 3

Problems and Solutions

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 nig PaK2 ( sRmabsisßBUEkKNitviTüafñakTI11 nig 12 )

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 KNitviTüaCMuviBiPBelak PaK3 EdlGñksikßakMBugkan´ enAkñúgéden£ xMJúáTáneroberogeLIgkñúgeKalbMngTukCaÉksar sRmab´ CaCMnYydl´GñksikßaEdlmanbMngeRtomRbLgsisßBUEkEpñkKNitviTüa nig eRtomRbLgykGaharUbkrN_nana nig müageTotkñúgeKalbMng cUlrYmelIksÞÜyvis&yKNitviTüaenARbeTskm<úCaeyIg[kanÉtrIkceRmIn EfmeTotedIm,Ibeg;InFnFanmnusß[mankanEteRcInedIm,ICYyGPivDÄn_ RbeTsCatirbseyIg . enAkñúgesoven£eyIgxMJúánxitxMRsavRCaveRCIserIsyklMhatýa¨g sRmaMgbMputykmkeFIVdMeNa£Rsayya¨gek,a£k,ayEdlGac[elakGñk gayyl´qabćgcaMGMBIsil,£énkareda£RsayTaMgGsén£ . büEnþeTa£Ca ya¨gNak¾eday kgV£xat nig kMhusqþgedayGectnaRákdCamanTaMg bec©keTs nig GkçraviruTæ . GaRs&yehtuen£ eyIgxMJúCaGñkeroberogrgcaM edayrIkrayCanic©nUvmtiri£KnEbbsSabnaBIsMNakGñksikßakñúgRKbmCÄdæan edIm,ICYyEklMG esovePAen£[ánkanEtsuRkiRtPaBEfmeTot . CaTIbBa©bén£eyIgxMJúGñkeroberogsUmeKarBCUnBrdlGñksikßaTaMgGs´ [mansuxPaBmaMmYn nig TTYlC&yCMn£RKb´Parkic© .

átdMbgéf¶TI 17 mifuna 2009 GñkniBnæ lwm plþún

EpñkRbFanlMhat

Problems

KNitviTüaCMuvijBiPBelak

EpñkRbFanlMhat´

- 1 -

n1-eK[ CacMnYnKt;FmμCati . cMeBaHRKb;cMnYnBit cUrRsayfa ³

1nn2n2 2cos21sin21 2-cUrRsaybBa¢ak;fa 333 )

b1

a1

)(ba(2ab

ba

cMeBaHRKb;cMnYnBitviC¢man a nig b . 3-cUrkMnt;RKb;GnuKmn_ IRIR:f EdlepÞógpÞat; ³

))y(f)x(f)(yx()yx(f 22 cMeBaHRKb; IRy;x 4-KNnaplbUk

101

0i2

ii

3i

x3x31

xS

Edl ...,3,2,1i;101

ixi .

5-cUrkMnt;témøEpñkKt;énplbUk ³

10000

1......

3

1

2

1S .

6-KNnatémøénplKuN )44cot1).....(3cot1)(2cot1)(1cot1(P oooo


- 2 -

z;y;x7-eK[ CacMnYnBitEdl

3zxyzxy

5zyx

cUrbgðajfa 3

13z1

8-cMeBaHRKb;cMnYnKt;viC¢man eK[ n

3 23 23 2 1n2n1n1n2n

1)n(f

KNna )999999(f)999997(f...)5(f)3(f)1(f 9-cUrbgðajfacMeBaHRKb;cMnYnKt;viC¢man eKman ³ n

n21

....2n

11n

1n2)1n2(

1...

4.21

2.11

10-bgðajfa ³

16

1)

20

27cos

2

1)(

20

9cos

2

1)(

20

3cos

2

1)(

20cos

2

1(

11-eK[RtIekaN mYymanmMukñúgCamMuRsYc . cUrRsayfa ³ ABC

2

23CcosBcosAcos

12-eK[ CacMnYnkMupøicEdl 21 z;z 0r|z||z| 21 .

bgðajfa 2

2

212

21

2

212

21

r

1

zzr

zz

zzr

zz


13-eKyk CacMnYnkMupøicEdlepÞógpÞat; n21 z....;;z;z

TMnak;TMng ³ 1n,...,2,1,0k;0z)kn(iz)1k( 1 kk

k-kMnt; ebIeKdwgfa 0z nn210 2z....zzz

x-cMeBaHtémø Edl)ankMnt;xagelIcUrbgðajfa ³ 0z

!n

)1n3(|z|....|z||z||z|

n2

n2

22

12

0

14-eK[ CacMnYnkMupøicedaydwgfa ³ 321 z;z;z

0zzzzzzzzz 133221321 . cUrbgðajfa |z||z||z| 321 15-eK[sIVúténcMnYnBit ³

1n;n

111

n

111a

22

n

cUrbgðajfa 20321 a

1....

a

1

a

1

a

1 CacMnYnKt; .

16-eK[ CacMnYnBitviC¢manEdl z;y;x 1xyz . cUrRsayfa ³

2

1

1x)1z(

1

1z)1y(

1

1y)1x(

1222222

- 3 -


17-eK[ CacMnYnBitviC¢manEdl c;b;a 1abc . cUrRsayfa ³ 0)aa(c)cc(b)bb(a 222 18-eK[ k CacMnYnKt;mYy ehIyeKyk ³ 11kk1kkn 3 23 2 cUrbgðajfa CacMnYnKt;mYy . 23 n3n

19-eK[GnuKmn_ IRIR:f eday ³

1xx

)1xx()x(f 36

32

cUrrktémøGb,brmaénGnuKmn_enH ? 20-cUrKNnatémøplKuN ³ )29tan3).....(2tan3)(1tan3(P ooo 21-eK[ nig . 0a...,,a,a n21 0x...,,x,x n21

cUrRsayfa ³

n21

2n21

n

2n

2

22

1

21

x....xx

)a...aa(

x

a...

x

a

x

a

- 4 -


22-eK[ CabIcMnYnBitviC¢manEdl . c;b;a 1abc

cUrbgðajfa

2

3

)ba(c

1

)ac(b

1

)cb(a

1333

23-eK[ . cUrRsaybBa¢ak;fa ³ 0z;y;x

2

1

y3x2z

z

x3z2y

y

z3y2x

x

24-eK[sIVút epÞógpÞat;lkçxNÐ ³ n21 a....;;a;a

...;|1a||a|;0a 121 nig |1a||a| 1nn . bgðajfa

2

1

n

a...aa n21

25-eK[ CabIcMnYnBitviC¢man . cUrbgðajfa ³ c;b;a

)abc

cba1(2

a

c1

c

b1

b

a1 3

26-eK[ CaRbEvgRCugrbs;RtIekaNmYy . c;b;a

cUrbgðajfa ³ cbabacacbcba 27- KNnaplbUk ³

n

1k1k1kkk

k

n)23)(23(

6S rYcTajrktémø . n

nSlim

- 5 -


28-KNnaplKuNxageRkam ³ n

242n

0kk

2n

2

xtan.....

4

xtan.

2

xtan.xtan

2

xtanP

nk

29-KNnaplKuNxageRkam ³

n

0k

2k

2n

k)

2

xtan1(P

30 eK[RtIekaN mYymanmMukñúgCamMuRsYc . ABC

cUrbgðajfa ³ CsinBsinAsin32CsinBsinAsin 222 31-cUrbgðajfa 1

iz32

iz6

luHRtaEt

3

1|z|

32-eK[ CacMnYnkMupøicEdlmanm:UDúlesμI 1 n321 z....;;z;z;z

eKtag

n

1k k

n

1kk )

z

1()z(Z .

cUrbgðajfa 2nZ0 33-eK[cMnYnkMupøic z Edl 1|z| . cUrbgðajfa ³ 4|z1||z1|2 2

- 6 -


34-eK[sIVúténcMnYnBit INnn )v( kMnt;eday ³ 5v0 nig TMnak;TMngkMenIn 0n;1v2v 2

n1n bgðajfa 22

nn2

1n1n )1vv(1vv rYcKNna CaGnuKmn_én n . nv

35-eK[sIVúténcMnYnBit INkk )u( kMnt;eday ³ u nig TMnak;TMngkMenIn 90

n

1p

pk

pn1k uCu

Edl )!pn(!p

!nCp

n .

cUrKNna ku CaGnuKmn_én k nig n 36-eK[sIVúténcMnYnBit INnn )u( kMnt;eday ³ nig TMnak;TMngkMenIn 1u0 1u4u2u n

2n1n

cUrKNna CaGnuKmn_én n nu

37-eK[ CamMukñúgrbs;RtIekaN mYy . C;B;A ABC

cUrbgðajfa 332

Ccot

2

Bcot

2

Acot

38-eK[ CamMuRsYckñúgrbs;RtIekaN mYy . C;B;A ABC

cUrbgðajfa 3)31()Ctan1)(Btan1)(Atan1(

- 7 -


39-eK[cMnYnKt;viC¢man n . eKdwgfa n Ecknwg 7 [sMNl; % ehIy nEcknwg [sMNl;# 8

k> etIcMnYn n enaHEcknwg [sMNl;b:un μan ? 56

x> rkcMnYn enaHedaydwgfa n 5626n5616 . 40-eK[GnuKmn_ f kMnt;elI IR ehIyepÞogpÞat;TMnak;TMng³

543

232 x1x4

x1

x1f

x1

1xfx

cUrKNnaGaMgetRkal³ . 1

0

dx.xfI

41-enAkñúgtMruyGrtUNrma:l;manTisedAviC¢man

k,j,i,0 manÉkta enAelIGkS½ eK[BIrcMnuc cm1 )0,2,0(A nig . Cabøg;mansmIkar )1,2,1(B P 04zy2x2 . cUrsresrsmIkarbøg; Q kat;tamcMnuc nig BehIypÁúMCamYy A

bøg; )anmMuRsYcmYymantémø )P(4

.

42-eK[ CabIcMnYnBitviC¢man . cUrRsayfa ³ c;b;a

)cabcab(9)2c)(2b)(2a( 222

- 8 -


43-eK[RtIekaN mYymanmMukñúgCamMuRsYc . ABC

k>bgðajfa 2

3CcosBcosAcos

x>bgðajfa 8

1CcosBcosAcos

K>bgðajfa 8

1

2

Csin

2

Bsin

2

Asin

44-eK[RtIekaN mYy . tag p CaknøHbrimaRt nig ABC R CakaMrgVg;carwkeRkARtIekaN . cUrbgðajfa ³ k>

R4

p

2

Ccos

2

Bcos

2

Acos

x> 2

Ccos

2

Bcos

2

Acos4CsinBsinAsin

45-eK[RtIekaN mYy . tag r nig ABC R erogKañCakaMrgVg; carwkkñúg nig carwkeRkARtIekaN . k> cUrbgðajfa ³

R

r1CcosBcosAcos

x> ebI CaRtIekaNEkgenaHcUrRsayfa ABC r)12(R 46-edaHRsaysmIkar ³

1x3x4.....x16x4x n

- 9 -


47-cUrbgðajfaenAkñúgRKb;RtIekaNeKmanTMnak;TMng ³ 2R2

abcCcoscBcosbAcosa

Edl CaRCug nig c;b;a RCakaMrgVg;carwkeRkARtIekaN . 48-KNnaplKuN ³

n

1k2k2

k2

n2tan1

x2tan1P Edl 2n2

|x|

49-eK[ nig x CacMnYnBitepÞógpÞat; ³ d;c;b;a

d

x4sin

c

x3sin

b

x2sin

a

xsin Edl Zk;kx

cUrbgðajfa )ca3(b)db4(a 4223

50-eK[ CaRtIekaNmYy . cUrbgðajfa ³ ABC

k> 12

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

x> 9

3

2

Ctan

2

Btan

2

Atan

51-eK[ CaRtIekaNmYy . cUrbgðajfa ³ ABC

2

Ctan.

