Traingle Function

eroberogeday lwm pl:ún briBaØabR&tKNitviTüa nig BaNiC¢kmµ

n 1

n n 1

2

n2 2 2 1

n

sin2tan .tan ....tan 2 .

2 2 sin 2

rkßasiTiæ 2008

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GñkshkarN_RtYtBinitübec©keTs

elak lwm qun

elak Esn Bisidæ

elak Titü em¨g

elakRsI Tuy rINa

elak RBwm suxnit

elak pl b�unqay

GñkrcnaRkb nig bec©keTskMBüÚT&r

kBaØa lI KuNÑaka

GñkRtYtBinitüGkçaraviruTæ

elak lwm miK:sir

© rkßasiTæi lwm pl:ún 2008


GarmÖkfa esovePA GnuKmn_RtIekaNmaRtEdlGñksikßakMBugkanénAkñúgéd

en¼xMJúáTánxitxMRsavRCav nigniBnæeLIgkñúgeKalbMNgTukCaÉksar

RsavRCavsRmab´GñksikßaEdlmanbMNgcgéc¼ cg´dwgGMBIemeronen¼

[kanÉtc,as´ . enAkñúgesovePA en¼ánRbmUlpþMúnUvRbFanlMhat´lð@

ya¨geRcIn nigmanlkçN¼xusEbøk@Kña.RbFanlMhatńImYy@xMJúáTán

xitxMeRCIserIsya¨gsRmitsRmaMgbMputRBmTaMgeFIVdMeNa¼Rsayya¨g

ek,a¼k,ayEdlGac[GñksikßagayylńigqabćgcaMGMBIviFIsaRsþeFIV

dMeNa¼RsaylMhatńImYy@ . büEnþeTa¼Caya¨gNak¾eday kgV¼xat

bec©keTs Kruekaslü nig kMhusGkçraviruTæRákdCaekItmaneLIg

edayGectnaCaBMuxaneLIy . GaRs&yehtuen¼xMJúáTCaGñkniBnæ

rgćaMTTYlnUvmtiri¼KnÉbbsSabnaBIsMNak´GñksikßakñúgRKb´mCÄdæan

edaykþIesamnsßrIkrayCanic©edIm,IEklMGesovePAen¼[kanÉtman

suRkitüPaBEfmeTot .

xMJúáTCaGñkniBnæsgÇwmfaesovePA GnuKmn_RtIekaNmaRt

mYyk,alen¼nwgcUlrYmnaMelakGñkeq<a¼eTArkC&yCMn¼kñúgkarsikßa nig

karRbLgRbECgnanaCaBMuxaneLIy .

sUm[GñksikßaTaMgGs´mansuxPaBlðman®áCJaQøasév nigman

sMNaglðkñúgqakCIvit nig karsikßa !

át´dMbgéf¶TI 7 Ex mkra qñaM 2008

GñkniBnæ nig RsavRCav lwm pl:ún


GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 1 -

emeronsegçb

GnuKmn_RtIekaNmaRt

1¿ TMnakTMngsMxan´@

enAelIrgVg´RtIekaNmaRteyIgtag CargVas´énmMu

)OM,ox(

.

eKán sinOQ,cosOP .

