KN³kmμkarniBnæ nig eroberog. KN³kmμkarRtYtBinitübec kets. KN³kmμkarRtYtBinitüGkçraviruTæ elak lwm mikásir

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2 KN³kmμkriBæ ig eroberog lwm pláú ig Es Bisidæ KN³kmμkrRtYtBiitübec kets elk lwm qu elk ; sun elk RBwm suitü elk Titü em:g elk Gwug smng elkrs Tuy rn elk pl b uqy KN³kmμkrRtYtBiitüGkçrviruTæ elk lwm mikásir krkmubüútr½ kbø l KuNÑk elk Gwug smng i

3 Grmáf esoepapkt8 EdlGñksikSkMBugk; eakñúgédeh EckecjCbEpñkEdlEpñkT CkRmglMht;eRCsers EpñkT CEpñkdMeNHRsy ig EpñkT ClMht;Guvtþ_ rl;brbflmht;myy²eakñúgesovepaeh eygjúmercsersyket lmht; NEdlmlkçN³Bik mkefvdmenhrsykmruy:gek,hk,ybmput eklbmngékreroberogcgrkgkwedm,tukcéksrcmyysrmb;gñk sikskñúgrkb;mcädæ ig mü:getotedm,culrymelksþüyvisy½knitvitükñúg RbeTskm<úCeyg[k;EtrkceRmqb;rhs½Rsbtmsm½yviTüsRsþTMeb esovepaehmil hysekhységehet kmhusqágedygectrkd CmTMgbec kets ig GkçrviruTæ ehtueheygjmúcgñkibæ rgcmcic Uv mtirihk;ebbsßbbgñksiksrkb;mcädæedykþrkryedm,eklmgesoveh [k;etmsurkitüpbrbesrelgefmetot CTbB b; JMúTGñkiBæsUmeKrBCUBrGñksikSTMgGs;msuPBl mrcjqøsév ig TTYleCKC½yCic kñúgkrsiks t;dmbg éf T 7 EFñÚ qñm GñkiBæ ig RsyRCv lwm pláú Tel : Emil: lim_phlku@ymilcom Website: wwwmthtodywordpresscom ii

4 krmglmht;!> curkmnt;rkb;gukm_ f : R R edydwgfsmpb y f( f(y f BitCic RKb;, y R tg[epñkkt;é ( cmebhrkb;cmybit (, currsyf si cos #> edhrsysmkr ³ 6 log ( log ( 8 log ( $> ekmgukm_ f ( Edl KN f [ f [ f [ f ( ]] ] %> ek[gukm_ f ( bsi c bcos Edl, b curbgðjf f ( cmebhrkb;cmybit b c - -

5 ig y (si cos si( t y y cos(cot (si cos y Edl ig,,, k> cmebhrkb; tg u cos y si ig v cos y si ^> ekmsvút ( ig ( y kmnt;edy currsyf ( u ig ( v sutæetcsvútfrnmrt >KN u ig v CGuKm_é ig K> Tjrk ig y CGuKm_é ig &> ek[gukm_ f kmnt;rkb; R edytmk;tmg ³ f ( f ( si cos f ( RKb; R currsyf - -

6 *> ek[gukm_ f kmnt;rkb; R edytmk;tmg ³ f ( f ( si cos kmnt;cmybit r ig edm,[gukm_ ( f ( r si( RKb; R R edytmk;tmg ³ (>ek[gukm_ f kmnt;rkb; 5 f ( si f ( cos currktémøtucbmput ig FMbMputé f (!> ek[gukm_ f ig gkmnt;rkb; R f ( g( si f ( g( cos f ( ig g ( currkgukm_!!> ek[gukm_ f kmnt;rkb; (, f Gcsresr ³ edytmk;tmg ³ edytmk;tmg ³ f ( f ( l( currk f (cos CGuKm_é t t - -

7 ek[gukm_ f kmnt;rkb; R edytmk;tmg ³ e f ( f ( 8 ( e f ( curknlmt lim!#> ek[gukm_ : R R f( ( yf( y ( yf( k> kmnt;rkgukm_ f ( f kmnt;edytmk;tmg ³ > kmnt; edm,[ (!$> cmebh ig b CcMYBit smkr f y y( b mb sy:gtic myyccmybit curkntémøtucbmputé b? MO 97!%> ek[gukm_el f kmt;edytmk;tmg ³ 7 f( 6 f( ig f( 5 Edl,,, 9f( currsyf f ( CRbPKsRmYlmi!^> ek[svútécmybit u ( kmnt;edy ³ y - -

8 7 u ig u u u RKb; bgðjfekgckmnt;cmybit Edl u (u cmebhrkb; ryckn u CGuKm_é!&> ek[cmykmupøic cos i si Edl R kñúgbøg;kmpøic (o, i, j ekeha M CcMucrUbPBé curkmnt;témøtucbmput ig FMbMputé r OM?!*> ek[cmykmupøic,, ehyepþógpþt;tmk;tmg ³ {, ig currsyf }!(>ek[cmykmupøic ig Edl currsyf - 5 -

9 @> ek[svútcmybit ( kmnt;edy ³,,,,, i i cmebhrkb; i CTRmg;RtekNmRtrYcTjrk ektgsvútcmykmupøic k> currsyf > curdk; CGuKm_ é K> TjrktYTUeTAésVút et ( CsVútYbb ek[svútécmykmupøic ( kmnt;edy ³ i ig Edl,,, i i k> tg w bgðjf ( w CsVútFrNmRtécMY kmupøic ryckn w CGuKm_é edysresrltæplctrmg; RtekNmRt > Tjbgðjf cos (cos isi - 6 -

10 ek[svútécmybit ( u ig ( v kmnt;edy ³ u v ig u v u v Edl u v u i v k> ekbiitüsvútécmykmupøic currsyf ( CsVútFrNmRtécMYkMupøic ryckn CGuKm_é edysresrltæplctrmg;rteknmrt > smedg u ek[svútécmybit u u v ig v CGuKm_é ( ig ( v kmnt;edy ³ u v u v uv u i v k> ekbiitüsvútécmykmupøic currsyf ryctjf > smedg u ig v CGuKm_é Edl - 7 -