2

BAtan

ba

ba

- 10 -


52- cUrbgðajfa ³

2

2

ba1)xcosbx)(sinxcosax(sin

53-eK[RtIekaN mYymanmMukñúgCamMuRsYc ABC

nig RCug nigépÞRkLa K . cUrRsayfa ³ c;b;a

2

cbaK4acK4cbK4ba

222222222222

54-RbsinebI )z1)(y1)(x1(xyx Edl 1z;y;x0 cUrbgðajfa

4

3)y1(z)x1(y)z1(x

55-eK[ CaRbEvgRCugrbs;RtIekaNmYyEdlman c;b;a

brimaRtesμI 2 . cUrRsayfa ³ 2abc2cba

2

3 222

56-eK[

n

1kn

2

3

2

2

22

k

2

n3

n....

3

3

3

2

3

1

3

kS

KNna CaGnuKmn_én n rYcTajrk . nS nn

Slim

- 11 -


57-RtIekaN mYymanRCug ABC cAB;bAC;aBC tag

2

cbap

CaknøHbrimaRt r nig R

CakaMrgVg;carwkkñúgnigkaMrgVg;carwkeRkAénRtIekaN . cUrRsaybBa¢ak;TMnak;TMngxageRkam ³ k> Rr4rpabcabc 22

x> Rr8r2p2cba 22222

K> p

r

2

Ctan

2

Btan.

2

Atan

X>p

rR4

2

Ctan

2

Btan

2

Atan

g> r

p

2

Ccot

2

Bcot

2

Acot

58-eK[ CabIcMnYnBitviC¢manEdl z;y;x 1zyx . cUrRsaybBa¢ak;fa 81

z

11

y

11

x

1

59-eK[ CabIcMnYnBitviC¢manEdl c;b;a 1cba 222 bgðajfa

abc

)cba(23

c

1

b

1

a

1 333

222

- 12 -

EpñkdMeNa¼Rsay

Solutions


lMhat´TI1 eK[ n CacMnYnKt;Fm μCati . cMeBaHRKb;cMnYnBit cUrRsayfa ³ 1nn2n2 2cos21sin21 .

dMeNa¼Rsay Rsayfa ³ 1nn2n2 2cos21sin21 tag nig Edl 2sin21x 2cos21y 0y;0x

eKman 22 cos21sin21yx

x4y

4yx

)cos(sin22yx 22

eday y enaH 4 b¤ x0 0x 4 yk n2n2 )cos21()sin21(T

T

T nn

nn

)x4(x)x(f

yx

Edl . 4x0

- 13 -


eyIgman 1n1n )x4(nnx)x('fdxdT

)x(g)4x2(n

)x(g])x4(x[n

])x4(x[n 1n1n

Edl 0)x4(...)x4(xx)x(g 2n3n2n . ebI eKTajb¤s 04x2 2x . cMeBaH eK)an 2x 1nnn 2)24(2)2(f taragGefrPaBén nn )x4(x)x(f

x 0 2 4 )x('f

- 14 -

)x(f .

.

1n2

tamtaragxagelIeKTaj [4;0]x2)x(f 1n . dUcenH 1nn2n2 2cos21sin21 RKb; IR


lMhat´TI2 cUrRsaybBa¢ak;fa 333 )

b1

a1

)(ba(2ab

ba

cMeBaHRKb;cMnYnBitviC¢man a nig b .

dMeNa¼Rsay Rsayfa ³

333 )b1

a1

)(ba(2ab

ba

edayKuNGgÁTaMgBIrénvismPaBnwg 3 ab eK)an ³

3 23 23 2 )ba(2ba tag 3 ax nig 3 by eK)an (*))yx(2yx 3 3322 tamvismPaB eKman ³ GMAM

2433336 yx3yxyxx nig 4233336 yx3yxyxy

bUkvismPaBTaMgBIrenHGgÁnigGgÁeK)an ³ 42246336 yx3yx3yyx4x

- 15 -


EfmGgÁTaMgBIrénvismPaBnwg eK)an 66 yx

322233

6422466336

)yx()yx(2

yyx3yx3x)yyx2x(2

eKTaj 3 3322 )yx(2yx naM[ (*)Bit . dUcenH 333 )

b1

a1

)(ba(2ab

ba

RKb; . 0b;0a

lMhat´TI3 cUrkMnt;RKb;GnuKmn_ IRIR:f EdlepÞógpÞat; ³

))y(f)x(f)(yx()yx(f 22 cMeBaHRKb; IRy;x

dMeNa¼Rsay kMnt;rkGnuKmn_ :f eKman ))y(f)x(f)(yx()yx(f 22

yk eK)an yx 0)0(f yk enaH 0y;1x )1(f))0(f)1(f()1(f yk eK)an f 1y;tx ))1(f)t(f)(1t()1t( 2

- 16 -


eday )1(f)1(f enaH )1())1(f)t(f)(1t()1t(f 2

yk eK)an 1y;tx )2())1(f)t(f)(1t()1t(f 2

pÞwm nig eK)an ³ )1( )2(

)1(f)t(f)1t()1(f)t(f)1t( eKTaj t)1(f)t(f edaytag IR)1(fk dUcenH kx)x(f CaGnuKmn_EdlRtUvrk .

lMhat´TI4 KNnaplbUk

101

0i2

ii

3i

x3x31

xS

Edl ...,3,2,1i;101

ixi .

dMeNa¼Rsay KNnaplbUk

101

0i2

ii

3i

x3x31

xS

eyIgman 1 33332 )1x(x)x1(xx3x3

- 17 -


tag 33

3

2

3

)x1(x

x

x3x31

x)x(f

eK)an 3

i3

i

3i

i)x1(x

x)x(f

ehIy 3

i3

i

3i

ix)x1(

)x1()x1(f

eK)an 1x)x1(

)x1(

)x1(x

x)x1(f)x(f

3i

3i

3i

3i

3i

3i

ii

eKTaj )x1(f1)x(f ii eday

101i

xi enaH 101

i101101

i1x1 i

eK)an

101

0i

101

0i

101

0ii )

101i101

(f1101

if)x(fS

S102101

i101f102S

101

0i

eKTaj 512

102S ( eRBaH

101

0i

101

0i 101i101

f101

if )

- 18 -


lMhat´TI5 cUrkMnt;témøEpñkKt;énplbUk ³

10000

1......

3

1

2

1S .

dMeNa¼Rsay kMnt;EpñkKt;rbs; S ³ eyIgBinitü 1k;

1kk

2

kk

2

k

1

eKTaj )1kk(2k

1

eK)an 198)110000(2)1kk(2S10000

2k

müa:geTot 1k;k1k

2

kk

2

k

1

eKTaj )k1k(2k

1

eK)an 197)210001(2)k1k(2S10000

2k

dUcenHEpñkKt;én S KW . 197

- 19 -


lMhat´TI6 KNnatémøénplKuN

)44cot1).....(3cot1)(2cot1)(1cot1(P oooo

dMeNa¼Rsay KNna

)44cot1).....(3cot1)(2cot1)(1cot1(P oooo eyIgBinitü

asinacosasin

asinacos

1acot1

eday )a45sin(2acosasin o ehtuenH

asin

)a45sin(2acot1

o

eyIg)an

o

o

o

o

44

1a

44

1a

o

asin)a45sin(

2acot1P

22ooo

ooo44

244sin......2sin.1sin

1sin.....43sin.44sin.2P

dUcenH P . 22ooo 2)44cot1).....(2cot1)(1cot1(

- 20 -


lMhat´TI7 eK[ CacMnYnBitEdl z;y;x

3zxyzxy

5zyx

cUrbgðajfa 3

13z1

dMeNa¼Rsay bgðajfa

313

z1 eKman 5zyx

eyIg)an z5yx

22

22

zz1025)yx(

)z5()yx(

eday 3zxyzxy

eyIg)an )yx(z3xy 2zz53xy

)z5(z3xy

eyIgman ( 0xy4)yx()yx 22

eKTaj ( 0)zz53(4)zz1025 22

- 21 -


0)13z3)(1z(

0)1z(13)1z(z3

0)13z13()z3z3(

013z10z3

0z4z2012zz1025

2

2

22

eKTaj 3

13z1 .

lMhat´TI8 cMeBaHRKb;cMnYnKt;viC¢man n eK[

3 23 23 2 1n2n1n1n2n

1)n(f

KNna )999999(f)999997(f...)5(f)3(f)1(f

dMeNa¼Rsay KNna )999999(f)999997(f...)5(f)3(f)1(f eKman

3 23 23 2 1n2n1n1n2n

1)n(f

3 233 2 )1n()1n)(1n()1n(

1

- 22 -


KuNPaKyknwgPaKEbgnwg 33 1n1n )1n1n(

21

)1n()1n(

1n1n)n(f 33

3333

33

eK)an 50)010(21

)999999(f...)3(f)1(f 3 6 dUcenH 50)999999(f)999997(f...)5(f)3(f)1(f

lMhat´TI9 cUrbgðajfacMeBaHRKb;cMnYnKt;viC¢man n eKman ³

n21

....2n

11n

1n2)1n2(

1...

4.21

2.11

dMeNa¼Rsay bgðajfa ³

n21

....2n

11n

1n2)1n2(

1...

4.21

2.11

tag n2)1n2(

1.....

4.21

2.11

T

)

n2

1....

4

1

2

1(2

n2

1.....

3

1

2

11

)n2

1

1n2

1(....)

4

1

3

1()

2

11(

- 23 -


n2

1.....

2n

1

1n

1

)n1

....31

21

1(n21

....31

21

1T

dUcenH n21

....2n

11n

1n2)1n2(

1...

4.21

2.11

lMhat´TI10 bgðajfa ³

16

1)

20

27cos

2

1)(

20

9cos

2

1)(

20

3cos

2

1)(

20cos

2

1(


16

1)

20

27cos

2

1)(

20

9cos

2

1)(

20

3cos

2

1)(

20cos

2

1(

tag )20

27cos

2

1)(

20

9cos

2

1)(

20

3cos

2

1)(

20cos

2

1(P

3

0n

n

203

cos21

eKman )2

asin43(

2

1

2

asin21

2

1acos

2

1 22 ehIy )

2

asin43(

2

asin

2

asin4

2

asin3

2

a3sin 22

- 24 -


naM[ 2

asin

2

a3sin

2

asin43 2

ehtuenH 2

asin

2

a3sin

2

1acos

2

1

yk 20

a 3n eK)an

403

sin

403

sin.