M

x 'x

o

y

P

Q


1


eKánTMnak´TMngsMxan´@énGnuKmn_rgVg´RtIekaNmaRtdUcxag

eRkam ½

1¿ 1cossin 22 4¿ 1cot.tan

2¿

cos

sintan 5¿

2

2

cos

1tan1

3¿

sin

coscot 6¿

2

2

sin

1cot1

2¿ rUbmnþplbUk nig pldk

1¿ acosbsinbcosasin)basin(

2¿ bsinasinbcosacos)bacos(

3¿ btanatan1

btanatan)batan(

4¿ acosbsinbcosasin)basin(

5¿ bsinasinbcosacos)bacos(

6¿ btanatan1

btanatan)batan(

3¿ rUmnþmMuDub

1¿ acosasin2a2sin

2¿ asin11acos2asinacosa2cos 2222

3¿ atan1

atan2a2tan

2

2


4¿ rUbmnþknø¼mMu

1¿ 2

acos1

2

asin 2

2¿ 2

acos1

2

acos2

3¿ acos1

acos1

2

atan 2

5¿ kenßam xtan,xcos,xsin CaGnuKmn_én 2

xtant

1¿ 2t1

t2xsin

2¿ 2

2

t1

t1xcos

3¿ 2

2

t1

t1xtan

6¿ kenßam a3tan,a3cos,a3sin

1¿ asin4asin3a3sin 3

2¿ acos3acos4a3cos 3

3¿ atan31

atanatan3a3tan

2

3

7¿ rUbmnþbMElgBIplKuNeTAplbUk

1¿ ])bacos()bacos([2

1bcosacos

2¿ ])bacos()bacos([2

1bsinasin

3¿ ])basin()basin([2

1bcosasin

3


4¿ ])basin()basin([2

1acosbsin

6¿ rUbmnþbMElgBIplbUkeTAplKuN

1¿ 2

qpcos

2

qpcos2qcospcos

2¿ 2

qpsin

2

qpsin2qcospcos

3¿ 2

qpcos

2

qpsin2qsinpsin

4¿ 2

qpcos

2

qpsin2qsinpsin

5¿ qcospcos

)qpsin(qtanptan

6¿ qcospcos

)qpsin(qtanptan

7¿ qsinpsin

)qpsin(qcotpcot

8¿ qsinpsin

)pqsin(qcotpcot

7¿ smIkarRtIekaNmaRt

1¿ smIkar vsinusin mancemøIy

Zk,k2vu

k2vu

2¿ smIkar vcosucos mancemøIy

Zk,k2vu

k2vu

3¿ smIkar vtanutan mancemøIy kvu 4


8¿ rUbmnþbEmøgFñÚEdlKYkt´sMKal´

1¿

cot)2

(tan

sin)2

(cos

cos)2

(sin

2¿

tan)(tan

cos)(cos

sin)(sin

3¿

cot)2

(tan

sin)2

(cos

cos)2

(sin

4¿

tan)(tan

cos)(cos

sin)(sin

5¿

Zk,tan)k(tan

cos)k2(cos

sin)k2(sin

5

GnuKmn_RtIekaNmaRt


9¿ RkahVikGnuKmn_RtIekaNmaRt

1¿ ExßekagGnuKmn_ xsiny

0 1

1

x

y

( C ) : y = sinx

2¿ ExßekagGnuKmn_ xcosy

0 1

1

x

y

( C ) : y = cosx

6

GnuKmn_RtIekaNmaRt


3¿ ExßekagGnuKmn_ xtany

0 1

1

x

y

( C ) : y = tanx

4¿ ExßekagGnuKmn_ xcoty

0 1

1

x

y

7

GnuKmn_RtIekaNmaRt


lMhat´ nig dMeNa¼Rsay

lMhatTI1

eK[ ba

ccos,

ac

bcos,

cb

acos

cUrRsayfa 12

tan2

tan2

tan 222

.

dMeNa¼Rsay

Rsayfa 12

tan2

tan2

tan 222

eyIgman 2

cos1

2cos,

2

cos1

2sin 22

eKán acb

acb

cb

a1

cb

a1

cos1

cos1

2tan 2

bac

bac

ac

b1

ac

b1

cos1

cos1

2tan 2

cba

cba

ba

c1

ba

c1

cos1

cos1

2tan 2

eyIgán cba

cba

cba

bac

cba

acb

2tan

2tan

2tan 222

cba

cbabacacb

2tan

2tan

2tan 222

1cba

cba

2tan

2tan

2tan 222

.

dUcen¼ 12

tan2

tan2

tan 222

8

GnuKmn_RtIekaNmaRt


lMhatTI2

eK[ a

btan .

cUrRsayfa 33

8

3

8

)ba(

1

b

sin

a

cos

.

dMeNa¼Rsay

Rsayfa 33

8

3

8

)ba(

1

b

sin

a

cos

eKman a

btan naM[

a

btan 2

eday

cos

sintan

eKán a

b

cos

sin2

2

smmUl ba

1

ab

cossin

a

cos

b

sin 2222

eKTaj ba

1

b

sin 2

naM[ )1(bIa(

b

b

sin43

8

ehIy ba

1

a

cos2

naM[ )2(bIa(

a

a

cos43

8

bUkTMnak´TMng )1( nig )2( Gg:nwgGg:eKán ½

343

8

3

8

)ba(

1

)ba(

ba

b

sin

a

cos

dUcen¼ 33

8

3

8

)ba(

1

b

sin

a

cos

.

9

GnuKmn_RtIekaNmaRt


lMhatTI3

eK[RtIekaN ABC mYyman c,b,a CargVas´RCugQm

erogKñaénmMu C,B,A .

tag p Caknø¼brimaRténRtIekaN .

k¿cUrRsayfa bc

)cp)(bp(

2

Asin

rYcTajrkTMnak´TMngBIreTotEdlRsedogKñaen¼ .

x¿cUrbgHajfa 8

1

2

Csin

2

Bsin

2

Asin

nig 2

3CcosBcosAcos .

dMeNa¼Rsay

Rsayfa bc

)cp)(bp(

2

Asin

eKman 2

Acos1

2

Asin 2

tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman

Acosbc2cba 222 eKTaj

bc2

acbAcos

222

A

B C

10

GnuKmn_RtIekaNmaRt


eKán bc4

acbbc2

2bc2

acb1

2

Asin

222

222

2

bc4

)cba)(cba(

bc4

)cb(a

2

Asin

222

eday p2cba ena¼ )bp(2cba,)cp(2cba

eKán bc4

)bp(2).cp(2

2

Asin 2

naM[ bc

)cp)(bp(

2

Asin

.

dUcen¼ bc

)cp)(bp(

2

Asin

.

eKGacTajTMnak´TMngRsedogKñaen¼dUcxageRkam ½

ab

)bp)(ap(

2

Csin,

ac

)cp)(ap(

2

Bsin

.

x¿bgHajfa 8

1

2

Csin

2

Bsin

2

Asin

tamsRmayxagelIeKman bc

)cp)(bp(

2

Asin

ab

)bp)(ap(

2

Csin,

ac

)cp)(ap(

2

Bsin

.

eKán )1(abc

)cp)(bp)(ap(

2

Csin

2

Bsin

2

Asin

tamvismPaBkUsIu .2

eKán )bp)(ap(2)bp()ap(

)bp)(ap(2c

)cp)(ap(2bap2

11

GnuKmn_RtIekaNmaRt


eKTaj )2(2

1

c

)bp)(ap(

dUcKñaEdr )3(2

1

a

)cp)(bp(

nig )4(2

1

b

)cp)(ap(

KuNTMnak´TMng )4(,)3(,)2( Gg:nwgGg:eKán ½

)5(8

1

abc

)cp)(bp)(ap(

tamTMnak´TMng )1( nig )5(

eKTaj 8

1

2

Csin

2

Bsin

2

Asin .

bgHajfa 2

3CcosBcosAcos

eyIgman

2

CBcos

2

CBcos2

2

Asin21CcosBcosAcos 2

2

Csin

2

Bsin

2

Asin41

)2

CBcos

2

CBcos(

2

Asin21

)2

CBcos

2

Asin(

2

Asin21

2

CBcos

2

Asin2

2

Asin21 2

eKTaj 2

Csin

2

Bsin

2

Asin41CcosBcosAcos

tamsRmayxagelIeKman 8

1

2

Csin

2

Bsin

2

Asin

ehtuen¼ 2

3)

2

1(41CcosBcosAcos 12

GnuKmn_RtIekaNmaRt


lMhatTI4

eK[RtIekaN ABCmYyman c,b,a CargVas´RCugQm

erogKñaénmMu C,B,A .

tag p Caknø¼brimaRténRtIekaN .

k¿cUrRsayfa bc

)ap(p

2

Acos

rYcTajrkTMnak´TMng

BIreTotEdlRsedogKñaen¼ .