11 @$> ek[svútécmybit ( kmnt;edy ³ d e d e,,, d e e e Edl CcMyKt;FmμCti k> KN ; ryctjrktémøé > KN CGuKm_é ryctjrktémø ig ( K> cmebh [,] curerbobefobtémø e ig bgðjf ( CsVútek edymicmc;kn X> bgðjfcmebhrkb; [,] ekm ryctjrkkesmgmé tmkrkn etsvút ( mlmtb ek[svút k> bgðjf > KN d Edl ( CsVútcuH CGuKm_é K> currsyf ( ( e e e d cmebh - 8 -

12 ryctjrk ek[gmgetrkl lim si cos d d 6 d Edl kmnt; edm,[ migrs½ywg ryckn é ek[gmgetrkl d Edl cmebhtémø kmnt; edm,[ migrs½ywg ryckn cmebhtémø é EdlrkeXjgel cos bsi #> currsyf d b Edl ig b CBrcMYBit - 9 -

13 #!> ek[svút ( u écmybitkmt;el N * edy ³ u ( ( k> kmt;cmybit,b, C A edm,[ A B C ( ( S u u u u > tg KN lim S K> cmebhrkb; N * ektg V u Edl g CGuKm_kMt;edy ig S' curbgðjf ig g( d g( ( ( V V V V S' lim S' S g( d ehytjrk S ' - -

14 KNGMgetRkl ³ si d si cos 6 ig ##> ek[ f CGuKm_kMt;kñúgceøH k> curbgðjf f (si d f(si d si d cos > Guvtþ_ curkn #$> ek[gukm_ f kmt;el ; b Edl ; b : f( b f( b b b curbgðjf f(d Guvtþ_ curkn f(d si d si cos 6 ; si d cos l( f(, t ( t f( ( t f(t t t #%> ek[gukm_ k> bgðjf Edl t - -

15 > KN f(d rct J K> Tjrktémø d #^> ekmsvút ( kmt;cmebhrkb;! ( e d edy ³ k-curknty -curbb k; CGuKm_é ryctj[f ³ e p P! K-cUrrklmt lim ryctjf lim e 788!!!! #&> ek[ f CGuKm_KUel, (d k> curbgðjf f(d, q,q > Guvtþ_ ³ f q - -

16 curkn cos d J e b b f (d f( b d ig d #*> curbgðjf Guvtþ_ ³ curkn log ( t d #(> eksμtf f CGuKm_mYykMt;el R ehyepþogpþt;tmk;tmg³ f f cos d curkn f $> ek[svútgmgetrkl si cos d Edl N k> curkn CGuKm_é > curknplbuk S k k CGuKm_é ryctjrktémøélmt lim S - -

17 $!> ek[svútgmgetrkl d Edl N k> currktmk;tmgrvg ig > KNplKuN P, CGuKm_é K> curknplbuk S Pk P P P k CGuKm_é ryctjrktémøélmt lim S rkrubmþkn CGuKm_é $@> ek[gmgetrkl ig curbgðjf si d si cos cos J d si cos J ryctjrktémø ig J d, N $#> ekegygmgetrkl * k-kn CGuKm_é -KNplbUk S rklmt lim S - -

18 ( $$> ekegygmgetrkl d cos ( d * ig J, N, cos k-bgðjf J -KN ig J CGuKm_é K-eRblTæplgelcUrbRgYmplbUk ³ S coscos cos cos cos cos $%> ek[rtekn ABC myymmmu A,B, C CmMuRsYc si A si B si C currsyf 8 cos A cos B cos C $^> ek]bmfsmkr 8 6 mb sb,, curkn S - 5 -

19 - 6 -

20 lmht;t curkmnt;rkb;gukm_ y f( f : R R edydwgfsmpb f(y, y f BitCic RKb; R tg[epñkkt;é ( MO dmenhrsy kmnt;rkb;gukm_ f : R R RKb; yk ig ektj f( f(, y R ekmsmpb -krn f( yk y ek f( f( f( eh f( b f( y CMYskñúg (* ek f( f( f( b f( f( M[ f ( CGuKm_efr tg f( c CMYskñúgsmkr (* ek c c c eh c, c duceh f( b f( c Edl [, c erbh c - 7 -

21 -krn f( yk y eh ( CMYskñúg (* ek f( f( f( f b f( k> cmebh f( eheygyk f(y f(y y R f( CMYskñúg (*ek > cmebh f( eheygyk ( yk, y kñúg (* ek f ( f( f( y ek f f( (** Ettm (** ek f( f( ehtuehektj f( mibiterbh f( srubmkekcemøy f(, R b f( c, R Edl c - 8 -

22 lmht;t cmebhrkb;cmybit (, currsyf si cos dmenhrsy Rsyf si cos tmvismpb Cuchy Schwr ek ³ si cos ( si cos edy si cos si( si si cos cos 6 ek duceh ( ( si cos - 9 -

23 lmht;t edhrsysmkr ³ 6 log ( log ( dmenhrsy edhrsysmkr ³ tg t ( smkrgcsresr ³ 6 t t log t t 6 8 t ( t ( t t 8 log ( yk u t ig v t t Edl t ek u v uv elkcker u uv v uv ( u v ektj u v b t t t t t t, t t ek log ( M[ t ek log ( M[ ; -cmebh -cmebh duceh - -

24 lmht;t ekmgukm_ f ( Edl KN f [ f [ f [ f ( ]] ] dmenhrsy KN f [ f [ f [ f ( ]] ] tg f ( ek f [ f [ f [ f ( ]] f ( ] f ( f[ f [ f [ f ( ]]] f ( f ( f ( ducehkrkn f [ f [ f [ f ( ]] ] KWRtUvkMNt;tY ésvútedlkmnt;edy f (, - -

25 - - smkrsmál;ésvútkw r r r ek r r r M[, r r tgsvútcmyy r r b ek b b b 6 M[ ( b CsVútFrNmRtmersug q igty b tmrubmþ q b b ( edy b ektj ( b b duceh ( ( ] ( [(

26 - - lmht;t5 ek[gukm_ b c b f cos si ( Edl, b curbgðjf c b f ( cmebhrkb;cmybit dmenhrsy bgðjf c b f ( tmvismpb ( ( ( b b b b yk b c b cos, si ig b c b b b cos, si ek ( cos si ( ( f b c b ektj cos (si ( b c f edy cos si RKb;cMYBit duceh c b f (