2

1

20

3cos

2

1n

1n

n

eK)an 16

1

40sin

4081

sin.

16

1

403

sin

403

sin.

2

1P

3

0nn

1n

eRBaH 40

sin)40

2sin(40

81sin

. dUcenH

16

1)

20

27cos

2

1)(

20

9cos

2

1)(

20

3cos

2

1)(

20cos

2

1(

- 25 -


lMhat´TI11 eK[RtIekaN mYymanmMukñúgCamMuRsYc . cUrRsayfa ³ ABC

223

CcosBcosAcos


223

CcosBcosAcos tamvismPaB Cauchy-Schwartz eyIg)an³

)1()CcosBcosA(cos3)CcosBcosAcos( 2 tag CcosBcosAcosT

2

CBcos

2A

sin22A

sin21

2CB

cos2

CBcos2

2A

sin21

2

2

eRBaH 2

Asin

2

A

2cos

2

CBcos

.

eday B nig CamMuRsYcenaH C2

C0;2

B0

eKTaj

42CB

4

naM[ 12

CBcos

- 26 -


ehtuenH 23

)2A

sin21(21

23

2A

sin22A

sin21T 22 b¤ )2(

23

CcosBcosAcos tamTMnak;TMng nig eKTaj)an³ )1( )2(

2

9)CcosBcosAcos( 2

dUcenH 2

23CcosBcosAcos .

lMhat´TI12 eK[ CacMnYnkMupøicEdl 21 z;z 0r|z||z| 21 .

bgðajfa 2

2

212

21

2

212

21

r

1

zzr

zz

zzr

zz


2

2

212

21

2

212

21

r

1

zzr

zz

zzr

zz

tag nig )x2sinix2(cosrz1 )y2siniy2co(rz2 Edl IRy;IRx .

- 27 -


eK)an ³

)yxcos(

)yxcos(.

r

1

])yxcos()yxsin(i2)yx(cos2[r

)yxcos()yxsin(i2)yxcos()yxcos(2

])y2x2sin(.i)y2x2cos([rr

])y2sinx2(sini)y2cosx2(cos[r

zzr

zz

2

2221

221

dUcKñaEdr )xysin(

)xysin(

r

1

zzr

zz

212

21

eK)an ³

)xy(sin

)xy(sin

)yx(cos

)yx(cos

r

1

zzr

zz

zzr

zz2

2

2

2

2

2

212

21

2

212

21

eday )yx(cos

)yx(cos

)yx(cos 22

2

eRBaH 1)yx(cos 2

ehIy )yx(sin)xy(sin

)xy(sin 22

2

eRBaH 1)yx(sin 2

1)yx(sin)yx(cos)xy(sin

)xy(sin

)yx(cos

)yx(cos 222

2

2

2

dUcenH 2

2

212

21

2

212

21

r

1

zzr

zz

zzr

zz

.

- 28 -


lMhat´TI13 eKyk z CacMnYnkMupøicEdlepÞógpÞat;TMnak;TMng n21 z....;;z;

1n,...,2,1,0k;0z)kn(iz)1k( 1 kk k-kMnt; ebIeKdwgfa 0z n

n210 2z....zzz


!n

)1n3(|z|....|z||z||z|

n2

n2

22

12

0

dMeNa¼Rsay k-kMnt; ebIeKdwgfa 0z n

n210 2z....zzz

eKman ( 0z)kn(iz)1k k1k eK)an

1k

kn.i

z

z

k

1k

)!pn(!p

!nC;Ci

z

z

1k

kn.i

z

z

pn

pn

p

0

p

1p

0k

)1p(

0k k

1k

eKTaj z ....,2,1,0p;Czi pn0

pp

- 29 -


- 30 -

neday b¤ n210 2z....zzz

n

0p

np 2)z(

man n0

n

0p

n

0p

ppn0p )i1(ziCz)z(

eK)an nn0 2)i1(z

eKTaj nn

n

0 )i1()i1(

2z

dUcenH . n0 )i1(z


!n

)1n3(|z|....|z||z||z|

n2

n2

22

12

0

edayGnuvtþn_vismPaB GMAM eyIg)an

2nn

21n

20n

20

2n

21

20 )C(....)C()C(zz...zz

!n

)1n3(

n

)1n(....)2n2()1n2(n2

!n

2

))1n)....(2n2)(1n2(n2!n

2

!n!n

)!n2(.2Cz

n

nn

n

nnn2

20

dUcenH !n

)1n3(|z|....|z||z||z|

n2

n2

22

12

0

.


lMhat´TI14 eK[ CacMnYnkMupøicedaydwgfa ³ 321 z;z;z

0zzzzzzzzz 133221321 . cUrbgðajfa | |z||z||z 321

dMeNa¼Rsay bgðajfa | |z||z||z 321 eKman 0zzzzzzzzz 133221321 eKTaj

)2(0)zz(zzz

)1(zzz

21321

321

yk ( CMnYskñúg ( eyIg)an ³ )1 )2

0zzz 2321 naM[ | |z|.|z||z 21

23

dUcKñaEdr | nig | |z|.|z||z 312

2 |z|.|z||z 322

1

eyIg)an ³ |z||z||z||z||z||z||z||z||z| 133221

23

22

21

b¤ (| 0|)z||z(||)z||z(||)z||z 213

232

221

dUcenH | . |z||z||z 321

- 31 -


lMhat´TI15 eK[sIVúténcMnYnBit ³

1n;n

111

n

111a

22

n

cUrbgðajfa 20321 a

1....

a

1

a

1

a

1 CacMnYnKt; .


20321 a

1....

a

1

a

1

a

1 CacMnYnKt;

eyIgman 1n;n

111

n

111a

22

n

eK)an 22

n

n

111

n

111

1

a

1

1n2n21n2n2

4

1

n

111

n

111

4

n

22

22

- 32 -


7)291(4

1

)76129(....)513()15(4

1

1n2n21n2n24

1

a

1 20

1n

2220

1n n

dUcenH 7a

1....

a

1

a

1

a

1

20321

CacMnYnKt; .

lMhat´TI16 eK[ CacMnYnBitviC¢manEdl z;y;x 1xyz . cUrRsayfa ³

2

1

1x)1z(

1

1z)1y(

1

1y)1x(

1222222


2

1

1x)1z(

1

1z)1y(

1

1y)1x(

1222222

eyIgman 2x2yx1y)1x( 2222

eday x xy2y22

- 33 -


eKTaj )1xxy(21y)1x( 22

naM[ 1xxy

1.

2

1

1y)1x(

122

eKman enaHeKGacyk 1xyz c

az,

b

cy;

a

bx

Edl . 0c;0b;0a

eK)an a

cba1

a

b

a

c1xxy

ehtuenH 1cba

a.

2

1

1y)1x(

122

RsaydUcKñaEdreK)an ³

3cba

c.

2

1

1x)1z(

1

2cba

b.

2

1

1z)1y(

1

22

22

eFIVplbUkvismPaB nig eK)an ³ )2(;)1( )3(

2

1

1x)1z(

1

1z)1y(

1

1y)1x(

1222222

- 34 -


lMhat´TI17 eK[ a CacMnYnBitviC¢manEdl abcc;b; 1 . cUrRsayfa ³

0)aa(c)cc(b)bb(a 222

dMeNa¼Rsay bgðajfa 0)aa(c)cc(b)bb(a 222 eday abc enaHeKGactag 1 2

2

2

2

2

2

x

zc;

z

yb;

y

xa

Edl . 0z;0y;0x

vismPaBxagelIsmmUl ³

Bit0z

xy

y

zx

y

zx

x

yz

x

yz

z

xy

0xy

z2

y

xz2

zx

y2

x

zy2

yz

x2

z

yx2

0xy

z

y

xz

zx

y

x

zy

yz

x

z

yx

0)y

x

y

x(

x

z)

x

z

x

z(

z

y)

z

y

z

y(

y

x

2

22

2

22

2

22

2

4

222

4

222

4

22

2

4

222

4

222

4

22

4

4

2

2

4

4

2

2

4

4

2

2

- 35 -


lMhat´TI18 eK[ k CacMnYnKt;mYy ehIyeKyk ³

11kk1kkn 3 23 2 cUrbgðajfa n CacMnYnKt;mYy . 23 n3

dMeNa¼Rsay bgðajfa n CacMnYnKt;mYy 23 n3

eKmansmPaB ³ )cabcabcba)(cba(abc3cba 222333

tag n1c;1kkb;1kka 3 3 22

eK)an ¬ eRBaH ¦ 0)n1(3)n1(k2 3 1ab

b¤ 0n33nn3n31k2 32 eKTaj 2k2n3n 23 CacMnYnKt;RKb;cMnYnKt; k . dUcenH n CacMnYnKt;mYy . 23 n3

- 36 -


lMhat´TI19 eK[GnuKmn_ IRIR:f eday ³

1xx

)1xx()x(f 36

32

cUrrktémøGb,brmaénGnuKmn_enH ?

dMeNa¼Rsay rktémøGb,brmaénGnuKmn_ )x(f

1xx

)1xx()x(f 36

32

eyIgsMKal;eXIjfa 1)o(f kMnt; . GnuKmn_Gacsresr ³

1)x

1x(3)

x

1x(

)1x

1x(

1x

1x

)1x

1x(

)x(f3

3

33

3

tag x

1xt Edl b¤ 2|t| [;2[]2;]t

eK)an 1t3t

)1t()t(g)x(f 3

3

.

eyIgman 23

3232

)1t3t(

)1t)(1t(3)1t3t()1t(3)x('g

- 37 -


23

22

23

2332

)1t3t(

)2t2t()1t(3

)1t3t(

)1ttt1t3t()1t(3)t('g

ebI eK)an 0)t('g 1t b¤ 02t2t2 smmUl 31t;31t;1t 321 eday [;2[]2;]t enaHeKTaj 31t eday 0

)1t3t(

)1t(323

2

RKb; t [;2[]2;]

enaH mansBaØadUc )t('g 2t2t2 . Rtg;cMnuc 31t GnuKmn_ bþÚrsBaØaBI eTA )t('g )( )(

naM[ mantémøGb,brmaRtg; )t(g 31t KW 332)32(3

32

3)31(g

dUcenHtémøGb,brmaén f KW 332 .