x¿TajbBa¢ak´fa 2222 p2

Ccos.ab

2

Bcos.ac

2

Acos.bc .

dMeNa¼Rsay

Rsayfa bc

)ap(p

2

Acos

eKman 2

Acos1

2

Acos2

tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman

Acosbc2cba 222 eKTaj

bc2

acbAcos

222

A

B C

13

GnuKmn_RtIekaNmaRt


eKán bc4

acbbc2

2bc2

acb1

2

Acos

222

222

2

bc4

)acb)(acb(

bc4

a)cb(

2

Acos

222

eday p2cba ena¼ )ap(2acb

eKán bc

)ap(p

bc4

)ap(2.p2

2

Acos2

¦ bc

)ap(p

2

Acos

.

dUcen¼ bc

)ap(p

2

Acos

.

eKTajánTMnak´TMngRsedogKñaen¼dUcxageRkam ½

ab

)cp(p

2

Ccos,

ac

)bp(p

2

Bcos

.

x¿TajbBa¢ak´fa 2222 p2

Ccos.ab

2

Bcos.ac

2

Acos.bc

tamsRmayxagelIeKman bc

)ap(p

2

Acos

ab

)cp(p

2

Ccos,

ac

)bp(p

2

Bcos

eKTaj a.pp)ap(p2

Acosbc 22

c.pp)cp(p

2

Ccosab

b.pp)bp(p2

Bcosac

22

22

222

2222

pp2p3

)cba(pp32

Ccos.ab

2

Bcos.ac

2

Acos.bc

dUcen¼ 2222 p2

Ccos.ab

2

Bcos.ac

2

Acos.bc .

14

GnuKmn_RtIekaNmaRt


lMhatTI5

eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc .

k¿cUrRsayfa Ctan.Btan.AtanCtanBtanAtan .

x¿TajbBa¢ak´fa 33CtanBtanAtan .

dMeNa¼Rsay

k¿Rsayfa Ctan.Btan.AtanCtanBtanAtan

eyIgman CBA ¦ CBA

eKán )Ctan()BAtan(

CtanBtanAtanCtanBtanAtan

CtanBtanAtan1

BtanAtan

dUcen¼ Ctan.Btan.AtanCtanBtanAtan .

x¿TajbBa¢ak´fa 33CtanBtanAtan

eday C,B,A CamMuRsYc ( tamsmµtikmµ )

eKTaj 0Ctan,0Btan,0Atan

tamvismPaBkUsIueyIgGacsresr ½

3 Ctan.Btan.Atan3CtanBtanAtan

eday Ctan.Btan.AtanCtanBtanAtan

eKán 3 CtanBtanAtan3BtanBtanAtan 15

GnuKmn_RtIekaNmaRt


27)CtanBtanA(tan

)CtanBtanA(tan27)CtanBtanA(tan2

3

dUcen¼ 33CtanBtanAtan .

lMhatTI6


cUrRsayfa 1CcosBcosAcos 222

dMeNa¼Rsay

Rsayfa 1CcosBcosAcos 222

tag CcosBcosAcosT 222

CcosBcosAcos21

)BAcos()BAcos(Ccos1

Ccos)BAcos(Ccos1

Ccos)BAcos(Ccos1

Ccos)BAcos()Ccos(1

Ccos)BAcos()BAcos(1

Ccos2

B2cosA2cos1

Ccos2

B2cos1

2

A2cos1

2

2

2

2

2

A

B C

16

GnuKmn_RtIekaNmaRt


eKán CcosBcosAcos21CcosBcosAcos 222

eday C,B,A CamMuRsYcena¼ 0Ccos,0Bcos,0Acos

naM[ 1CcosBcosAcos21

dUcen¼ 1CcosBcosAcos 222 .

lMhatTI7


cUrRsayfa 2CsinBsinAsin 222 .

dMeNa¼Rsay

Rsayfa 2CsinBsinAsin 222

tag CsinBsinAsinT 222

CcosBcosAcos22

)BAcos()BAcos(Ccos2

Ccos)BAcos(Ccos2

Ccos)BAcos(Ccos2

Ccos)BAcos()BAcos(2

Ccos2

B2cosA2cos2

Ccos12

B2cos1

2

A2cos1

2

2

2

eday C,B,A CamMuRsYcena¼ 0Ccos,0Bcos,0Acos

dUcen¼ 2CsinBsinAsin 222 .

17

GnuKmn_RtIekaNmaRt


lMhatTI8

eK[ ba

1

b

xsin

a

xcos 44

Edl 0ba,0b,0a .

cUrRsaybBa¢ak´fa 44

10

4

10

)ba(

1

b

xsin

a

xcos

.

dMeNa¼Rsay

RsaybBa¢ak´fa 44

10

4

10

)ba(

1

b

xsin

a

xcos

eyIgman ba

1

b

xsin

a

xcos 44

eyIgán ab)xsinaxcosb)(ba( 44

0)xcosbxsina(

0xcosxsinab2xcosbxsina

01xcosxsin2)xcosx(sinabxcosbxsina

0)1xcosx(sinabxcosbxsina

0abxsinabxcosbxsinaxcosab

22

224222

222224222

444242

442424

eKTaj ba

1

ba

xsinxcos

b

xsin

a

xcos 2222

eKán ba

1

a

xcos 2

naM[ )1(

)ba(

a

a

xcos54

10

ehIy ba

1

b

xsin 2

naM[ )2(

)ba(

a

b

xsin54

10

bUksmIkar )1( nig )2( eKán

454

10

4

10

)ba(

1

)ba(

ba

b

xsin

a

xcos

dUcen¼ 44

10

4

10

)ba(

1

b

xsin

a

xcos

.