27 lmht;t6 ig y (si cos si( t y y cos(cot (si cos y Edl ig,,, k> cmebhrkb; tg u cos y si ig v cos y si ekmsvút ( ig ( y kmnt;edy currsyf ( >KN u ig K> Tjrk ig dmenhrsy k> Rsyf ( ekm ³ u ig ( v sutæetcsvútfrnmrt v CGuKm_é ig y CGuKm_é ig u ig v sutæetcsvútfrnmrt ( (si cos si ( t y - -

28 KuNGgÁTMgBrwg cos cos ek ³ cos(si cos si(cos si y ekm ³ y cos(cot (si KuNGgÁTMgBrwg y si si ek ³ cos(cos si cos y si(si cos y buksmkr ( ig ( GgÁ ig GgÁeK ³ cos y si cos( cos y si u cos y si edy ektj ersug q u dksmkr u cos u cos M[ u CsVútFrNmRtm ( ( ig ( GgÁ ig GgÁeK ³ cos y si si ( cos y si v cos y si edy ektj ersug q v v si v si M[ v CsVútFrNmRtm ( ( ( - 5 -

29 >KN u ig v CGuKm_é ig ekm u cos y si ek u u qu coscos ehy v cos y si cos ek v v q si cos cos v cos cos, v cos si duceh u K> Tjrk ig y CGuKm_é ig edy u cos y si ig v ek ektj ehy ektj cos y si u v cos cos cos si cos cos si u v y si cos cos si y si - 6 -

30 lmht;t7 ek[gukm_ f kmnt;rkb; R edytmk;tmg ³ f ( f ( si cos f ( RKb; R currsyf dmenhrsy Rsyf f ( ekm f f ( ( si cos ( kñúg ( ek ³ f ( f ( si( cos( f ( f ( si cos f ( f ( si cos ( ( ig f ( si cos CMYs edy dksmkr ( GgÁ igggáek ehtueh f ( si cos duceh f ( RKb; R - 7 -

31 lmht;t8 ek[gukm_ f kmnt;rkb; R edytmk;tmg ³ f ( f ( si cos kmnt;cmybit r ig edm,[gukm_ ( f ( r si( RKb; R dmenhrsy kmnt;cmybit r ig ekm f ( f ( CMYs edy ek ³ f Gcsresr ³ si cos f ( f ( si cos f ( f ( si 6 cos dksmkr ( ig ( GgÁ igggáek f ( si cos b f ( si cos ( ( ( - 8 -

32 CMYs edy f ( f ( ek ³ si( cos si cos( f ( ( cos si f ( (si cos si cos 6 6 f ( si( 6 f ( r si( r ; 6 edyerbobefobcmyywg duceh - 9 -

33 lmht;t9 ek[gukm_ f kmnt;rkb; 5 f ( si f ( currktémøtucbmput ig FMbMputé f ( dmenhrsy rktémøtucbmput ig FMbMputé f ( R edytmk;tmg ³ cos 5 f ( si f ( cos ( CMYs edy kñúg ( ek ³ 5 f ( si( f ( cos( 5 f ( f ( si f ( si f ( cos buksmkr (5 9si 5 si 5 cos f ( 5 9si 5 ( ig ( ek ³ si cos f ( 5 cos (5 si f ( 5 cos (5 si f ( (5 9si ( - -

34 - - ( ( 5 si ( cos cos si ( ( 5 cos ( si (5 si 5 cos ( si (5 si (5 si (5 cos ( f f f f f f f tmvismpb Schwr Cuchy ek ³ ( ( 9 6 si ( cos f f tm ( ig ( ek ³ ( ( ( ( 9 6 ( 5 ( 9 6 ( 5 f f f f f f f duceh ( mi f ig ( m f

35 lmht;t ek[gukm_ f ig gkmnt;rkb; f ( g( si f ( g( cos f ( ig g ( currkgukm_ dmenhrsy rkgukm_ ( ekm CMYs edy f ig g ( R edytmk;tmg ³ f ( g( si ( f ( g( cos ( kñúg ( ek ³ f ( g( cos( f ( g( si ( ( & ( ek f ( si buksmkr - -

36 ektj f ( si tm ( ektj g( si f ( si CMYs edy duceh f ek g( si( cos ( si ; g( cos - -

37 - - lmht;t ek[gukm_ f kmnt;rkb; (, edytmk;tmg ³ l( ( ( f f currk (cos f CGuKm_é t t Edl dmenhrsy rk (cos f CGuKm_é t t ekm l( ( ( f f ( l( ( ( l ( ( f f f f CMYs edy kñúg ( ek ³ ( 6l( ( ( l( ( ( f f f f

38 buksmkr ( ig ( ek ³ f ( l( f ( l( cos cos si l( cos l(t l t edy t f (cos ehtueh f (cos l( t duceh l t - 5 -

39 lmht;t ek[gukm_ f kmnt;rkb; f ( f ( 8 ( e f ( lim curknlmt dmenhrsy f ( KNlmt lim ekm f( CMYs edy f( f( f( R edytmk;tmg ³ 8 e (e kñúgtmk;tmg f( 9f( 8 (e (e efvpldksmkr & ( 8f( 6 f( e e ( ek ³ e ( ek ³ 8(e e e e ( ( - 6 -

40 - 7 - eyg e e lim f( lim e e lim e e e lim e e lim duceh f( lim

41 lmht;t ek[gukm_ f : R R kmnt;edytmk;tmg ³ f( ( yf( k> kmnt;rkgukm_ f ( > kmnt; edm,[ ( dmenhrsy k> kmnt;rkgukm_ f ( ( yf( y ( yf( y y( f y ( yf( y yk, y CMYskñúg ( ek ³ f( M[ f( tg y u ig y v TMk;TMg ( Gcsresr ³ uf (v vf(u eh uv(u EcksmkreHwg uv ek ³ v y y( u v u v y y ( - 8 -