∞∞∞∞∞∞

- 38 -


lMhat´TI20 cUrKNnatémøplKuN ³

)29tan3).....(2tan3)(1tan3(P ooo

dMeNa¼Rsay KNnatémøplKuN ³

29

1k

o

ooo

ktan3

)29tan3).....(2tan3)(1tan3(P

eKman o

oo

o

oo

kcos

ksinkcos3

kcos

ksin3ktan3

o

oo

kcos

)k30cos(2

eK)an

29

1ko

oo

kcos

)k30cos(2P

29oooo

oooo29

229cos28cos.....2cos1cos

1cos2cos.....28cos29cos2

dUcenH 29ooo 2)29tan3).....(2tan3)(1tan3(

- 39 -


lMhat´TI21 eK[ a nig x . cUrRsayfa ³ 0a...,,a, n21 0x...,,x, n21

n21

2n21

n

2n

2

22

1

21

x....xx

)a...aa(

x

a...

x

a

x

a


n21

2n21

n

2n

2

22

1

21

x....xx

)a...aa(

x

a...

x

a

x

a

tag n21

2n21

n

2n

2

22

1

21

n x...xx

)a...aa(

x

a...

x

a

x

aT

cMeBaH n eK)an 1 00x

a

x

aT

1

21

1

21

1 Bit

cMeBaH eK)an ³ 2n

21

221

2

22

1

21

2 xx

)aa(

x

a

x

aT

Bit0

)xx(xx

)xaxa(

)xx(xx

)aa(xx)xaxa)(xx(

2121

21221

2121

221211

222

2121

- 40 -


]bmafavaBitdl;tYTI k KW 0Tk Bit eyIgnwgRsayfavaBitdl;tYTI 1k KW 0T 1k Bit eKman 0Tk ¬ kar]bmaxagelI ¦ eK)an 0

x...xx

)a...aa(

x

a...

x

a

x

a

k21

2k21

k

2k

2

22

1

21

eKTaj k21

2k21

k

2k

2

22

1

21

x...xx

)a...aa(

x

a...

x

a

x

a

EfmGgÁTaMgBIrnwg 1k

21k

x

a

eK)an ³

1k

21k

k21

2k21

1k

21k

k

2k

2

22

1

21

x

a

x...xx

)a...aa(

x

a

x

a...

x

a

x

a

eday 1k21

21k21

1k

21k

k21

2k21

x...xx

)a...aa(

x

a

x...xx

)a...aa(

eKTaj)an ³

1kk21

21kk21

k

21k

k

2k

2

22

1

21

xx...xx

)aa...aa(

x

a

x

a...

x

a

x

a

naM[ 0T 1k Bit . dUcenH

n21

2n21

n

2n

2

22

1

21

x....xx

)a...aa(

x

a...

x

a

x

a

.

- 41 -


lMhat´TI22 eK[ a CabIcMnYnBitviC¢manEdl abcc;b; 1 . cUrbgðajfa

2

3

)ba(c

1

)ac(b

1

)cb(a

1333

dMeNa¼Rsay Rsayfa

2

3

)ba(c

1

)ac(b

1

)cb(a

1333

eKman ³

)ba(c)ac(b)cb(ac

1

b

1

a

1

)ba(cc

1

)ac(bb

1

)cb(aa

1

T

)ba(c

1

)ac(b

1

)cb(a

1T

2222

333

eday 222

)cabcab(abc

cabcab

c

1

b

1

a

1

ehIy )cabcab(2)ba(c)ac(b)cb(a eKTaj

2

3

2

)abc(3

2

cabcabT

3 2

dUcenH

2

3

)ba(c

1

)ac(b

1

)cb(a

1333

.

- 42 -


lMhat´TI23 eK[ . cUrRsaybBa¢ak;fa ³ 0z;y;x

2

1

y3x2z

z

x3z2y

y

z3y2x

x

dMeNa¼Rsay Rsayfa

2

1

y3x2z

z

x3z2y

y

z3y2x

x

tag y3x2z

z

x3z2y

y

z3y2x

xT

)zxyzxy(5zyx

)zyx(T

yz3xz2z

z

xy3yz2y

y

xz3xy2x

xT

222

2

2

2

2

2

2

2

eK)an 2

1

)zxyzxy(5zyx

)zyx(

2

1T 222

2

0)zxyzxy(5zyx

)xz()zy()yx(

2

1

2

1T 2222

222

eKTaj)an 2

1T .

dUcenH 2

1

y3x2z

z

x3z2y

y

z3y2x

x

.

- 43 -


lMhat´TI24 eK[sIVút a epÞógpÞat;lkçxNÐ ³ n21 a....;;a;

...;|1a||a|;0a 121 nig | |1a||a 1nn . bgðajfa

2

1

n

a...aa n21

dMeNa¼Rsay Rsayfa

2

1

n

a...aa n21

eyIgman |1a||a| 1nn naM[ 1a2aa 1n

21n

2n

eK)an

n

1kk

2k

1n

1k1k

1n

1k

21k

2k )1a2a()1a2a()a(

n

1k

n

1kk

2k

n

1k

2k

21n n)a(2)a()a(a

eKTaj 2

n

2

naa

21n

n

1kk

dUcenH 2

1

n

a...aa n21 .

- 44 -


lMhat´TI25 eK[ a CabIcMnYnBitviC¢man . cUrbgðajfa ³ c;b;

)abc

cba1(2

a

c1

c

b1

b

a1 3

dMeNa¼Rsay Rsayfa )

abc

cba1(2

a

c1

c

b1

b

a1 3

tag

a

c1

c

b1

b

a1T

)abc

cba1(213

abc

)cba(2

1abc

cba

abc

)cba(21

abc

)cba(3

1abc

c3

abc

b3

abc

a3

1c

c

b

c

a

c

c

b

b

b

a

b

c

a

b

a

a

ab

c

c

a

a

b

a

c

c

b

b

a2

33

333

333

dUcenH )abc

cba1(2

a

c1

c

b1

b

a1 3

.

- 45 -


lMhat´TI26 eK[ a CaRbEvgRCugrbs;RtIekaNmYy . cUrbgðajfa ³ c;b;

cbabacacbcba


cbabacacbcba eday CaRbEvgRCugrbs;RtIekaNenaHeK)an ³ c;b;a

0bac;0acb;0cba tamvismPaB SchwartzCauchy eK)an ³

)3(c2bacacb

)2(a2baccba

)1(b2acbcba

bUkvismPaB nig eKTTYl)an ³ )2(;)1( )3(

)cba(2)bacacbcba(2 dUcenH

cbabacacbcba .

- 46 -


lMhatTI27 KNnaplbUk ³

n

1k1k1kkk

k

n)23)(23(

6S

rYcTajrktémø n

. nSlim

dMeNa¼Rsay KNnaplbUk ³

n

1k1k1kkk

k

n)23)(23(

6S

eKman )23)(23(

6

23

3

23

31k1kkk

k

1k1k

1k

kk

k

eK)an

n

1k1k1k

1k

kk

k

n23

3

23

3S

dUcenH 1n1n

1n

n23

33S

.

ehIy 21323

33limSlim 1n1n

1n

nn

n

dUcenH n

. 2Slim n

- 47 -


lMhatTI28 KNnaplKuNxageRkam ³

n242

n

0kk

2n

2

xtan.....

4

xtan.

2

xtan.xtan

2

xtanP

nk

dMeNa¼Rsay KNnaplKuN ³

n242

n

0kk

2n

2

xtan.....

4

xtan.

2

xtan.xtan

2

xtanP

nk

eKman a2sin

asin2

acosasin2

asin2

acos

asinatan

22

yk k2

xa eK)an

k

k2

k

2

x2sin

2

xsin2

2

xtan

eKTaj x2sin2

xsin

.2

2

x2sin

2

xsin

.2Pn

2

12n

0kk

2

k2

2n

1n

1n

k

1k

k

dUcenH x2sin2

xsin

.2P . n2

)12(n

1n

1n

- 48 -


lMhatTI29 KNnaplKuNxageRkam ³

n

0k

2k

2n

k)

2

xtan1(P

dMeNa¼Rsay KNnaplKuN ³

n

0k

2k

2n

k)

2

xtan1(P

eyIgman k

2

k

k2

k2

k2

k2

2

xcos

2

x2cos

2

xcos

2

xsin

2

xcos

2

xtan1

eK)an n

2

n

0kk

2

1k2

n

2

xcos

x2cos

2

xcos

2

xcos

P1n1k

k

dUcenH n

2n

2

xcos

x2cosP

1n .

- 49 -


lMhatTI30 eK[RtIekaN mYymanmMukñúgCamMuRsYc . cUrbgðajfa ³ ABC

CsinBsinAsin32CsinBsinAsin 222


CsinBsinAsin32CsinBsinAsin 222 tamRTwsþIbTsIunUs

R2Csin

c

Bsin

b

Asin

a ¬ :R kaMrgVg;carwkeRkARtIekaN ¦

eKTaj )I(

CsinR2c

BsinR2b

AsinR2a

tamRTwsþIbTkUsIunUs )II(Acosbc2cba 222

yk ( CYskñúg ( eK)an ³ )I )II

AcosCsinBsin2CsinBsinAsin

AcosCsinBsinR8CsinR4BsinR4AsinR4222

2222222

- 50 -


eKTaj CsinBsin2

AsinCsinBsinAcos

222

ehtuenH )1(AsinCsinBsin2

AsinCsinBsinAcot

222

dUcKñaEdr )2(BsinAsinCsin2

BsinAsinCsinBcot

222

ehIynwg )3(CsinBsinAsin2

CsinBsinAsinCcot

222

bUkTMnak;TMng nig eKTTYl)an ³ )2(;)1( )3(

)4(CsinBsinAsin2

CsinBsinAsinCcotBcotAcot

222

müa:geToteKman CBA b¤ CBA eK)an )Ctan()BAtan( Ctan

BtanAtan1

BtanAtan

eKTaj CtanBtanAtanCtanBtanAtan KuNGgÁTaMgBIrnwg eK)an ³ CcotBcotAcot

1AcotCcotCcotBcotBcotAcot

tamvismPaB cMeBaHRKb; GMAM 0z;y;x

eKman

zx2xz

yz2zy

xy2yx

22

22

22

- 51 -


eK)an )zxyzxy(2)zyx(2 222

b¤ zxyzxyzyx 222

EfmGgÁTaMgBIrnwg zx2yz2xy2 eK)an ³ )zxyzxy(3)zyx( 2

edayyk Ccotz;Bcoty;Acotx eK)an 3)CcotBcotA(cot 2

naM[ )5(3CcotBcotAcot tamTMnak;TMng nig eK)an ³ )4( )5(

3CsinBsinAsin2

CsinBsinAsin 222

dUcenH CsinBsinAsin32CsinBsinAsin 222

- 52 -


lMhatTI31 cUrbgðajfa 1

iz32

iz6

luHRtaEt

3

1|z|


3

1|z|

eKman 1iz32

iz6

lkçxNÐ 2 b¤ 0iz3 3

i2z

eK)an | |iz32||iz6

9

1|z|

9

1zz

3zz27

zz9iz6zi641zi6iz6zz36

)zi32)(iz32()iz6)(iz6(

|iz32||iz6|

2

22

dUcenH 3

1|z| .