18

GnuKmn_RtIekaNmaRt


lMhatTI9

cUrKNna 7

4sin

7

2sin

7sinS 222

dMeNa¼Rsay

KNna 7

4sin

7

2sin

7sinS 222

eyIgán 2

7

8cos1

27

4cos1

27

2cos1

S

)7

8cos

7

4cos

7

2cos(

2

1

2

3

tag 7

8cos

7

4cos

7

2cosT

7

cos7

3cos

7

5cos

)7

(cos)7

3(cos)

7

5(cos

KuNGg:TaMgBIrnwg 7

sin2 eKán ½

7sin

7cos2

7sin

7

3cos2

7sin

7

5cos2

7sinT2

tamrUbmnþ )basin()basin(bsinacos2

7sin)

7sin(

7

6sin

7sinT2

7

2sin)

7

2sin

7

4(sin)

7

4sin

7

6sin(

7sinT2

7sin

7cos2

7sin

7

3cos2

7sin

7

5cos2

7sinT2

eKTaj 2

1T naM[

4

7)

2

1(

2

1

2

3S

dUcen¼ 4

7

7

4sin

7

2sin

7sinS 222

. 19

GnuKmn_RtIekaNmaRt


lMhatTI10

cUrKNna 7

8sin

7

4sin

7

2sinS

dMeNa¼Rsay

eKman 7

sin7

8sin,

7

3sin

7

4sin,

7

5sin

7

2sin

ehIy 7

8sin

7sin

7

2sin

nig 07

4sin

eKán 07

sin7

3sin

7

5sinS

elIkGg:TaMBIrCakaereKán ½

7sin

7

3sin2

7sin

7

5sin2

7

3sin

7

5sin2

7sin

7

3sin

7

5sinS 2222

tag 7

sin7

3sin

7

5sinM 222

)7

5cos

7cos

7

3cos(

2

1

2

3

)7

5cos()

7cos()

7

3cos(

2

1

2

32

7

2cos

7

6cos

7

10cos

2

3

yk 7

cos7

3cos

7

5cosT

KuNGg:TaMgBIrnwg 7

sin2 eKán ½

7sin

7cos2

7sin

7

3cos2

7sin

7

5cos2

7sinT2

tamrUbmnþ )basin()basin(bsinacos2

20

GnuKmn_RtIekaNmaRt


7sin)

7sin(

7

6sin

7sinT2

7

2sin)

7

2sin

7

4(sin)

7

4sin

7

6sin(

7sinT2

7sin

7cos2

7sin

7

3cos2

7sin

7

5cos2

7sinT2

eKTaj 2

1T naM[

4

7

4

1

2

3M

tag 7

sin7

3sin2

7sin

7

5sin2

7

3sin

7

5sin2N

0)

7sin(.sin2

7

8cos

7

6cos

7

4cos

7

2cos

7

6cos

7

4cos

7

8cos

7

2cos

eKán 4

70

4

7NMS2

eday 0S

ena¼ 2

7S .

dUcen¼ 2

7

7

8sin

7

4sin

7

2sinS

.

lMhatTI11

cUrRsayfa *INn,0)1(7

ncos4

7

ncos)1(4

7

ncos8 1n2n3

dMeNa¼Rsay

eKman INn,7

n3n

7

n4

eKán )7

n3n(sin

7

n4sin

21

GnuKmn_RtIekaNmaRt


0)1(7

ncos4

7

ncos)1(4

7

ncos8

)1(7

ncos4.)1(

7

ncos4

7

ncos8

])7

ncos1(43[.)1(

7

ncos4

7

ncos8

)7

nsin43(

7

nsin.)1()1

7

ncos2(

7

ncos

7

nsin4

)1()7

nsin4

7

nsin3(0)1

7

ncos2(

7

ncos

7

nsin4

)ncos(7

n3sin

7

n3cos)nsin(

7

n2cos

7

n2sin2

1n2n3

n2n3

2n3

2n2

n32

dUcen¼ *INn,0)1(7

ncos4

7

ncos)1(4

7

ncos8 1n2n3

lMhatTI12

cUrKNna 9

7cos

9

4cos

9cosS 333

dMeNa¼Rsay

KNna 9

7cos

9

4cos

9cosS 333

tamrUbmnþ acos3acos4a3cos 3 ¦ a3cos4

1acos

4

3acos3

kenßamEdl[GacsresrCa ½

)3

7cos

3

4cos

3(cos

4

1)

9

7cos

9

4cos

9(cos

4

3S

tag 9

7cos

9

4cos

9cosM

eday 9

13cos

9

4cos

9

13cos

9

7cos

9cosM

KuNnwg

3sin2

eKán 22

GnuKmn_RtIekaNmaRt


0)9

7cos(sin2

9

16sin

9

2sin

2

3M2

9

10sin

9

16sin

9

4sin

9

10sin)

9

2sin(

9

4sin

2

3M.2

3sin

9

13cos2

3sin

9

7cos2

3sin

9cos2

3sinM2

eKTaján 0M

tag 2

3

2

1

2

1

2

1

3

7cos

3

4cos

3cosN

eKán 8

3N

4

1M

4

3S

dUcen¼ 8

3

9

7cos

9

4cos

9cosS 333

.

lMhatTI13

eK[kenßam

7

5cos

7

3cos

7cosS 333

nig 7

5cos

7

3cos

7cosT 444

k¿cUrRsayfabIcMnYn7

5cos,

7

3cos,

7cos

Ca¦srbs´smIkar

01x4x4x8:)E( 23 .

x¿Tajrktémø ½

7

5cos

7cos

7

5cos

7

3cos

7

3cos

7cosN,

7

5cos

7

3cos

7cosM

nig 7

5cos

7

3cos

7cosP

.