42 f(v v f(u u u f(u u u f(v v v v f( ( f( f( ( tmtmk;tmgehbb k;f CGuKm_efrRKb; f ektj duceh > kmnt; edm,[ f( ek ( tg cos 8cos cos 6cos ek k,k,, 6 5 {,, cos, cos, 8 8 ektj } duceh 5 cos 8-9 -

43 lmht;t cmebh ig b CcMYBit smkr b mb sy:gtic myyccmybit curkntémøtucbmputé b? MO 97 dmenhrsy KNtémøtUcbMputé b ekm b EckGgÁTMgBrésmkreH wg ek ³ ( tg b ( b ( b smkrehgcsresr ³ b b b ( - -

44 tmvismpb ( b ( tm & ( Cuchy Schwr ekm ³ b ( ek ³ ( ( b ( ( b ( [ ( ] b 9 b 5 t edy t 9 b t 5 t 9 f(t t 5 t 9 (t (t f'(t (t (t yk ( eh ehy ek tggukm_ ek t - -

45 ektj f (t CGuKm_ekRKb; t tmlkçn³égukm_ekek f(t f( 9 9 Et f( 5 eh ektj b f(t duceh témøtucbmputé b esμwg f(t 5 - -

46 lmht;t5 ek[gukm_el f kmt;edytmk;tmg ³ 7 f( f( ig f( 5 currsyf ( dmenhrsy Rsyf ( smkrsmál; 9r 9r 6 Edl,,, 9f( f CRbPKsRmYlmi f CRbPKsRmYlmi ³ tgsvútcmyy ek r 6 r 9r r r 6 r 6 (r r f( f( edy f( f( 6 9f( - -

47 eh f( 6 9f( 9f( 9f( 8 6f( 9f( f( 9f( f( f( ekm 9 efr M[ ( CsVútBVþmplsgrYm 9 Et d ig 7 f( eh ( d 5 9( eh f( f( f( ek 9 6 ehy - -

48 ( f( 9 6 ( tg GCD(, d ek dp Edl GCD(p,q dq ekm ( ( dp (p dq qd ektj Eckdc;wg d eh d M[ GCD(, duceh f( CRbPKsRmYlmi - 5 -

49 lmht;t6 ek[svútécmybit ( u kmnt;edy ³ 7 u ig u u u RKb; bgðjfekgckmnt;cmybit Edl u (u cmebhrkb; ryckn u CGuKm_é dmenhrsy kmnt;cmybit ekm u u u ( ehy u (u ( ( CMYskñúg ( ek ³ yk u u u u ( (u u u u smkrehbitcic cmebhrkb;témø luhrtet ³ - 6 -

50 duceh M;[ KN u CGuKm_é ³ cmebh ek u (u l(u v v v ektj l( u l(u ( tg v l(u tm ( ek M[ ( v CsVútFrNmRt mplefobrym q ig l(u l ek v v q edy v l(u ektj duceh v l l l v u - 7 -

51 lmht;t7 ek[cmykmupøic cos i si kñúgbøg;kmpøic (o, i, j ekeha M CcMucrUbPBé curkmnt;témøtucbmput ig FMbMputé r OM? dmenhrsy kmnt;témøtucbmput ig FMbMputé r OM ekm ³ OM Edl R b ( cos ( si OM cos si b OM cos si ek r tmvismpb cos si cos si Cuchy Schwr ekm ³ cos si b cos si ektj r RKb; R duceh ig r mi r m - 8 -

52 - 9 - lmht;t8 ek[cmykmupøic,, ehyepþógpþt;tmk;tmg ³ ig currsyf }, { dmenhrsy Rsyf }, { ekm ek b tg ek ³ ( ( ( (

53 ek ( b -eb b ek ( ( b ( -eb ek ( ( {, duceh } - 5 -

54 lmht;t9 ek[cmykmupøic ig Edl currsyf dmenhrsy Rsyf tmvismpbrtekn b b ek ³ ( ehy duceh - 5 -

55 lmht;t ek[svútcmybit ( kmnt;edy ³,,,,, i i cmebhrkb; i CTRmg;RtekNmRtrYcTjrk ektgsvútcmykmupøic k> currsyf > curdk; CGuKm_ é K> TjrktYTUeTAésVút et ( CsVútYbb et? dmenhrsy i k> Rsyf cmebhrkb; ekm i i ek edy - 5 -

56 - 5 - ek i i ( i i ( i i duceh i > dk; i CTRmg;RtekNmRt ³ ek isi cos i i Tjrk CGuKm_é ³ edy i eh ( CsVútFrNmRtécMY kmupøicedlmersug isi cos i q igty si i cos i i

57 tmrubmþ cos isi q (cos isi tmrubmþdwm½rek K> TjrktYTUeTAésVút ekm ek ( edy i ( i cos isi ( ( ek tmtmk;tmg & ( duceh ehy si ( CsVútYbEdlmYb p 6 si - 5 -

58 lmht;t ek[svútécmykmupøic ( kmnt;edy ³ i ig Edl,,, i i k> tg w bgðjf ( w CsVútFrNmRtécMY kmupøic ryckn w CGuKm_é edysresrltæplctrmg; RtekNmRt > Tjbgðjf cos (cos isi dmenhrsy k> bgðjf ( w CsVútFrNmRtécMYkMupøic ³ ekm w ek w duceh w i i ( i ( CsVútFrNmRtécMYkMupøic i w

59 KN w CGuKm_é ³ ek w w q edy ig ek duceh i w cos isi 6 6 i q cos isi 6 6 w (cos isi 6 6 w cos isi rubmþdwmr½ 6 6 cos (cos isi w eh w cos isi 6 6 cos isi cos cos (cos isi cos (cos isi > Tjbgðjf ekm duceh

60 lmht;t ek[svútécmybit ( u ig ( v kmnt;edy ³ u v ig u v u v Edl u v u i v k> ekbiitüsvútécmykmupøic currsyf ( CsVútFrNmRtécMYkMupøic ryckn CGuKm_é edysresrltæplctrmg;rteknmrt > smedg u ig dmenhrsy k> Rsyf ekm u v CGuKm_é ( CsVútFrNmRtécMYkMupøic ³ i v ek u i v duceh u v i u i (u v iv ( CsVútFrNmRtécMYkMupøic i