- 53 -


lMhatTI32 eK[ CacMnYnkMupøicEdlmanm:UDúlesμ I . n321 z....;;z;z;z 1

eKtag

n

1k k

n

1kk )

z

1()z(Z .

cUrbgðajfa 0 2nZ

dMeNa¼Rsay bgðajfa 2nZ0

eday z CacMnYnkMupøicEdlmanm:UDúlesμ I n321 z....;;z;z; 1

enaHeKGactag kkk xsin.ixcosz Edl n,....,3,2,1k;IRxk eK)an

n

1k k

n

1kk )

z

1()z(Z

0xsinxcos

xsinixcosxsin.ixcos

2n

1kk

2n

1kk

n

1kkk

n

1kkk

eK)an 0Z

- 54 -


müa:geTottamvismPaB SchwartzCauchy

n

1kk

22n

1kk xcosnxcos

nig

n

1kk

22n

1kk xsinnxsin

eKTaj )x(sinnxcosnZn

1kk

2n

1kk

2

2

n

1kk

2k

2

nn.nZ

xsinxcosnZ

dUcenH . 2nZ0

sMKal; ³ eKGacRsay 2nZ tammYyrebobeTotdUcxageRkam eday 1|z| k enaH

kk z

1z RKb;

- 55 -

n.....;;2;1 k

eK)an

n

1k k

n

1kk )

z

1()z(Z

2

2n

1kk

2n

1kk

n

1kk

n

1kk

n

1kk

n

1kk

n|z|z

zzzz

eKTaj)an . 2nZ


lMhatTI33 eK[cMnYnkMupøic Edl |z 1|z . cUrbgðajfa ³

4|z1||z1|2 2

dMeNa¼Rsay bgðajfa 4|z1||z1|2 2 tag tsin.itcosz

eK)an |2

tsin|2tsin)tcos1(|z1| 22

ehIy |tcos|2t2sin)t2cos1(|z1| 222 |

2

tsin21|2 2

eK)an

|

2

tsin21||

2

tsin|2|z1||z1| 22

edayyk 1x1;2

tsinx ehIytagGnuKmn_ f

kMnt;eday Edl |x21||x|)x(f 2 1x1 cMeBaH eK)an 1x1 2)x(f

2

2

dUcenH 4|z1||z1|2 2 .

- 56 -


lMhatTI34 eK[sIVúténcMnYnBit ( INnn )v kMnt;eday ³

5v0 nig TMnak;TMngkMenIn v 0n;1v2 2n1n

bgðajfa 22nn

21n1n )1vv(1vv

rYcKNna CaGnuKmn_én n . nv

dMeNa¼Rsay bgðajfa 22

nn2

1n1n )1vv(1vv tag 1vvw 2

1n1nn eday 0n;1v2v 2n1n

eK)an 1)1v2(1v2w 22n

2nn

22

nn

2n

2nn

2n

2n

4n

2n

)1vv(

)1v(1vv2v

v4v41v2

dUcenH 22nn

21n1n )1vv(1vv

KNna CaGnuKmn_én n ³ nv

tag )1vvln(t 2nnn

- 57 -


eK)an )1vvln(t 21n1n1n

eday 22nn

21n1n )1vv(1vv

eKTaj n2

nn1n t2)1vvln(2t naM[ CasIVútFrNImaRtmanersug )t( n 2q nigtY )25ln(t0 . tamrUbmnþ n2nn

0n )25ln()25ln(2qtt eday )1vvln(t 2

nnn eKTaj)an )1()25(1vv

n22nn

eday 1)1vv)(1vv( 2nn

2nn

eKTaj 1vv

11vv

2nn

2nn

)2()25()25(

11vv

n

n2

2

2nn

bUksmIkar nig eKTaj)an ³ )1( )2(

nn 22n )25()25(v2

dUcenH 2

)25()25(v

nn 22

n

.

- 58 -


lMhatTI35 eK[sIVúténcMnYnBit INkk )u( kMnt;eday ³

9u 0 nig TMnak;TMngkMenIn

n

1p

pk

pn1k uCu

Edl )!pn(!p

!nC . p

n

cUrKNna ku CaGnuKmn_én k nig n

dMeNa¼Rsay KNna ku CaGnuKmn_én k nig n ³ eyIgman

n

1p

pk

pn1k uCu

eday pk

n

0p

pk

pn

n

1p

pk

pn )u1(1uC1uC

eK)an pk1k )u1(1u

eKTaj )u1ln(p)u1ln( k1k TMnak;TMngenHbBa¢ak;fa })u1ln({ k CasIVútFrNImaRtman pleFobrYm p nigtYdMbUg 10ln)u1ln( o eK)an 10lnp)u1ln( k

k naM[ 110ukp

k .

- 59 -



lMhatTI36 eK[sIVúténcMnYnBit ( INnn )u kMnt;eday ³ lMhatTI36

1u0 1u0 nig TMnak;TMngkMenIn u 1u4u2 n2

n1n eK[sIVúténcMnYnBit ( INnn )u kMnt;eday ³

cUrKNna u CaGnuKmn_én n cUrKNna u CaGnuKmn_én n nn

nig TMnak;TMngkMenIn u 1u4u2 n2

n1n

dMeNa¼Rsay KNna CaGnuKmn_én n ³ KNna CaGnuKmn_én n ³ nunu

dMeNa¼Rsay

eKman u eKman u 1u4u2 n2

n1n 1u4u2 n2

n1n

KuNGgÁTaMgBIr nwg 2 eK)an ³ KuNGgÁTaMgBIr nwg 2 eK)an ³ 2u8u4u2 n

2n1n 2u8u4u2 n

2n1n

EfmGgÁTaMgBIrnwg eK)an ³ EfmGgÁTaMgBIrnwg eK)an ³ 22

2n1n )1u(4)1u(2

2n1n )1u(4)1u(2

tag tag )1u(2lnv nn )1u(2lnv nn

eK)an eK)an )1u(2lnv 1n1n )1u(2lnv 1n1n

)1u(2ln2v

)1u(4lnv

n1n

2n1n

naM[ ( CasIVútFrNImaRtmanpleFobrYm q)vn 2

- 60 - - 60 -


nigtY )4ln()]1u(2[lnv 00 tamrUbmnþ 1n2nn

0n 2ln4ln2qvv

eday )1u(2lnv nn eKTaj 1n2

n 2)1u(2

dUcenH . 12u 12

n

1n

lMhat´TI37 eK[ CamMukñúgrbs;RtIekaN mYy . C;B;A ABC

cUrbgðajfa 332

Ccot

2

Bcot

2

Acot

dMeNa¼Rsay bgðajfa 33

2

Ccot

2

Bcot

2

Acot

- 61 -

eKman CBA naM[ 2

C

22

B

2

A

eK)an

2

C

2tan

2

B

2

Atan

2

Ccot

2

Btan

2

Atan1

2

Btan

2

Atan


12

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

2

Btan

2

Atan1)

2

Btan

2

A(tan

2

Ctan

2

Ctan

1

2

Btan

2

Atan1

2

Btan

2

Atan

KuNGgÁTaMgBIrnwg 2

Ccot

2

Bcot

2

Acot eK)an ³

2

Ccot

2

Bcot

2

Acot

2

Ccot

2

Bcot

2

Acot eday C;B;A0 enaH

22

C;

2

B;

2

A0

eK)an 02

Ccot;0

2

Bcot;0

2

Acot

tamvismPaB eK)an ³ GMAM

2

Ccot

2

Bcot

2

Acot27

2

Ccot

2

Bcot

2

Acot

2

Ccot

2

Bcot

2

Acot3

2

Ccot

2

Bcot

2

Acot

2

Ccot

2

Bcot

2

Acot3

2

Ccot

2

Bcot

2

Acot

3

3

3

dUcenH 332

Ccot

2

Bcot

2

Acot .

- 62 -


lMhat´TI38 eK[ CamMuRsYckñúgrbs;RtIekaN mYy . C;B;A ABC

cUrbgðajfa 3)31()Ctan1)(Btan1)(Atan1(

dMeNa¼Rsay bgðajfa 3)31()Ctan1)(Btan1)(Atan1( eday CamMuRsYcenaH C;B;A 0Ctan;0Btan;0Atan

tamvismPaB RKb; eKman³ GMAM 0z;0y;0x

33

3 2223

)xyz1()z1)(y1)(x1(

xyzzyx3xyz31)z1)(y1)(x1(

xyz)zxyzxy()zyx(1)z1)(y1)(x1(

yk Ctanz;Btany;Atanx eK)an ³ 3)CtanBtanAtan1()Ctan1)(Btan1)(Atan1(

eKman )Ctan()BAtan( Ctan

BtanAtan1

BtanAtan

eKTaj CtanBtanAtanBtanBtanAtan xyzzyx

- 63 -



3

3

xyz3xyz

xyz3zyx

eKTaj 33xyz b¤ 33CtanBtanAtan eK)an 33 )331()Btan1)(Btan1)(Atan1( dUcenH 3)31()Ctan1)(Btan1)(Atan1( .

lMhat´TI39 eK[cMnYnKt;viC¢man n . eKdwgfa n Ecknwg 7 [sMNl; % ehIy nEcknwg [sMNl;# 8

k> etIcMnYn n enaHEcknwg [sMNl;b:un μan ? 56

x> rkcMnYn enaHedaydwgfa n 5626n5616 .

dMeNa¼Rsay k> etIcMnYn n enaHEcknwg [sMNl;b:un μan ? 56

tamsm μtikmμeKdwgfa n Ecknwg [sMNl; % naM[man 7

- 64 -


INq1 Edl )1(5q7n 1 ehIymüa:geTot nEcknwg 8 [sMNl; # enaHnaM[man

INq2 Edl )2(3q8n 2 tam ¬!¦ nig ¬@¦ eK)anRbBn½æ

)2(3q8n

)1(5q7n

2

1

b¤

)4(21q56n7

)3(40q56n8

2

1

dksmIkar ¬# ¦ nig ¬$¦ eK)an 19)qq(56n 21 tag INq,qqq 21 eK)an . 19q56n

TMnak;TMngenHmann½yfa cMnYn n enaHEcknwg [sMNl; !( . 56

x> rkcMnYn enaHedaydwgfa n 5626n5616 eKman naM[ 19q56n 562619q565616 b¤

56

5607q

56

5597 b¤

56

7100q

56

5399

naM[ . 100q

cMeBaH eK)an 100q 5619195600n . dUcenHcMnYn n enaHKW 5619n .