K¿KNna 7

5cos

7

3cos

7cosQ 222

rYcTajrktémøSnigT23


dMeNa¼Rsay

k¿RsayfabIcMnYn 7

5cos,

7

3cos,

7cos

Ca¦srbs´smIkar

01x4x4x8:)E( 23

tag 3,2,1n,7

1n2cosx n

Ca¦smIkar )E( eKán

)*(0

7

)1n2(sin

7

)1n2(3sin

7

)1n2(sin

7

)1n2(4sin

0

7

)1n2(sin

7

)1n2(sin4

7

)1n2(sin3

7

)1n2(2sin2

7

)1n2(4sin

7

)1n2(sin2

7

)1n2(2sin

.4

0)7

)1n2(sin43(

7

)1n2(2cos

7

)1n2(cos4

0)4

)1n2(sin1(41)1

7

)1n2(cos2(

7

)1n2(cos4

017

)1n2(cos4

7

)1n2(cos4

7

)1n2(cos8

3

2

22

23

eday 07

)1n2(sin:*INn

ehtusmIkar )*( smmUl ½

02

)1n2(cos

7

)1n2(sin2

07

)1n2(3sin

7

)1n2(4sin

00 epÞógpÞat´ .

dUcen¼ 7

5cos,

7

3cos,

7cos

Ca¦srbs´smIkar )E(

24

GnuKmn_RtIekaNmaRt


x¿Tajrktémø P,N,M

snµtfa 7

5cosx,

7

3cosx,

7cosx 321

tamRTwsþIbTEvütGnutþn_kñúgsmIkar 01x4x4x8 23

eKán ½

2

1

a

cxxxxxx

7

5cos

7cos

7

5cos

7

3cos

7

3cos

7cosN

2

1

a

bxxx

7

5cos

7

3cos

7cosM

313221

321

nig 8

1

a

dxxx

7

5cos

7

3cos

7cosP 321

.

dUcen¼

2

1

7

5cos

7cos

7

5cos

7

3cos

7

3cos

7cosN

2

1

7

5cos

7

3cos

7cosM

nig 8

1

7

5cos

7

3cos

7cosP

.

K¿KNna 7

5cos

7

3cos

7cosQ 222

eyIgán 2

3

2

2

2

1 xxxQ

4

5)

2

1(2

4

1N2M

)xxxxxx(2)xxx(

2

3132212

321

dUcen¼ 4

5

7

5cos

7

3cos

7cosQ 222

.

Tajrktémø S nig T

eyIgán 7

5cos

7

3cos

7cosS 333

25

GnuKmn_RtIekaNmaRt


¦ 3

3

3

2

3

1 xxxS

eday 7

5cosx,

7

3cosx,

7cosx 321

Ca¦srbs´

)E( ena¼eKán

)3(01x4x4x8

)2(01x4x4x8

)1(01x4x4x8

3

2

3

3

3

2

2

2

3

2

1

2

1

3

1

bUksmIkar )3(,)2(,)1( Gg:nwgGg:eKán ½

03M4Q4S8

03)xxx(4)xxx(4)xxx(8 321

2

3

2

2

2

1

3

3

3

2

3

1

eKTaj 2

1

8

3

8

7

8

3

22

1

4

5

8

3

2

MQS

dUcen¼ 2

1

7

5cos

7

3cos

7cosS 333

.

müa¨eTot 4

3

4

2

4

1444 xxx

7

5cos

7

3cos

7cosT

edayKuNsmIkar )3(,)2(,)1( erogKñanwg 321 x,x,x

eKán

)'3(0xx4x4x8

)'2(0xx4x4x8

)'1(0xx4x4x8

2

2

3

3

3

4

3

2

2

2

3

2

4

2

1

2

1

3

1

4

1

bUksmIkar )'3(,)'2(,)'1( Gg:nwgGg:eKán ½

0MQ4S4T8

0)xxx()xxx(4)xxx(4)xxx(8 321

2

322

2

1

3

3

3

2

3

1

4

3

4

2

4

1

eKTaj 4

3

8

1

8

7

8

M

2

QST

.

dUcen¼ 4

3

7

5cos

7

3cos

7cosT 444

26

GnuKmn_RtIekaNmaRt


lMhatTI14

eda¼RsaysmIkar ½ 2246

1275611264 xxxx .

dMeNa¼Rsay

eda¼RsaysmIkar

)(112756112642246 xxxx

lk&çx&NÐ 0x1 2 ¦ ]1,1[x

eyIgman )cos(cos aaa 34

188

3144

4334

4334

33

24

22224

4224

33

aa

aaaa

aaaa

aaaaaa

aaaa

coscos

)sin(cos)cos(cos

sinsincoscos

)sinsin(sin)coscos(cos

sinsincoscos

)cos(cos aaa 45

aa

aaaaaa

aaaaaa

aaaaaa

aaaa

coscoscos

)coscos)(cos(coscoscos

)cos(cossincoscoscos

cossinsin)coscos(cos

sinsincoscos

52016

21488

12488

222188

44

35

3235

2235

24

1184832

13421326

246

232

aaa

aaaa

coscoscos

)coscos(coscos

aaaa

aaaaaaa

aaaaa

aaaaaaa

coscoscoscos

)coscos)(sinsin(sincoscos

cossinsincoscos

sinsincoscos)cos(cos

75611264

344326

3326

6667

357

33

27

GnuKmn_RtIekaNmaRt


yk tx cos Edl ],[ 0t smIkar )(1 sresr ½

tttt

tttt

sincoscoscos

coscoscoscos

275611264

1275611264

246

2246

KuNGg:TaMgBIrnwg 0tcos eKán ½

)cos(cos

sincos

cossincoscoscoscos

tt

tt

tttttt

22

7

27

275611264357

eKTaj

Z'k;k,'k2t22

t7

k2t22

t7

smmUl

Z'k;k,9

'k2

10t

9

k2

18t

eday ],[ 0t eKTajsMNMutémø t dUcxageRkam ½

}10

7;

10

3;

18

17;

18

13;

18

9;

18

5;

18{t

eday 0tcos ena¼

2

t

dUcen¼smIkar )(1 mansMNMu¦sdUcxageRkam ½

}cos;cos;cos;cos;cos;cos{10

7

10

3

18

17

18

13

18

5

18

x .