61 KN CGuKm_é ³ ek q Et u iv ig q ek i cos i cos isi (cos isi cos isi duceh > smedg u ig ekm u edy duceh u rubmþdwm½r v CGuKm_é i v cos isi cos ig v si isi

62 lmht;t ek[svútécmybit ( u ig ( v kmnt;edy ³ u v ig u v u v uv u i v k> ekbiitüsvútécmykmupøic currsyf ryctjf > smedg u ig v CGuKm_é dmenhrsy k>rsyf ryctjf ekm u i v ek u iv u u (u duceh v iv iu v iu Edl ³ v (iv

63 mü:getoteb eh eb eh 8 eb eh k ]bmfvbitdl;tyt k KW k k eygwgrsyfvbitdl;tyt k KW k k ekm k k Ettmkr]bm k k ek k k ( Bit duceh > smedg u ig v CGuKm_é ekm edy u iv i ek duceh u (cos isi (cos isi (cos isi cos ; v si - 6 -

64 lmht;t ek[svútécmybit ( kmnt;edy ³ d e d e,,, d e e e Edl CcMyKt;FmμCti k> KN ; ryctjrktémøé > KN CGuKm_é ryctjrktémø ig ( K> cmebh [,] curerbobefobtémø e ig bgðjf ( CsVútek edymicmc;kn X> bgðjfcmebhrkb; [,] ekm ryctjrkkesmgmé tmkrkn etsvút ( mlmtb et? dmenhrsy k> KN ; ryctjrktémøé d e d ek e e e e e d - 6 -

65 ( e d d e duceh ek e e duceh l( ( e 'd d e e l( e l( e l l( e e e ehy l( l( e > KN CGuKm_é ryctjrktémø ig ekm ek duceh e e d ig d e e ( e e d e d e e e ( ; e cmebhrkb; - 6 -

66 eb ek e edy l( ek e l( eb ek e e e e (e l( (e e l( ektj duceh ( K> erbobefobtémø e ig e ( ekm e e e (e cmebh [,] ekm e ( ek e e e (e ( e e ig duceh e e bgðjf ( CsVútek edymicmc;kn ( edy e e RKb; [,] ek ³ ( e e duceh e e eh e ( d ³ e M[ ( CsVútek e e d

67 X> bgðjfcmebhrkb; [,] ekm cmebh [,] e ektj e e ek e TjrkkeSmGmé tmkrkn edy duceh edy e d ³ e e e e e e e d d e d e e e RKb; e lim eh lim eh - 6 -

68 lmht;t5 ek[svút Edl d k> bgðjf ( CsVútcuH > KN CGuKm_é K> currsyf rycrk dmenhrsy k> bgðjf lim RKb; [,] ( ( CsVútcuH ( ekm EckGgÁTMgBrwg ek ³ duceh d ( CsVútcuH d cmebh

69 > KN CGuKm_é d d d ( duceh K>Rsyf ( ( rycrk lim edy ( CsVútcuHeHcMeBHRKb; ekm duceh ( ( ig lim

70 lmht;t6 KNGMgetRkl dmenhrsy KN si cos si cos 6 d tg t eh d dt cmebh ek t cmebh ek t GMgetRklGcsresr ³ ( tsi( tcos ( tsitcos sitcos duceh tdt cos d ( t( dt tdt sitcos 7 t 6 tdt

71 lmht;t7 KNGMgetRkl d dmenhrsy KN d ek d ( ( d l( l l d ( ( ' ( duceh l(

72 lmht;t8 ek[gmgetrkl d Edl kmnt; edm,[ migrs½ywg ryckn é EdlrkeXjgel dmenhrsy kmnt; edm,[ migrs½ywg d ekm Edl t eh d dt tg eb eh t ig eb eh t ek (t (t dt 5 t t dt cmebhtémø edm,[ migrs½ywg luhrtet 5 eh 5 cmebh 5 t dt t 5 l 5 5 ek 5 l t duceh 5 ig 5 5 l

73 lmht;t9 ek[gmgetrkl d Edl kmnt; edm,[ migrs½ywg ryckn cmebhtémø é EdlrkeXjgel dmenhrsy kmnt; edm,[ migrs½ywg ekm d Edl t eh d dt tg eb eh t ig eb eh t ek (t (t dt t t dt edm,[ migrs½ywg luhrtet eh tdt t duceh ig l( l t t l( - 7 -

74 lmht;t cos bsi currsyf d b Edl ig b CBrcMYBit dmenhrsy cos bsi Rsyf d b ekm cos bsi d tmvismpb ekm cos bsi b ektj cos bsi Cuchy Schwr RKb; R cos bsi b d d d rct rct( cos bsi d b edy duceh d - 7 -

75 lmht;t ek[svút ( u écmybitkmt;el N * edy ³ u ( ( k> kmt;cmybit A,B, C edm,[ ( ( A B C > tg S u u u u KN lim S K> cmebhrkb; N * ektg V u Edl g CGuKm_kMt;edy ig S' curbgðjf ig g( d g( ( ( V V V V S' lim S' S g( d ehytjrk S ' - 7 -

76 - 7 - dmenhrsy k> kmt;cmybit C,B, A Edl C B A ( ( eyg C( B( ( A( smmul A C B (A C B (A ektj C B A C B A A M[ C, B, A > KN lim S m k k u u u u u S ( ( ( ( (k (k (k k (k k k k k k duceh ( ( lim S lim

77 K> bgðjf S' S g( d cmebhrkb; N * eygmv u Edl g CGuKm_kMt;edy ig S' V V V eyg S ' V duceh mü:getotekm mlmmduc ek edy eyg S' g( g( d ( ( V u g( d k k u g(d g(d g( d k S' S g( d g( ( ( u ( ( g( ( S tmsrmygel ( ( d ( ( ( - 7 -

78 ( ( l ( ( l l l l( l( l( ( ( l l l ( ( duceh ( ( l ( ( l ' S ig l S' lim

79 lmht;t KNGMgetRkl ³ 6 d cos si si ig 6 d cos si si dmenhrsy KNGMgetRkl 6 d cos si si tg u M[ d du eb 6 eh u ehy eh 6 u ek du ( u ( cos u ( si u ( si 6 J d si cos cos du u si u cos u cos 6 6

