lwm pl:ún nig Esn Bisidæ · sRmab´fñak´TI11 nig sisßBUEkKNitviTüa ) 4- dMeNa£RsayKMrU cMnYnkMupøic lImIt edrIev ( sRmab´fñak´TI. 12) 5- segUbmnþKNçitvbriTüa ( sRmab´fñak´TI11-12
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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[kan´EtrIkceRmIn EfmeTotedIm,Ibeg;InFnFanmnusß[mankanEteRcInedIm,ICYyGPivDÄn_ RbeTsCatirbseyIg . enAkñúgesoven£eyIgxMJúánxitxMRsavRCaveRCIserIsyklMhat´ya¨g sRmaMgbMputykmkeFIVdMeNa£Rsayya¨gek,a£k,ayEdlGac[elakGñk gayyl´qab´cgcaMGMBIsil,£énkareda£RsayTaMgGs´en£ . b¨uEnþeTa£Ca ya¨gNak¾eday kgV£xat nig kMhusqþgedayGectnaRákdCamanTaMg bec©keTs nig GkçraviruTæ . GaRs&yehtuen£ eyIgxMJúCaGñkeroberogrgcaM edayrIkrayCanic©nUvmtiri£KnEbbsSabnaBIsMNakGñksikßakñúgRKbmCÄdæan edIm,ICYyEklMG esovePAen£[ánkanEtsuRkiRtPaBEfmeTot . CaTIbBa©b´en£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
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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(
dMeNa¼Rsay bgðajfa ³
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhat´TI11 eK[RtIekaN mYymanmMukñúgCamMuRsYc . cUrRsayfa ³ ABC
223
CcosBcosAcos
dMeNa¼Rsay bgðajfa ³
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 -
KNitviTüaCMuvijBiPBelak
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
dMeNa¼Rsay bgðajfa ³
2
2
212
21
2
212
21
r
1
zzr
zz
zzr
zz
tag nig )x2sinix2(cosrz1 )y2siniy2co(rz2 Edl IRy;IRx .
- 27 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
x-cMeBaHtémø Edl)ankMnt;xagelIcUrbgðajfa ³ 0z
!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 -
KNitviTüaCMuvijBiPBelak
- 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
x-cMeBaHtémø Edl)ankMnt;xagelIcUrbgðajfa ³ 0z
!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
.
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhat´TI15 eK[sIVúténcMnYnBit ³
1n;n
111
n
111a
22
n
cUrbgðajfa 20321 a
1....
a
1
a
1
a
1 CacMnYnKt; .
dMeNa¼Rsay bgðajfa
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 -
KNitviTüaCMuvijBiPBelak
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
dMeNa¼Rsay bgðajfa
2
1
1x)1z(
1
1z)1y(
1
1y)1x(
1222222
eyIgman 2x2yx1y)1x( 2222
eday x xy2y22
- 33 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
dMeNa¼Rsay Rsayfa ³
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 -
KNitviTüaCMuvijBiPBelak
]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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhat´TI26 eK[ a CaRbEvgRCugrbs;RtIekaNmYy . cUrbgðajfa ³ c;b;
cbabacacbcba
dMeNa¼Rsay Rsayfa ³
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhatTI30 eK[RtIekaN mYymanmMukñúgCamMuRsYc . cUrbgðajfa ³ ABC
CsinBsinAsin32CsinBsinAsin 222
dMeNa¼Rsay bgðajfa ³
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhatTI31 cUrbgðajfa 1
iz32
iz6
luHRtaEt
3
1|z|
dMeNa¼Rsay bgðajfa
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
tamvismPaB eK)an ³ GMAM
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
- 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
.
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
tamvismPaB eK)an ³ GMAM
- 72 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
tamvismPaB eK)an ³ GMAM
3 CcosBcosAcos3CcosBcosAcos 3
3
CcosBcosAcosCcosBcosAcos
eday 2
3CcosBcosAcos ¬sRmayxagelI¦
eKTaj 8
1
32
3
CcosBcosAcos
3
dUcenH 8
1CcosBcosAcos .
- 74 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhat´TI46 edaHRsaysmIkar ³
1x3x4.....x16x4x n dMeNa¼Rsay edaHRsaysmIkar
1x3x4.....x16x4x n
elIkGgÁTaMgBIrCakaereK)an ³
1x23x4.....x16x4
1x2x3x4.....x16x4x
n
n
elIkGgÁTaMgBIrCakaereK)an ³
1x43x4....x16
1x4x43x4.....x16x4
n
n
eFIVrebobenHCabnþbnÞab;eK)an ³ 1x2.2x43x4 nnn ¤ 1x2n
eKTaj n4
1x Cab¤ssmIkar .
- 81 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
dMeNa¼Rsay bgðajfa
2
Ctan.
2
BAtan
ba
ba
eKman BsinR2b;AsinR2a ¬ RkaMrgVg;carwkeRkA¦
- 87 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
lMhat´TI52 cUrbgðajfa ³
2
2
ba1)xcosbx)(sinxcosax(sin
dMeNa¼Rsay bgðajfa ³
12
ba1)xcosbx)(sinxcosax(sin
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
KNitviTüaCMuvijBiPBelak
Bit02
ba1t
2
ba
0ab2
ba1t)ba(t
2
ba
22
22
2
dUcenH 2
2
ba1)xcosbx)(sinxcosax(sin
.
lMhat´TI53 eK[RtIekaN mYymanmMukñúgCamMuRsYc nig RCug ABC c;b;a
nigépÞRkLa . cUrRsayfa ³ K
2
cbaK4acK4cbK4ba
222222222222
dMeNa¼Rsay bgðajfa ³
2
cbaK4acK4cbK4ba
222222222222
eKman Bsinca2
1Asinbc
2
1Csinab
2
1K
tag 222222222 K4acK4cbK4baT BcosacAcoscbCcosba 222222222
- 90 -
KNitviTüaCMuvijBiPBelak
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
dMeNa¼Rsay bgðajfa ³
4
3)y1(z)x1(y)z1(x
- 91 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
.
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
tamrUbmnþehrugeK)an ³ pr)cp)(bp)(ap(pS
elIkGgÁTaMgBIrCakaereK)an ³
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
dMeNa¼Rsay bgðajfa
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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 -
KNitviTüaCMuvijBiPBelak
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
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