28

GnuKmn_RtIekaNmaRt


lMhatTI15

eda¼RsaysmIkar ½

3232326321 )(tan)(tan)(tantan xxxx

dMeNa¼Rsay

eda¼RsaysmIkar ½

3232326321 )(tan)(tan)(tantan xxxx ¿

lk&çx&NÐ Zkkx

,2

.

tag 02

txt ,tan smIkarsresr ½

0)3t3t)(3t(

0)3t(6)9t3t)(3t(

0)18t6()27t(

0)9t6t(2

27t27t9t9t15t9t3

27t27t9t)8t12t6t()1t3t3t(t

)3t()2t()1t(t

2

2

3

3

2323

2323233

3333

eKTaj 3t nig 0332

tt Kµan¦seRBa¼ 0129

cMeBa¼ 3t eKán 32

xtan

033

032

))(tan(tan

tan

xx

x

eKán 03 xtan ¦ 3xtan naM[ Zkkx

,3

ehIy 03 xtan ¦ 3xtan naM[ Zkkx

,3

29

GnuKmn_RtIekaNmaRt


lMhatTI16

eKman CaFñÚrmYyEdlKitCara¨dügéhIyepÞógpÞat´

20

.

eK[Exßekag )(P smIkar sincos 122 xxy .

1¿cUrkMnt´témø edIm,I[Exßekag )P( b¨¼nwgGk&ßGab´sIus )ox'x(

rYcsgÉxßekag )(P TaMgena¼ .

2¿bgHajfaeRkABIkrNIkñúgsMNYrTI1 Exßekag )P( kat´Gk&ß

Gab´sIus )ox'x( ánBIrcMnuc 'M nig ''M Edlman

Gab´sIusviC¢man .

3¿etIeKRtUv[témø bünµanxø¼eTIbGab´sIus 'x nig ''x

éncMnuc 'M nig ''M epÞógpÞat´TMnak´TMng 222 ''' xx

4¿cUrrkTMnak´TMngKµanGaRs&ynwg rvagGab´sIus 'x

nig ''x .

dMeNa¼Rsay

1¿ Exßekag )P( b¨¼nwgGk&ßGab´sIus )ox'x( ½

kUGredaenkMBUléná¨ra¨bUl )(P KW cosa

bxS

2

ehIy sincoscos 1222

Sy

sinsin

sincos2

21

30

GnuKmn_RtIekaNmaRt


Exßekag )P( b¨¼nwgGk&ßGab´sIus )ox'x( kalNa 0Sy

eKán 02

sinsin

¦ 0)1(sinsin naM[ 0sin nig 1sin

eday 2

0

ehtuen¼eKTaj 2

0

, .

sg´Exßekag )(P ½

-ebI 0 eKán 22112 )( xxxy

-ebI 2

eKán 2xy

2 3 4-1-2

2

3

4

-1

0 1

1

x

y

2¿ Gab´sIuséncMnuc 'M nig ''M

Gab´sIuséncMnuc 'M nig ''M KWCa¦srbs´smIkar ½

Mapsoft ContentScaler Tryout

31

GnuKmn_RtIekaNmaRt


0sin1cosx2x 2

DIsRKImINg´bRgYménsmIkarKW sincos' 12

)sin(sin

sinsin

1

2

eKman sin nig sin1 viC¢manCanic©RKb´

20 , .

eKTaján 01 )sin(sin' naM[ )(P kat´Gk&ß

Gab´sIusCanic©Rtg´BIrcMnuc 'M nig ''M .

müa¨geTotplKuN nigplbUkén¦s sin1P

nig cos2S .

suTæEtviC¢manRKb´

20 , dUcen¼ 'x

nig ''x suTæEtviC¢man .

3¿ lk&çx&NÐ 222 ''' xx

eKman PSxx 2222 '''

)1sinsin2(2)sin1sin22(2

)sin1cos2(2)sin1(2cos422

22

eday 2''x'x 22 eKTaján

0)sin21(sin

0sinsin2

11sinsin2

2)1sinsin2(2

2

2

2


32

GnuKmn_RtIekaNmaRt


eday 2

0

ehtuen¼eKTaj 6

0

,

4¿TMnak´TMngKµanGaRs&ynwg rvagGab´sIus 'x nig ''x

eKman cos2S nig sin1P

eKTaján 2

Scos nig P1sin

edayRKb´cMnYnBit eKman 1sincos 22

eKán 1)P1(4

S 22

0)2P(P4S

0P8P4S

4)P1(4S

2

22

22

dUcen¼ 0)2''x'x(''x'x4)''x'x( 2


33

GnuKmn_RtIekaNmaRt


lMhatTI17

eKmansmIkardWeRkTIBIr ½

013

22

12

xxE )

cos(:)( Edl

20

.

eK«bmafasmIkar )(E man¦sBIrEdltageday atan

nig btan .

k¿ kMnt´témø edIm,I[ 4

ba .

x¿ eda¼RsaysmIkar )(E cMeBa¼témø EdlánrkeXIj

K¿ eRbIlTæplxagelIcUrTajrktémø®ákdén 12

tan .

dMeNa¼Rsay

k¿ kMnt´témø edIm,I[ 4

ba

eday atan nig btan Ca¦srbs´ )(E ena¼eKmanTMnak´TMng

)(cos

tantan 11

2

ba nig )(tantan 213

2ba

tamrUbmnþ )3(btanatan1

btanatan)batan(

ykTMnak´TMng )1( nig )2( CMnYskñúg )3( eKán ½

cos)232(

)1cos2(3

)13

2(1

cos

12

)batan( eday 4

ba

34

GnuKmn_RtIekaNmaRt


eKTaján 1cos)232(

)1cos2(3

2

3cos

cos2cos323cos32

eday 2

0

dUcen¼eKTaj 6

.

x¿ eda¼RsaysmIkar )(E ½

ebI 6

ena¼ )E( Gacsresr 01

3

22

3

22 xx )(

3)32(4

3)347(4

3316

328

43

84

3

834

)13

2(4)2

3

2(

2

2

eKTaj¦s

32)3

3242

3

2(

2

1x

3

1)

3

3242

3

2(

2

1x

2

1

dUcen¼ 32x,3

1x 21 .