80 eyg ³ d d cos si cos si J J duceh d cos si si 6 ig d cos si cos J 6

81 lmht;t ek[ f CGuKm_kMt;kñúgceøH k> curbgðjf f (sid ; f(sid si d cos > Guvtþ_ curkn dmenhrsy k bghjf f (sid f(sid tg u M[ du d cmebh eh u ig cmebh eh u ek f(si d ( uf si( u ( du f(si d ( uf(siudu f(si d f(siudu uf (siudu f(si d f(si d f(si d f (sid f(sid duceh

82 > Guvtþ_ si si ekm d d ek duce¼ f(si d cos si si d rct(cos cos si d cos

83 lmht;t ek[gukm_ f kmt;el ; b Edl ; b : f( b f( b b b curbgðjf f(d f(d Guvtþ_ si curkn d cos dmenhrsy b b b bgðjf f(d f(d tg u b M[ du d cmebh eh u b ehy b eh u b ek f(d ( b uf( b u( b du - 8 -

84 - 8 - b b b b b b b b b b b f(d b ( f(d f(d f(d b ( d f( f(d b ( f(d d b f( b ( f(d udu b uf( b ( f(d duceh b b f(d b f(d Guvtþ_ KN d cos si tg cos si f( edy cos cos ek cos si (cos si f( eygm f( cos si ( cos si( f(

85 tmsrmygeleyg si si d d cos cos tg cmebh eh eyg duceh U cos M[ du si d U ig eh u du U U rcsi rcsi rcsi si d cos du U - 8 -

86 lmht;t5 l( ek[gukm_ f(, t ( t k> bgðjf f( ( t f(t t Edl t > KN t f(d rct J K> Tjrktémø d dmenhrsy t ( t k> bgðjf f( ( t f(t t t l( m f( eyg t f( t l( ( t t t t l( t ( t ( t ( t - 8 -

87 t l l( t l l( t f( ( t t ( t t ( t t ( t l( t f( ( t t t t t ( t f( ( t f(t t t duceh > KN tg t t f(d ( t M[ d d cmebh eh t ehy eh t t t ek f( dt f( t ( t t ( t t ( t edy f( ( t f(t ek t t ( t ( t f(t dt t ( t dt f(tdt t l( duceh f(d 8 dt - 8 -

88 rct K> Tjrktémø J d tg u rct du d M[ dv d V l( l( J rct l J l l 8 rct J d (l 8 eyg d duceh

89 lmht;t6 ekmsvút ( kmt;cmebhrkb; edy ³! ( e d k-curknty -curbb k; CGuKm_é ryctj[f ³ e p P! K-cUrrklmt lim ryctjf lim e 788!!!! dmenhrsy k-ekm ( e d ( e u tg dv d! du d e M[ d ek ( e e ( d e e duceh e v e

90 -bb k; CGuKm_é ekm ( M[ tg e d! ( e d (! u ( du ( ( M[ dv e d v e ( e ( e d (! (! ( e d (!! (! (! ek duceh Tj[f e p P! (! :! :! ekm cmebh cmebh >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

91 cmebh :! edyefvplbuktmk;tmgehggá ig GgÁ ek ³ edy e e!!!!! e e!!!! p p! duceh K-rklmt lim cmebh, ekm e e ig ( ek ( e ( e( e M[ ( d ( e d ( edy! ( d! ( e ektj!(!( kln M[ duceh lim!(! d

92 Tjf lim e 788!!!! ekm e p p! M[ e p p! ek lim lim e e p p! erbh lim duceh lim e 788!!!!

93 lmht;t7 ek[ f CGuKm_KUel, (d k> curbgðjf f(d, q,q > Guvtþ_ ³ curkn f q cos d J e ig d dmenhrsy (d k>bgðjf f(d, q,q ekm f q f(d f(d f(d q q q tg t M[ d dt igcmebh, M[ t, t f (d f( tdt q f( tdt q f( d ek t t q q q q edy ( f CGuKm_KUeH f( f(,, - 9 -

94 ektj f (d q q f( d q etacyskñúg! ek ³ f(d q f(d f(d q q q (q f(d f(d q duceh f (d f(d, q,q q > Guvtþ_ ³ KNGMgetRkl cos d edy cos d si cos duceh d cos CGuKm_KUeHeK ³ - 9 -

95 KN J d e edy f( CGuKm_KUeRB¼ f( f( eyg J d d (9 8 9 ( d e duce¼ J - 9 -

96 lmht;t8 b b curbgðjf f (d f( b Guvtþ_ ³ d curkn log( t d dmenhrsy b b bgðjf f (d f( b d tg b t M[ d dt ig cmebh,b M[ t b, b b ek f (d f( b t( dt f( b b b b f (d f( b d duceh Guvtþ_ ³ KN log ( t d tdt - 9 -

97 eyg M[eKTj log t( d t log d t t t log d t log d t log log( t d log d log( td log( t d - 9 -

98 lmht;t9 eksμtf f CGuKm_mYykMt;el R ehyepþogpþt;tmk;tmg³ f f cos curkn f d dmenhrsy KNGMgetRkl f d eygm f(d f(d tg t M[ d dt ig cmebh, f(d M[ t, ek f (d f( t( dt f( tdt f( d

99 ektj f( d f(d f( f( d edy f f cos si si f d ek si d si d cos duceh

100 lmht;t ek[svútgmgetrkl si cos d Edl N k> curkn CGuKm_é > curknplbuk S k k CGuKm_é ryctjrktémøélmt lim S dmenhrsy k> KN CGuKm_é eygm ³ si cos d si cos cos d si ( si cos d U si M[ du cos d tg cmebh [, ] M[ U [, ]

101 eyg ³ ( ( ( ( U U du U du U du U ( U duceh ( ( > KNplbUk k k S tmsrmygeleygm ³ ( (

102 eyg ³ ( ( 8 ( ( ( ( ( ( k k k k k k k k S k k k duceh ( ( 8 ( ( S KNtémøélmt lim S eygm S eyg lm S lim duceh S lim

103 lmht;t ek[svútgmgetrkl d Edl N k> currktmk;tmgrvg ig > KNplKuN P, CGuKm_é K> curknplbuk S Pk P P P k CGuKm_é ryctjrktémøélmt lim S rkrubmþkn CGuKm_é dmenhrsy k> rktmk;tmgrvg ig eygm d ig tg U dv d M[ d du d V d - -