- 65 -


lMhat´TI40 eK[GnuKmn_ kMnt;elI IRf ehIyepÞogpÞat;TMnak;TMng³

543

232 x1x4

x

xf

x1

1xfx

1

1

naGaMgetRkal³ .

dMeNa¼

1

0

dx.xfIcUrKN

- 66 -

Rsay KNnaGaMgetRkal³

1

0

dx.xfI

tag 3tx naM[ dx dt.t3 2 nigcMeBaH 1,0x naM[ 1, 0t

eK)an ) 1

0

1

0

23 dtt3).t(fdx.x(fI

naM[ 1

0

32 1dt).t(ftI3

1

TotebIeKtag t1

t1x

müa:ge naM[ 2)t1(

dt2dx

cMeBaH 1,0x naM[ 0,1 t

eK)an ³ 1

).x(fI

0

0

1

1

022 dt).

t1

t1(f

)t1(

12)

)t1(

dt2.(

t1

t1fdx


- 67 -

eKTaj)an

1

02

2dt).t1

t1(f

)t1(

1I

2

1

bUkTMnak;TMng ¬!¦ nig ¬@¦ eK)an ³ dt.)

t1

1(f

)t1(

1)t(ftI

2

1I

3

1

02

32

t1

tamsm μtikmμeKman ³

543

232 f

x1

1xfx

x1x4x1

x1

eK)an ³

6

63

6

164)t1(

6

1dt.)t1(t4I

6

5 1

0

1

0

64543

dUcenH 5

63dx).x(fI

1

0

.


lMhat´TI41

- 68 -

enAkñúgtMruyGrtUNrma:l;manTisedAviC¢man

k,j,i,0 manÉkta cm1 enAelIGkS½ eK[BIrcMnuc 0(A )0,2,

.nig 2,1(B P Cabøg;mansmIkar x2 )1, 04zy2 . cUrsresrsmIkarbøg; Q kat;tamcMnuc A nig ehIypÁúMCamYy B

bøg; )P( )anmMuRsYcmYymantémø 4

.

dMeNa¼Rsay

øg;sresrsmIkarb Q tag zbyax:Q 0dc CasmIkarEdlRtUvrk.

bøg; kat;tamcMnuc n epÞó ar

eday ig B enaHkUGredaencMnuc Q A A nig B gpÞat;nwgsmIk bøg; Q . eK)an

d)1(c)2(b)1

d)0(c)2(b)0( ¤ 0(a

0a b

0dcb2a

0db2

naM[eKTaj)an

)2(ca

)1(2

db


- 69 -

müa:geTotebIeyIgtag CamMupÁúMedaybøg; P nig Q

QP

QP

n.n

n.ncos

enaHeK)an

eday 1,2,2nP nig )c,b,a(nQ

CaviucTr½ m:al; øgNr énb ; an ³

P

nig Q . eK)

22222 cba3c1cos

222

cb2a2

ba.22

cb2a2

eday 4

¬bMrab;¦

enaHeKTaj 2

2

cba3

cb2a2222

!¦ nig ¬@¦ CYskñúg ¬# K)an ³ naM[ )cb22 2 )3()cba(9a2( 222 yksmIkar ¬ ¦ e

)c4

dc(9)cdc2(2 2

222

)4

d c2(9)dc(2 22 2

2222 d

4

9c18)dcd2c(2


04

dcd4c16

04

dcd4c16

0d4

9c18d2cd4c2

22

22

2222

02

dc4

2

naM[

8

dc .

tamTMnak;TMng ¬!¦ nig ¬@¦ eKTaj 2

db nig

8

da .

yktémø 2

db,

8

da nig

8

dc CYskñúgsmIkarbøg; Q

eK)an ³ 0dz

8

dy

2

dx

8

d:Q

smmUl 08zy4x:Q . dUcenHsmIkarbøg; EdlRtUvrkKW ³ Q 08zy4x:Q

- 70 -


lMhat´TI42 eK[ a CabIcMnYnBitviC¢man . cUrRsayfa ³ c;b;

)cabcab(9)2c)(2b)(2a( 222 dMeNa¼Rsay Rsayfa ³

)1()cabcab(9)2c)(2b)(2a( 222

eRCIserIs [2

;0]C,B,A

Edl

Ctan2c

Btan2b

Atan2a

eK)an

Ccos

2)Ctan1(22c

Bcos

2)Btan1(22b

Acos

2)Atan1(22a

222

222

222

naM[ CcosBcosAcos

8)2c)(2b)(2a( 222

222

tag T cabcab

T )AtanCtanCtanBtanBtanA(tan2

- 71 -


CcosBcosAcos

])CBAcos(CcosBcosAcos[2CcosBcosAcos

)BcosAsinCsinAcosCsinBsinCcosBsinA(sin2T

vismPaB smmUlnwg ³ )1(

CcosBcosAcos

)]CBAcos(CcosBcosA[cos18

CcosBcosAcos

8222

eKTaj)an ³

9

4)CBAcos(CcosBcosAcosCcosBcosAcos

tag 3

CBA .

tamvismPaB nig eyIg)an ³ GMAM Jensen

33

cos3

CcosBcosAcosCcosBcosAcos

eKTaj 9

4)3cos(coscos 33

eday cos3cos43cos 3 eK)an

9

4)cos3cos3(cos 33

27

4)cos1(cos 24


- 72 -


3

222

222

3

cos12

cos

2

cos

)cos1.(2

cos.

2

cos

naM[ 27

4)cos1(cos 24 Bit .

dUcenH Bit. )cabcab(9)2c)(2b)(2a( 222

lMhat´TI43 eK[RtIekaN mYymanmMukñúgCamMuRsYc . ABC

k>bgðajfa 2

3CcosBcosAcos

x>bgðajfa 8

1CcosBcosAcos

K>bgðajfa 8

1

2

Csin

2

Bsin

2

Asin

dMeNa¼Rsay k>bgðajfa

2

3CcosBcosAcos

eKman CBA eK)an )Ccos()BAcos(

- 73 -


b¤ CcosBsinAsinBcosAcos b¤ BcosAcosBsinAsinCcos tag )CcosBcosA(cos23T

0)1BcosA(cos)BsinA(sin

)BcosAcosBsinAsinBcosA(cos2322

dUcenH 2

3CcosBcosAcos

x>bgðajfa 8

1CcosBcosAcos

eday CamMuRsYcenaH C;B;A 0Ccos;Bcos;Acos


3 CcosBcosAcos3CcosBcosAcos 3

3

CcosBcosAcosCcosBcosAcos

eday 2

3CcosBcosAcos ¬sRmayxagelI¦

eKTaj 8

1

32

3

CcosBcosAcos

3

dUcenH 8

1CcosBcosAcos .

- 74 -


K>bgðajfa 8

1

2

Csin

2

Bsin

2

Asin

eKman 2

BAcos

2

BAcos2BcosAcos

eday 2

Csin

2

C

2cos

2

BAcos

nig 2

Csin21Ccos 2 eK)an ³

2

Csin2

2

BAcos

2

Csin21CcosBcosAcos 2

2

Csin

2

Bsin

2

Asin41

2

BAcos

2

BAcos

2

Csin21

2

Csin

2

BAcos

2

Csin21

eday 2

3CcosBcosAcos ¬sRmayxagelI¦

eKTaj 2

3

2

Csin

2

Bsin

2

Asin41

dUcenH 8

1

2

Csin

2

Bsin

2

Asin .

- 75 -


lMhat´TI44 eK[RtIekaN mYy . tag p CaknøHbrimaRt nig ABC R CakaMrgVg;carwkeRkARtIekaN . cUrbgðajfa ³ k>

R4

p

2

Ccos

2

Bcos

2

Acos

x> 2

Ccos

2

Bcos

2

Acos4CsinBsinAsin

dMeNa¼Rsay k>

R4

p

2

Ccos

2

Bcos

2

Acos

tag cAB;bAC;aBC tamRTwsþIbTsIunUs ³

Acosbc2cba 222 eday 12

Acos2Acos 2

)12

Acos2(bc2cba 2222

eKTaj bc4

)acb)(acb(

bc4

a)cb(

2

Acos

222

bc

)ap(p

bc4

)a2p2(p2

naM[ bc

)ap(p

2

Acos

- 76 -


dUcKñaEdr ac

)bp(p

2

Bcos

nig

ab

)cp(pCcos

eK)an abc

)cp)(bp)(ap(pp

2

Ccos

2

Bcos

2

Acos

eday R4

abc)cp)(bp)(ap(pS

eK)an R4

1

abc

)cp)(bp)(ap(p

dUcenH

R4

p

2

Ccos

2

Bcos

2

Acos .

x> CcosBcosAcos4CsinBsinAsin tamRTwsþIbTsIunUs ³

R2Csin

c

Bsin

b

Asin

a

eKTaj R2

cCsin;

R2

bBsin;

R2

aAsin

eK)an R

p

R2

p2

R2

cbaCsinBsinAsin

eday R4

p

2

Ccos

2

Bcos

2

Acos

dUcenH 2

Ccos

2

Bcos

2

Acos4CsinBsinAsin

- 77 -


lMhat´TI45 eK[RtIekaN mYy . tag nig r RABC erogKañCakaMrgVg;

rwkeRk carwkkñúg nig ca ARtIekaN . k> cUrbgðajfa ³

R

r1Ccos BcosAcos

bI CaRtIekaNEkgenaHcUrRsayfa ³ x> e ABC

- 78 -

r)12R (

dMeNa¼Rsay

R

r1CcosBcosAk> bgðajfa cos

eKman 2

BAcos

Acos

2

BAcos2Bcos

2

Csin

2

C

2cos

2

BAeday cos

2

Csin21Cnig cos 2 eK)an ³

2

Csin2

2

BAcos

2

Csin2Acos 1CcosBcos 2


2

Csin

2

Bsin

2

Asin41

2

BAcos

2

BAcos

2

Csin21

2

Csin

2

BAcos

2

Csin21

eK)an 2

Csin

2

Bsin

2

Asin41CcosBcosAcos

tamRTwsþIbTsIunUs ³ Acosbc2cba 222 eday

2

Asin21Acos 2

)2

Asin21(bc2cba 2222

eKTaj bc4

)cba)(cba(

bc4

)cb(a

2

Asin

222

bc

)cp)(bp(

bc4

)b2p2)(c2p2(

naM[ bc

)cp)(bp(

2

Asin

RsaydUcKñaEdr ³

ab

)bp)(ap(

2

Csin;

ac

)cp)(ap(

2

Bsin

eK)an ³

abc

)cp)(bp)(ap(41CcosBcosAcos

- 79 -


eday )cp)(bp)(ap(pR4

abcprS

eKTaj

SR4abc

Srp

prS

p

S)cp)(bp)(ap(

2

naM[ R4

r

abc

)cp)(bp)(ap(

dUcenH

R

r1CcosBcosAcos

x> ebI CaRtIekaNEkgenaHcUrRsayfa ABC r)12(R ]bmafa ABC CaRtIekaNEkgRtg; enaH A C

2B;

2A

eday R

r1CcosBcosAcos

eK)an R

r1CcosC

2cos0

R

r1C

4sin2

R

r1CcosCsin

eday 1C4

sin

enaH 2

R

r1

naM[ r)12(12

rR

dUcenH r)12(R .

- 80 -


lMhat´TI46 edaHRsaysmIkar ³

1x3x4.....x16x4x n dMeNa¼Rsay edaHRsaysmIkar

1x3x4.....x16x4x n

elIkGgÁTaMgBIrCakaereK)an ³

1x23x4.....x16x4

1x2x3x4.....x16x4x

n

n


1x43x4....x16

1x4x43x4.....x16x4

n

n

eFIVrebobenHCabnþbnÞab;eK)an ³ 1x2.2x43x4 nnn ¤ 1x2n

eKTaj n4

1x Cab¤ssmIkar .

- 81 -


lMhat´TI47 cUrbgðajfaenAkñúgRKb;RtIekaNeKmanTMnak;TMng ³

2R2

abcCcoscBcosbAcosa

Edl a CaRCug nig c;b; RCakaMrgVg;carwkeRkARtIekaN . dMeNa¼Rsay bgðajfa 2R2

abcCcoscBcosbAcosa

tag CcoscBcosbAcosaT eKman CsinR2c;BsinR2b;AsinR2a eK)an )C2sinB2sinA2(sinRT

23 R2

abc

R8

abc.R4CsinBsinAsinR4

)BAcos()BAcos(CsinR2

Ccos)BAcos(CsinR2

CcosCsin2)BAcos(Csin2R

CcosCsin2)BAcos()BAsin(2R

dUcenH 2R2

abcCcoscBcosbAcosa .