K¿ eRbIlTæplxagelITajrktémø®ákdén 12

tan


121

xx ,

eKTaj 3

1atan nig 32 btan

eday 3

1atan naM[

6

a ehIy

4

ba

naM[ 124

ab dUcen¼ 32

12

tan . 35

GnuKmn_RtIekaNmaRt


lMhatTI18

eK[GnuKmn_ cxbxacxbxaxf 2222 sincoscossin)(

Edl cba ,, CabIcMnYnBitviC¢man .

cUrRsayfa cba

xfcbca

2

2)(

rYcbBa¢ak´témøGtibrma nig Gb,brmaén )(xf .

dMeNa¼Rsay

Rsayfa cba

xfcbca

2

2)(

eyIgman )(sincoscossin)( 12222 cxbxacxbxaxf

eday cba ,, CabIcMnYnBitviC¢manena¼ 0 )(: xfIRx

elIkGg:TaMgBIrén )(1 CakaereKán ½

)()sincos)(cossin()(

sincoscossin)(

22222222

222222

cxbxacxbxacbaxf

cxbxacxbxaxf

tamvismPaBkUsiuRKb´cMnYnBit 0BA,

eKman BABA .2 ¦ BABA .2

eKán cbacxbxacxbxa 222222

)sincos)(cossin(

tamTMnak´TMng 2 eKTaján ½

)()( cba

cbacbaxf

2

4222

naM[ )()( 32

2 cba

xf

36

GnuKmn_RtIekaNmaRt


yk )sincos)(cossin()( cxbxacxbxaxP 2222

xsinxcos)ab()cb)(ca()x(P

xcos)ab(xcos)ab()cb)(ca()x(P

xcos)ab(xcos)ab)(cb(xcos)ab)(ca()cb)(ca()x(P

]xcos)ab()cb([]xcos)ab()ca[()x(P

]c)xcos1(bxcosa[]cxcosb)xcos1(a[)x(P

222

4222

4222

22

2222

eyIgman IRxxxab ,cossin)( 0222

eKTaján IRxcbcaxP ,))(()(

TMnak´TMng )(2 eKGacsresr ½

22

2

2

2

2222

)()(

))(()()()(

))(()()(

cbcaxf

cbcacbcaxf

cbcacbaxPcbaxf

eKTaj )()( 4cbcaxf

tamTMnak´TMng )(3 nig )(4 eKTaján ½

cba

xfcbca

2

2)( cMeBa¼RKb´ IRx .

dUcGnuKmn_mantémøGtibrmaesµI cba

M

2

2

nigmantémøGb,brmaesµI cbcam .

37

GnuKmn_RtIekaNmaRt


lMhatTI19

rktémøGb,brmaénGnuKmn_ ½

878

272

22

22

)cot(tancottan)(

)cot(tancottan)(

xxxxxQ

xxxxxP

Edl 2

0

x .

dMeNa¼Rsay

rktémøGb,brmaénGnuKmn_ ½

27222 )cot(tancottan)( xxxxxP Edl

20

x

tag xxz cottan Edl 2z

eKán 22222 xxxxz cottan)cot(tan

eKTaj 2222 zxx cottan

eyIgán 241272222 )()( zzzzP

eday 2z ehtuen¼eKán 25241 )(zP

dUcen¼témøGb,brmaén )(xP KW 25m .

müa¨geToteday 87)xcotx(tan8xcotxtan)x(Q 22

eKán 69)4z(87z82z)z(Q 22

eday 2z ehtuen¼edIm,I[ Q Gb,brmalu¼RtaEt

4z .dUcen¼témøGb,brmaén )(xQ KW 69m .

38

GnuKmn_RtIekaNmaRt


lMhatTI20

eda¼RsaysmIkar ½

223124

4

)(cos)(sin xx

dMeNa¼Rsay

eda¼RsaysmIkar ½

223124

4

)(cos)(sin xx

tamrUbmnþ )basin()basin(bcosasin2

smIkarxagelIGacsresrCabnþbnÞab´xageRkam ½

2

2)

3x2sin(

211)3

x2sin(2

)21(6

sin)3

x2sin(2

223)12

x4

xsin()12

x4

xsin(2

2

eKTaj

Zk,k243

x2

k243

x2

dUcen¼ Zkkxkx

;,24

5

24 .

39

GnuKmn_RtIekaNmaRt


lMhatTI21

eKmanGnuKmn_ 2

2

2

2

2

2 11

xx

xxxf

coscos

sinsin)(

cUrrktémøtUcbMputénGnuKmn_en¼ .

dMeNa¼Rsay

rktémøtUcbMputénGnuKmn_en¼

2

2

22

2

2 11)

sin(sin)

cos(cos)(

xx

xxxf

)sin

)(sin(

)sin

(cossin)sin(cos

)cossin

)(sin(cos

)cossin

sincos()sin(cos

)cossin

()sin(cos

sinsin

coscos

xx

xxxxx

xxxx

xx

xxxx

xxxx

xx

xx

2

1612

2

114

2

16124

114

4

114

12

12

4

2

4

22222

44

44

44

44

44

44

44

4

4

4

4

edayeKman 1x2sin 2 naM[

2

1x2sin

2

11 2

nig

17x2sin

161

4 .

eKTaj 2

25

2

174)

x2sin

161)(x2sin

2

11(4

4

2

40

GnuKmn_RtIekaNmaRt


eyIgán 2

25

2

1612

2

114

4

2 )

sin)(sin()(

xxxf

dUcen¼témøtUcbMputénGnuKmn_KW 2

25m .

lMhatTI22

eK[BIrcMnYnBit a nig b .

cUrRsaybBa¢k´fa 22 ba|xsinbxcosa|:IRx .