104 - - eyg d d d d d ( d ( tg d dv U M[ V d ( du eyg d ( ektj ( ( M[ duceh,

105 - - > KNplKuN, P CGuKm_é eygm P M[ P edy M[ ektj P P erbh P MeGy P P eyg ( k ( k k k k k k k k k P P P P 5 P P P P P P k k k k k k P P ( k ( k ( k ( k k k M[eKTj ( ( 6 P ( ( 6 P erbh P edy rcsi d

106 ig ek duceh d ( 6 P ( ( ( ( P ( ( K> KNplbUk S Pk P P P eygm k P ( ( ( ( ( S k k(k (k (k ( ( ( ( eyg duceh S ( ( ( 8 ( ( ig ( 8 ( ( lim S 8 X> rkrubmþkn CGuKm_é tmsrmygeleygm, - -

107 -krn p cmyess p eyg p p b ektj M[ duceh p p p p p 6 p 5 7 p 5 p p p 6 p p 6p p 579(p 6p p 579(p -krn p cmyku p eyg p p b ektj M[ edy p p p p p 5 p 6 p p 6 8 p p 5 p 6 8 p 5(p p 68(p 5(p duceh p 68(p - -

108 lmht;t ek[gmgetrkl ig si d si cos cos J d si cos curbgðjf J ryctjrktémø ig dmenhrsy bgðjf J ryctjrktémø ig J eygm si si cos tg t M[ d dt cmebh M[ t ehy M[ t d J - 5 -

109 si si cos si ( t ( dt si ( t cos ( t cos t dt J cos t si t J duceh mü:getoteyg ³ J J duceh si J si cos J d J d cos si cos d - 6 -

110 lmht;t ekegygmgetrkl * d, N k-kn CGuKm_é -KNplbUk S ryctjrklmt lim S dmenhrsy k-kn CGuKm_é * ( d, N d ( ( duceh ( ( -KNplbUk eygm S ( ( - 7 -

111 - 8 - eyg ( 5 ( ( S ( duceh S Tjrklmt lim S eyg lim S lim

112 lmht;t ekegygmgetrkl ( d cos ( d * ig J, N, cos k-bgðjf J -KN ig J CGuKm_é K-eRblTæplgelcUrbRgYmplbUk ³ S coscos cos cos cos dmenhrsy k-bgðjf eyg ³ J cos d d ( d ( d cos cos cos cos duceh J -KN ig J CGuKm_é ( ( eyg d t t( t( cos - 9 -

113 ( d ig J t t( duceh K-KNplbUk S cos coscos eyg ³ si cos(cos( cos cos (, J t si( cos cos( cos S coscos coscos cos k kcoskcos(k si k si( duceh S si coscos( cos cos si J - -

114 lmht;t5 ek[rtekn ABC myymmmu A,B, C CmMuRsYc si A si B si C currsyf 8 cos A cos B cos C dmenhrsy si A si B si C Rsyf 8 cos A cos B cos C si A si B si C tg cos A cos B cos C t A t B t C cos A cosb cosc tmvismpb Cuchy Schwr ek ³ (t A tb tc ( cos A cosb cosc tggukm_ f( t Edl (, ek f'( t cos f''( t ( t tmvismpb Jese ek ³ f(a f(b f(c A f( B C - -

115 A B C b t A tb tc t( t ektj (t A tb tc 7 ( tggukm_ g( cos Edl (, ek g'( si g''( cos M[ ( tmvismpb Jese ek ³ g CGuKm_e:g, (, A B C g(a g(b g(c g( A B C cos A cosb cosc cos( ektj ( cos A cosb cosc KuNvismPB & ( ( GgÁ ig GgÁeK ³ (ta tb tc 7 8 ( cos A cosb cosc si A si B si C cos A cos B cos C tm ( & ( ektj 8 duceh 8 cos - -

116 lmht;t6 ek]bmfsmkr 8 6 mb sb,, curkn S dmenhrsy KN S edy,, Cb ssmkr 8 6 ek ( ( ( mü:getot, RtUvepÞógpÞt;smkr 8 6 ek 8, 8 6 ( 8 6 (5 8 6 (6 buksmkrehggá ig GgÁeK ³ 8 ( 6( (7 - -

117 - - yktmk;tmg ( CYskñúg 7 ( ektj ³ (8 8 KuNsmkr (5,(6, ( wg,, erogkñek ³ buksmkrehggá ig GgÁeK ³ (9 ( 6( 8 S edy ( ( b ( 8 ( yktmk;tmg 8&( ( CMYskñúg 9 ( ek ³ 8 6( 8 S M[ S duceh S

118 !> ek[, y, Edl y y edhrsysmkr 9 y #> ek[rteknbrmrcug ³ 7 9, 9, ig 9, 9,b Edl b curkmnt;rkb;kucmykt;vic m (,b mépþrklesμkñ $> ek[ctuekne:g ABCD myy bþt; (AD ig (BC kt;kñrtg;cmuc P M ig N CcMuckNþléGgát;RTUg [AC] ig [BD] erogkñ currsyfrklépþéctuekn ABCD esμwg dgé RkLépÞRtekN MNP? - 5 -

119 %>ek[ctuekne:g ABCD myymrcug AB, BC b b c d CD c ig DA d tg p currsyfkesmépþrklrbs;ctuekn ABCD kmnt;edy ³ SABCD A C (p (p b(p c(p d bcdcos CRbPKsRmYlmiRKb; N 5 ^> currsyf &> currsyfcmymyy²gerkmeh CKUbécMYKt; ³ ,,, *>curkmnt;rkb;kucmykt;vic m (,y, edm,[ ³ y y ( y( CkerRkd (> edhrsyrbb½æsmkr ³ log( y log(y log( y y!> edhrsyrbb½æsmkr log(y log log y log(y log y log log( log log - 6 -

120 - 7 -!!>kmnt;rkb;cemøycmybitérbb½æsmkr ³ y y y!@>curkmnt; 5 5 by ebcmybit,b, ig y epþógpþt; ³ by 6 by 7 by by!#> edhrsyrbbæ½smkr y y y!$> edhrsyrbb½æ y ( (y y (