- 82 -


lMhat´TI48 KNnaplKuN ³

n

1k2k2

k2

n2tan1

x2tan1P Edl 2n2

|x|

dMeNa¼Rsay

KNnaplKuN

n

1k2k2

k2

n2tan1

x2tan1P

tamrUbmnþ atan1

atan1a2cos 2

2

eK)an x2tan1

x2tan1x2cos k2

k21k

ehIy x2cos

x2cos

x2cos

2sin2cos2tan1 k2

1k

k2

k2k2k2

eK)an x2cos

x2cos

)x2tan1(

x2tan11k2

k2

2k2

k2

dUcenH x2cos

x2cos

x2cos

x2cosP 1n2

2n

1k1k2

k2

n

.

- 83 -


lMhat´TI49 eK[ a nig x CacMnYnBitepÞógpÞat; ³ d;c;b;

d

x4sin

c

x3sin

b

x2sin

a

xsin Edl Zk;kx

cUrbgðajfa a )ca3(b)db4( 4223

dMeNa¼Rsay bgðajfa a )ca3(b)db4( 4223

tag td

x4sin

c

x3sin

b

x2sin

a

xsin

eKTaj

dtx4sin

ctx3sin

btx2sin

atxsin

eKman sin x2cosx2sin2x4

sin )tb1(tb4td

)x2sin1(x2sin4x4

x2cosx2sin4x4sin

222222

222

222

eKTaj )1(b4

d1

b

1t 2

2

22

- 84 -


müa:geTot xsin4xsin3x3sin 3

)ta43(atct

)xsin43(xsinx3sin22

2

eKTaj 2)a

c3(

a4

1t 2

2 pÞwm nig eK)an ³ 1 2

34

22

22

2

2

a4

ca3

b4

db4

a

c3

a4

1

b4

d1

b

1

KuNGgÁTaMgBIrnwg eK)an 43ba )ca3(b)db4(a 4223

dUcenH . )ca3(b)db4(a 4223

- 85 -


lMhat´TI50 eK[ CaRtIekaNmYy . cUrbgðajfa ³ ABC

k> 12

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

x> 9

3

2

Ctan

2

Btan

2

Atan

dMeNa¼Rsay bgðajfa k> 1

2

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

tamrUbmnþ btanatan1

btanatan)batan(

eK)an

2

C

2tan

2

BAtan

2

Btan

2

Atan1)

2

Btan

2

A(tan

2

Ctan

2

Ctan

1

2

Ccot

2

Btan

2

Atan1

2

Btan

2

Atan

dUcenH 12

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan .

- 86 -


x> 9

3

2

Ctan

2

Btan

2

Atan

tamvisPaB eK)an ³ GMAM

3

2

3

2

2

Ctan

2

Btan

2

Atan31

2

Ctan

2

Btan

2

Atan3

2

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

eKTaj 9

3

27

1

2

Ctan

2

Btan

2

Atan

dUcenH 9

3

2

Ctan

2

Btan

2

Atan .

lMhat´TI51 eK[ CaRtIekaNmYy . cUrbgðajfa ³ ABC

2

Ctan.

2

BAtan

ba

ba


2

Ctan.

2

BAtan

ba

ba

eKman BsinR2b;AsinR2a ¬ RkaMrgVg;carwkeRkA¦

- 87 -


eK)an )BsinA(sinR2

)BsinA(sinR2

ba

ba

2

Ctan

2

BAtan

2

Ccos

2

Csin

.2

BAtan

ba

ba

)2

C

2sin(

)2

C

2cos(

.2

BAtan

ba

ba

2

BAcos

2

BAsin2

2

BAcos

2

BAsin2

ba

ba

dUcenH

2

Ctan.

2

BAtan

ba

ba

.

- 88 -


lMhat´TI52 cUrbgðajfa ³

2

2



12


2

- 89 -

-ebI cos enaH 0x 2

2

2

ba1xsin

Bit -ebI cos eyIgEckGgÁTaMgBIrén 0x 1 nwg xcos2

xcos

1

2

ba1)bx)(tanax(tan 2

2

xtant tag enaH 222 t1xtan1

xcos

1

eK)an 22

t12

ba1)bt)(at(

222

22 t2

ba

2

bat1abt)ba(t


Bit02

ba1t

2

ba

0ab2

ba1t)ba(t

2

ba

22

22

2

dUcenH 2

2


.

lMhat´TI53 eK[RtIekaN mYymanmMukñúgCamMuRsYc nig RCug ABC c;b;a

nigépÞRkLa . cUrRsayfa ³ K

2

cbaK4acK4cbK4ba

222222222222


2

cbaK4acK4cbK4ba

222222222222

eKman Bsinca2

1Asinbc

2

1Csinab

2

1K

tag 222222222 K4acK4cbK4baT BcosacAcoscbCcosba 222222222

- 90 -


2

cba

2

bacacbcba

ca2

bacca

bc2

acbbc

ab2

cbaab

BcoscaAcosbcCcosabT

222

222222222

222222222

dUcenH

2

cbaK4acK4cbK4ba

222222222222

lMhat´TI54 RbsinebI )z1)(y1)(x1(xyx Edl 1z;y;x0 cUrbgðajfa

4

3)y1(z)x1(y)z1(x


4

3)y1(z)x1(y)z1(x

- 91 -


eKman 04

1xx)

2

1x( 22 cMeBaHRKb; x

eKTaj 4

1)x1(x eday 1x0 enaHeKTaj)an ³

4

1)x1(x0 . RsaydUcKñaEdreK)an ³

4

1)y1(y0 nig

4

1)z1(z0

eyIg)an 64

1)z1)(y1)(x1(xyz

eday )z1)(y1)(x1(xyz enaHeKTaj ³

64

1xyz 2 naM[

8

1xyz .

tag )y1(z)x1(y)z1(xT )xzyzxy()zyx( eday )z1)(y1)(x1(xyz b¤ xyz)zxyzxy()zyx(1xyz b¤ xyz21)zxyzxy()zyx( eKTaj

4

3

8

121xyz21T

dUcenH 4

3)y1(z)x1(y)z1(x

- 92 -


lMhat´TI55 eK[ a CaRbEvgRCugrbs;RtIekaNmYyEdlman c;b;

brimaRtes μ I . cUrRsayfa ³ 2

2abc2cba2

3 222

dMeNa¼Rsay bgðajfa 2abc2cba

2

3 222 edaybrimaRtrbs;RtIekaNenHes μ I enaHRCugTaMgbI 2 c;b;a

rbs;RtIekaNsuTæEttUcCag 1 . eyIg)an

2

1Asinbc

2

1S

tamrUbmnþehrug )cp)(bp)(ap(pS eday 1p

enaH 2

1)c1)(b1)(a1(S

eKTaj 4

1)c1)(b1)(a1(0

b¤ 4

1abc)cabcab()cba(10

b¤ 4

1abc)cabcab(210

- 93 -


b¤ 4

5abc)cabcab(1

b¤ 2

5abc2)cabcab(22

eKman )cabcab(2cba)cba( 2222

eKTaj ³

abc2)cabcab(24abc2cba

)cabcab(2abc2)cba(abc2cba222

2222

eday 2

5abc2)cabcab(22

eK)an 24abc2cba2

54 222

dUcenH 2abc2cba2

3 222 .

- 94 -


lMhat´TI56 eK[

n

1kn

2

3

2

2

22

k

2

n3

n....

3

3

3

2

3

1

3

kS

KNna CaGnuKmn_én n rYcTajrk n

nS nSlim

dMeNa¼Rsay KNna CaGnuKmn_én n ³ nS

n

1kn

2

3

2

2

22

k

2

n3

n....

3

3

3

2

3

1

3

kS

tag k

2

k3

kt cMeBaHRKb; k 1

eK)an kk

2

k

2

k1k3

1k2

3

k

3

)1k(tt3

yk kk1kk3

1k2tt3T

eK)an kkkk1k3

2

3

1k2

3

3k2TT3

b¤ kk1k1k2k3

2)tt3()tt3(3

b¤ kk1k2k3

2tt6t9

- 95 -


edayeKman ³

- 96 -

kk1k1k2kk1k2k t4)tt(3)tt(9tt6t9 eKTaj )tt(

4

3)tt(

4

9

3

1.

2

1t k1k1k2kkk

eK)an

n

1kk1k

n

1k1k2k

n

1kkk tt

4

3tt

4

9

3

1

2

1S

121n2nn

11n22n

n

t4

3t

4

9t

4

3t

4

9)

3

11(

4

1

)tt(4

3)tt(

4

9

3

11

3

11

.6

1

eday k

2

k3

kt

eK)an 2n

2

2n1n

2

1n213

)2n(t;

3

)1n(t;

9

4t;

3

1t

3

1.

4

3

9

4.

4

9

3

)1n(.

4

3

3

)2n(.

4

9

3.4

1

4

1S 1n

2

2n

2

nn

dUcenH n

2

n3

3n3n.

2

1

2

3S

eday 03

3n3nlim n

2

n

dUcenH

2

3Slim n

n

.


lMhat´TI57 eK[RtIekaN mYymanRCug ABC cAB;bAC;aBC tag

2

cbap

CaknøHbrimaRt r nig R CakaMrgVg;carwkkñúg

nigkaMrgVg;carwkeRkAénRtIekaN . cUrRsaybBa¢ak;TMnak;TMngxageRkam ³ k> Rr4rpabcabc 22

x> Rr8r2p2cba 22222

K> p

r

2

Ctan

2

Btan.

2

Atan

X>p

rR4

2

Ctan

2

Btan

2

Atan

g> r

p

2

Ccot

2

Bcot

2

Acot

dMeNa¼Rsay RsaybBa¢ak;TMnak;TMngxageRkam ³ k> bc Rr4rpabca 22

- 97 -


tamrUbmnþehrugeK)an ³ pr)cp)(bp)(ap(pS


223

2

22

prabcp)cabcab(p)cba(p

pr)cp)(bp)(ap(

rp)cp)(bp)(ap(p

eday ehIy p2cba R4

abcprS b¤ pRr4abc

eK)an 233 prpRr4p)cabcab(p2p

eKTaj Rr4prp

pRr4pprcabcab 22

32

dUcenH Rr4rpabcabc 22

x> Rr8r2p2cba 22222

eKman )cabcab(2)cba(cba 2222

eday nig Rr4rpabcabc 22 p2cba eK)an )Rr4rp(2p4cba 222222

dUcenH Rr8r2p2cba 22222

- 98 -


K> p

r

2

Ctan

2

Btan.