121 !%> edhrsyrbb½æsmkr!^> edhrsyrbb½æsmkr!&> ek[gukm_ f( 6 9(y y y 9( 9( by ( y by c (y c ( f : R R edydwgf ³ f( f( f( 7 currsyf f CGuKm_Yb!*> curkmnt;gukm_ f ( ebekdwgf ³ f( f( cos RKb; R!(> curkmnt;gukm_ ( f( f ebekdwgf ³ f( cos RKb; curkmnt;gukm_ ( f( f ebekdwgf ³ f( si RKb; R - 8 -

122 @!> curkmnt;gukm_ f ( ebekdwgf ³ f( si f( RKb; ek[gukm_ f : R R kmnt;edytmk;tmg ³ cos si f( f( cos f( cmebhrkb; currsyf ek[ (,b, c CcMYBitviC m currsyf ³ b 9 c b b c c ( b c cot cot cot 7 7 curbgðjf ek[rtekn ABC myyedydwgf ³ si AsiB sibsic sicsi A k ( si A( sib( sic ( k currsyf ABC mmmumyyekg kñúgrkb;rtekncurrsyf r pckøhbrimrt ig r CkMrgVg;crwkkñúgéRtekN - 9 -

123 @*> bgðjfkñúgrkb;rteknekm ³ cos A cosb b cosc c 8 6p bgðjfkñúgrkb;rteknekm ³,b, c CRCug ig pckøhbrimrté ABC b c A B C (si si si bc c b,b, c CRCugé ABC Edl #> ek[ u ( CsVútcMYBitkMNt;edy ³ u o ig u u 8u Edl N currsyf u ( ( Edl, Cb ssmkr 8 ( CsVútcMYBitkMNt;edy ³ #!> ek[ u u ( ( ( ( l( k> Rsyf RKb; > RKb; k ig bgðjf ³ k k l( k k - -

124 K> TjrkkeSmGmé u rycknlmtvkln #@> ek[ ( u CsVútcMYBitkMNt;edy ³ u ig u u l Edl KN u CGuKm_é ( CsVútcMYBitkMNt;edy ³ ##> ek[ u l( u ig u u Edl, KN u CGuKm_é #$> ek[ ( l(cos y dy k> bgðjf ( ( ( l ( l(cos y dy d 6 8 f( 6, Z > erbtmk;tmggelcurkn #%> currsyf #^> ek[gukm_el curkmt; edm,[ f( CkerRkd? - -

125 b #&> ek[ (,,, b R curbgðjfmbrcmybit ig ( EdlepÞógpÞt; [ ( ] ( r r, r, ( r ( Tjbgðjf ( ( rycbb k;fcmebhrkb;cmybit ek ( #*>ek[esekg (c : y igbþt; y m kmt; medm,[bþt; ( d kt;esekg ( c BrcMucqøúHKñeFob wgbþt;buhtmyyégks½kugrede #(>KNGMgetRkl ³ d t ig $> ekegygmgetrkl³ t t ( J t k-knkesm -KNlmtékeSm d cot t kln t d - -

126 $!> ekegysvútcmybit, Nedy k-knty ig -Rsyf CsVútFrNmRt ryckn K-rkplbUk S $@> ekegygmgetrkl e d k-curknty -curbgðjf³ K-cUrRsybB k;vismpb ³ cos cos d CGuKm_é lim S ryckn, N,e,788 e e, X-KNlmt lim ig lim - -

127 $#>ekegygmgetrkl ³ e cos d ig J e si d, N k-kn J wg JCGuKm_é -TjeGyUvtMél wgj $$> ek[gmgetrkl d, N ig t d, N ig t d k> KNtY ig > rktmk;tmgrvg $%> ek[gmgetrkl k> KNtY ig > rktmk;tmgrvg K> KN K $^> ek[gmgetrkl cos d, k> KNtY ig > rktmk;tmgrvg K> KN K N ig cos d - -

128 $&> ek[cmy N currsyf N CkerRkd $*>ek[cmykmupøicbr Z ig Z k> curbgðjf Z Z Z Z > CTUeTAcMeBHRKb;cMYkMupøic Z, Z,Z,, Z curbgðjf ³ Z Z Z Z Z Z $(>ek[cmykmupøic Z (cos i(si cos si currktémøbrmrbs;m:udúlécmykmupøicgel %> ek[ Z d Z CcMYkMupøicBrmm:UDúlesμ! iggkuym:g; erogkñ, k> currklkç½nðel, edm,[ Z Z usbsuü Z > kñúglkç½nðgelehcurrsyf Z Z CcMYBit ZZ - 5 -

129 %!> K[keSmplbUk ³ cos cos cos C cos cos cos si si si S cos cos cos ( k,k Z U Z C is U k> curbgðjf Edl U it rycsresr ZCrgBiCKNit > TjrkkeSmbRgÜmé C ig S %@-k> curkntémørkdé cos, si, t > cursresr Z i 5 5 CTRmg;RtekNmRt ryckn Z %#-ek]bmfsmkr (E b c mb sbrccmykmupøicedltgerogkñedy Z ig Z igcmebhrkb;cmykt;rwlthv ektg S Z Z 5 5 5,,,b,c R S bs cs 5 5 i 5 i 5 M k> currsybb k;f > Guvtþ_ edymibøtkn - 6 -

130 %$-ek[cmykmupøic i k-cursresr CTMrg;BiCKNit ryccursresr ig CTMrg;RtekNmRt K-TjrktMélRkdé cos ig si 8 8 %%-ek[kesm ³ C cos cos cos( S si si si( k> curbgðjf C i SCplbUkésVútFrNmRtcMYkMupøicmYy > Tjbgðjf ³ C ( si( cos si ig %^- k> curbgðjf ³ S i i i i e e e e i e 5 > TjrktémøéplbUk ³ k k C cos d S si k 5 k 5 ( si( si si - 7 -

131 %&> ekm 6 8 A cos 6cos cos cos B si 6si si si Z cos isi 9 9 curbgðjf A ib ( Z? k> ektg > cursresr ( Z CTRmg;RtekNmRt K> TjrkkeSmbRgYmé Aig B - 8 -