2

Atan

tamRTwsþIbTsIunUs ³ Acosbc2cba 222 eday

2

Asin21Acos 2

)2

Asin21(bc2cba 2222

eKTaj bc4

)cba)(cba(

bc4

)cb(a

2

Asin

222

bc

)cp)(bp(

bc4

)b2p2)(c2p2(

naM[ bc

)cp)(bp(

2

Asin

müa:geTot Acosbc2cba 222 eday 1

2

Acos2Acos 2

)12

Acos2(bc2cba 2222

eKTaj bc4

)acb)(acb(

bc4

a)cb(

2

Acos

222

bc

)ap(p

bc4

)a2p2(p2

naM[ bc

)ap(p

2

Acos

- 99 -


eK)an )ap(p

)cp)(bp(

bc

)ap(pbc

)cp)(bp(

2

Atan

RsaydUcKñaEdreKTaj ³

)cp(p

)bp)(ap(

2

Ctan;

)bp(p

)cp)(ap(

2

Btan

eFIViviFIKuNeK)an ³ 3p

)cp)(bp)(ap(

2

Ctan

2

Btan

2

Atan

p

r

p

pr

p

S

p

)cp)(bp)(ap(p

22

2

dUcenH p

r

2

Ctan

2

Btan.

2

Atan .

X>p

rR4

2

Ctan

2

Btan

2

Atan

tag 2

Ctan

2

Btan

2

AtanT

edayeKman )ap(p

)cp)(bp(

2

Atan

)cp(p

)bp)(ap(

2

Ctan;

)bp(p

)cp)(ap(

2

Btan

- 100 -


enaHeK)an ³

S

abacbcp)bacacb(p3

)cp)(bp)(ap(p

)bp)(ap()cp)(ap()cp)(bp(

)cp(p

)bp)(ap(

)bp(p

)cp)(ap(

)ap(p

)cp)(bp(T

2

pr

)cabcab(p)cba(2p3T

2

eday nig Rr4rpabcabc 22 p2cba

p

R4r

pr

Rr4rpp4p3T

2222

dUcenH p

rR4

2

Ctan

2

Btan

2

Atan

g> r

p

2

Ccot

2

Bcot

2

Acot

- 101 -

eKman CBA naM[ 2

C

22

BA

tamrUbmnþ

btanatan1

btanatan)batan(

eK)an

2

C

2tan

2

BAtan


2

Btan

2

Atan1)

2

Btan

2

A(tan

2

Ctan

2

Ctan

1

2

Ccot

2

Btan

2

Atan1

2

Btan

2

Atan

12

Atan

2

Ctan

2

Ctan

2

Btan

2

Btan

2

Atan

KuNGgÁTaMgBIrnwg 2

Ccot

2

Bcot

2

Acot eK)an ³

2

Ccot

2

Bcot

2

Acot

2

Ccot

2

Bcot

2

Acot eday

p

r

2

Ctan

2

Btan

2

Atan

dUcenH r

p

2

Ccot

2

Bcot

2

Acot

- 102 -


lMhat´TI58 eK[ CabIcMnYnBitviC¢manEdl z;y;x 1zyx . cUrRsaybBa¢ak;fa ³ 81

z

11

y

11

x

1

dMeNa¼Rsay RsaybBa¢ak;fa ³

81z

11

y

11

x

1

eKman xyz

)z1)(y1)(x1(1

z

11

y

11

x

1

eday x naM[ 1zy

yxz1

zxy1

zyx1

eK)an xyz

)yx)(xz)(zy(1

z

11

y

11

x

1

tamvismPaB eKman ³ GMAM

zx2xz;yz2zy nig xy2yx eK)an xyz8)yx)(xz)(zy(

- 103 -


naM[ 8xyz

)yx)(xz)(zy(

dUcenH 81z

11

y

11

x

1

.

lMhat´TI59 eK[ CabIcMnYnBitviC¢manEdl c;b;a 1cba 222 bgðajfa

abc

)cba(23

c

1

b

1

a

1 333

222


abc

)cba(23

c

1

b

1

a

1 333

222

tag abc

cba.23

c

1

b

1

a

1T

333

222

eday enaHeKTaj ³ 1cba 222

2

22

22

22

22

22

2 c

ba1

c

1;

b

ca1

b

1;

a

cb1

a

1

eFIVviFIbUksmPaBTaMgenHeK)an ³

22

222

222

2222 b

1

a

1c

a

1

c

1b

c

1

b

1a3

c

1

b

1

a

1

- 104 -


kenSam T Gacsresr ³

ab

c

ca

b

bc

a2

b

1

a

1c

a

1

c

1b

c

1

b

1aT

222

222

222

222

0b

1

a

1c

a

1

c

1b

c

1

b

1a

22

22

22

dUcenH abc

)cba(23

c

1

b

1

a

1 333

222

.

sUmrg´caMGanPaKTI4

- 105 -


lMhat´Gnuvtþn_

1-KNnaplbUk ³

000...1000

n...

1000

3

100

2

10

1

10

kS

2222n

1kk

2

n

rYcTajrklImItén kalNa nS n . 2-eK[ CasIVútcMnYnBitkMnt;eday ³ }x{ n

.......;135135x

5135x;135x;5x

4

321

cUrbgðajfa nn

xlim

man rYckMnt;témørbs;va . 3-edaHRsaykñúgsMNMucMnYnBiténsmIkar ³

2

x...xxnxn...4x21x n212

n21

4-eK[ CabIcMnUnBitviC¢man . cUrRsayfa ³ c;b;a

c

1

b

1

a

1

2

1

ac

1

cb

1

ba

1

5-bgðajfa nnnn 1n

2n

n

1...

3

1

2

11

.

- 106 -


6-cUrbgðajfa 2n2)2n(klog n2

1k2

n

RKb; n . 2

7-eK[ CC:f CaGnuKmn_mYykMnt;eday 2z)iz(f).z(f

cMeBaHRKb; . Cz

cUrbgðajfa 0)z(f)z(f cMeBaHRKb; Cz . 8-cUrbgðajfacMeBaHRKb;cMnYnKt;viC¢man cMnYn ³ n

24

2121721217nn

CacMnYnKt; nig minEmnCakaerR)akd . 9-eKyk CasIVút EdlkMnt;eday ³ 1nnu Fibonacci

nig 1uu 21 n1n2n uuu bgðajfacMeBaHRKb; rvag nig 6n nu 1nu manmYyCakaer R)akd . 10-bgðajfacMeBaHRKb; *INn cMnYn 5!n minEmnCakaer R)akd . 11-edaHRsaykñúgsMMNMucMnYnKt;énsmIkar ³ )xy1(4)xy1)(yx(2)1y)(1x( 22

- 107 -


12-eKkMnt;sIVút eday 1nna 1a1 nig cMeBaHRKb;cMnYnKt; eKman 1n 2a3a2a 2

nn1n cUrbgðajfa CacMnYnKt;cMeBaHRKb; . na n

13-eK[ nig cMeBaHRKb;cMnYnKt; 97aa 21 2n

)1a)(1a(aaa 21n

2n1nn1n .

k>bgðajfa CakaerR)akd na22

x> na222 CakaerR)akd 14-cUrbgðajfa 5

7

3cot

7

2cot

7cot 222

15-KNna

1n

1kn x)kncos()kxsin(S

16-eK[ CacMnYnsßitenAkñúgcenøaH n210 a...;;a;a;a

2;0

Edl 1n)4

atan(...)4

atan()4

atan( n10

. bgðajfa 1n

n210 natan....atan.atan.atan . 17-eK[ CaRtIekaNmYyEdlepÞógpÞat; ³ ABC

2222

r7

s6

2

Ccot3

2

Bcot2

2

Acot

Edl s CaknøHbrimaRt nig CakaMrgVg;carwkeRkARtIekaNenH. r

- 108 -


cUrbgðajfaRtIekaN ABC dUceTAnwgRtIekaN TmYyEdlman rgVas;RCugTaMgGs;CacMnYnKt;KμantYEckrYmrYckMnt;RCugTaMgenH ? 18-cUrbgðajfakñúgRKb;RtIekaNeKmanvismPaB ³

2

Csin

2

Bsin

2

Asin4

ab

c

ca

b

bc

a 222222

19-cUrbgðajfakñúgRKb;RtIekaNeKmanvismPaB ³ 3333 p16

81

c

Ccos

b

Bcos

a

Acos

Edl CargVas;RCug nig p CaknøHbrimaRt . c;b;a

20-eK[ CacMnYnkMupøicEdl 321 z;z;z 0r|z||z||z| 321 cUrRsaybBa¢ak;fa ³

2231332123121 r

1

|zz||zz|

1

|zz||zz|

1

|zz||zz|

1

21-KNna

n

1kn )!kn()!1k(

1k)!kn(klim

22-cUrbgðajfa 2)23cot1)(22cot1( oo

23-eK[ ...,3,2,1k;)xcosx(sink

1)x(f kk

k cUrbgðajfa

12

1)x(f)x(f 64 cMeBaHRKb; . x

- 109 -


24-eK[RtIekaN mYy . cUrbgðajfa ³ ABC

k> 8

1

2

Csin

2

Bsin

2

Asin

x> 4

3

2

Csin

2

Bsin

2

Asin 222

K> 4

9

2

Ccos

2

Bcos

2

Acos 222

X> 8

33

2

Ccos

2

Bcos

2

Acos

25-k>cUrbgðajfa ³

x3

tanx3

tanxtan

x3tan

x>KNna

n

1kkk 3

x

3tan.

3

x

3tan

26-bgðajfaRtIekaN mYyCaRtIekaNsm)atluHRtaEt ABC

2

cbaAcoscCcosbBcosa

27-eK[ CabIcMnYnBitviC¢man . cUrbgðajfa ³ c;b;a

)1c)(1b)(1a()1cabcab( 2222

28-cUrbgðajfa ³

o2

o

oooooo 1sin

1cos

90sin89sin

1...

3sin2sin

1

2sin1sin

1

- 110 -


29-eK[ CaRtIekaNmYymanRCug ehIy Ca ABC c;b;a x

cMnYnBitviC¢man . cUrbgðajfa ³ xxxxxx cba

2

1CcoscBcosbAcosa

30-eK[ CacMnYnBitviC¢man . cUrbgðajfa ³ z;y;x

k> 2

33

z1

z

y1

y

x1

x222

ebI . xyzzyx

x> 2

33

z1

z

y1

y

x1

x222

ebI 1z;y;x0 nig 1zxyzxy . 31-eK[ CaRtIekaNmYymanmMukñúgCamMuRsYc . ABC

cMeBaHRKb; eKtag ³ 3;2;1n

CcosBcosAcos)CcosBcosA(cos2x nnn3nn

cUrbgðaj 2

3xxx 321 .

32-eK[ CaRtIekaNmYymanmMukñúgCamMuRsYc . bgðajfa ³ ABC

CcotBcotAcotCcotBcotAcot6CcotBcotAcot 333

1x3cosx2cosxcos 222 33-edaHRsaysmIkar .

- 111 -

Éksareyag 1-360 Problems for Mathematical Contests 2-103 Trigonometry Problems 3-Complex Number from A to Z 4-Mathematical Olympiad in China 5-The IMO Compendium

lwm pl:ún nig Esn Bisidæ aCMuviijBPBelak 43-eK[RtIekaN ABC...

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