Gnuvtþn_ ½ rktémøGtibrma nig Gb,brmaén

222120 xxxf sincos)(

dMeNa¼Rsay

RsaybBa¢k´fa 22 ba|xsinbxcosa|:IRx

eyIgeRCIserIsvuicTr& )b;a(U

nig )xsin;xcos(V

tamniymn&y

cos.||V||.||U||V.U Edl CamMurvag

BIrviucT&ren¼ .

eKán |cos|||V||.||U||cos.||V||.||U||V.U

edayeKman 1|cos|:IR

eKán ||V||.||U||V.U

eday

1xcosxsin||V||

ba||U||

xsinbxcosaV.U

22

22 41

GnuKmn_RtIekaNmaRt


dUcen¼ 22 ba|xsinbxcosa|:IRx .

rktémøGtibrma nig Gb,brmaén 222120 xxxf sincos)(

tamrUbmnþxagelIeyIgman 292120212022 |sincos| xx

eKTaj 29212029 xx sincos

naM[ IRxxf ,)( 517

dUcen¼GnuKmn_mantémøGtibrma 51 nig Gb,brma 7

lMhatTI23

)x(f CatémøBiténGnuKmn_ f EdlcMeBa¼RKb´ IRx

eKman xxxfxf sincos)()( 32 .

cUrRsayfa 2)(xf cMeBa¼RKb´ IRx

dMeNa¼Rsay

Rsayfa 2)(xf cMeBa¼RKb´ IRx

eKman )(sincos)()( 132 xxxfxf

edayCMnYs x eday x kñúgTMnak´TMng )(1 eKán

)2(xsinxcos3)x(f2)x(f

eyIgánRbB&næsmIkar ½

42

GnuKmn_RtIekaNmaRt


xxxf

xxxfxf

xxxfxf

sincos)(

sincos)()(

sincos)()(

333

2

1

32

32

eKTaján xxxf sincos)(

)x4

sin(2

)4

cosxsinxcos4

(sin2

)xsin2

2xcos

2

2(2)x(f

eday 1)x4

sin(:IRx

.

dUcen¼ 2)x(f cMeBa¼RKb´ IRx

lMhatTI24

eKman )(xf GnuKmn_kMnt´elI IR eday ½

xxfxf 223 cos)(cos)(sin ¿

k-cUrkMnt´rkGnuKmn_ )(xf .

x-eda¼RsaysmIkar 2

1111

)tan()tan()tan().tan(

tftftftf

( t CaGBaØténsmIkar ) .

43

GnuKmn_RtIekaNmaRt


dMeNa¼Rsay

k-kMnt´rkGnuKmn_ )(xf

CMnYs x eday x

2 eKán xxfxf 223 cos)(sin)(cos

eyIgánRbB&næ

xxfxf

xxfxf

223

223

cos)(cos)(sin

cos)(cos)(sin

bMát´ )(sin xf eKTaján xx

xf 2

2

21cos

cos)(cos

dUcen¼ 2xxf )( .

x-eda¼RsaysmIkar

2

1111

)tan()tan()tan().tan(

tftftftf

lk&çx&NÐ Zkkt

;2

2

1111

22

22 )tan()tan()tan()tan(

tttt

dMeNa¼RsaysmIkaren¼eKáncMelIy

Zkktkt

,;3

.

lMhatTI25

eK[Exßekag sin)sin(sin)(:)( 5122 xxxfyP

Edl 0 .

kMnt´témø edIm,I[Exßekag )(P sSitenAelIGkß&

Gab´sIusCanic© . 44

GnuKmn_RtIekaNmaRt


dMeNa¼Rsay

kMnt´témø

eKman sin)sin(sin)(:)( 5122 xxxfyP

edIm,I[ )(P sSitenAelIGkß&Gab´sIusCanic©lu¼RtaEt

IRxxf ,)( 0

eBalKWeKRtUv[

0

0

'fa

eKman [;];sin 00fa

ehIy )sin(sin)sin(' 512

))(sinsin('

sinsin'

sinsinsinsin'

112

132

521

2

22

ebI 0' smmUl 12

1 sin eday 0

eKTaján 26

.

dUcen¼edIm,I[Exßekag )(P sSitenAelIGkß&Gab´sIus

Canic©lu¼RtaEteK[l&kçx&NÐ 26

.

45

GnuKmn_RtIekaNmaRt


lMhatTI26

eda¼RsaysmIkar ½

2

1

2

3

16

9

2

1

16

1 2424 xxxx coscoscoscos

dMeNa¼Rsay

eda¼RsaysmIkar ½

)(coscoscoscos 12

1

2

3

16

9

2

1

16

1 2424 xxxx

smIkar 1 Gacsresr ½

)(coscos

coscos

22

1

4

3

4

1

2

1

4

3

4

1

22

2

2

2

2

xx

xx

tag xt 2cos Edl 10 t smIkar )(2 Gacsresr

)(32

1

4

3

4

1 tt

elIGkß& )'( oxx eRCIserIscMnuc )(,)(,)(4

3

4

1BAtM

tam 3 eKán 2

1MBMA eday

2

1AB

eKán ABMBMA naM[ M enAkñúg AB

eKTaj 4

3

4

1 t smmUl

4

3

4

1 2 xcos

smmUl 2

3

2

1 |cos| x eKTaj 46

GnuKmn_RtIekaNmaRt


Zkkkxk

kx

';,''63

36

lMhatTI27

eK[RtIekaN ABC mYy .

k¿ cUrRsayfa 1 ACCBBA cotcotcotcotcotcot .

x¿ cUrRsayfa AAA cotcotcot 2212

.

K¿ eKdwgfamMu CBA ;; beg;ItánCasIVútFrNImaRt

mYYyEdlmanersugesµInwg 2q .

cUrRsaybBa¢ak´fa ½

8111

222

CBA sinsinsin .

dMeNa¼Rsay

k¿Rsayfa 1 ACCBBA cotcotcotcotcotcot

eyIgman CBA ¦ CBA

eKán )tan()tan( CBA

47

GnuKmn_RtIekaNmaRt


1

1

1

1

111

11

1

BACBCACBA

BA

CBA

BA

CBA

BA

cotcotcotcotcotcotcotcotcot

cotcot

cotcot

.cot

cotcot

tantantan

tantan

dUcen¼ 1 ACCBBA cotcotcotcotcotcot .

x¿Rsayfa AAA cotcotcot 2212

eyIgman A

AA

21

22

tan

tantan

1

2

11

2

2

12

2

A

A

A

AA cot

cot

cot

cotcot

dUcen¼ AAA cotcotcot 2212 .

K¿RsaybBa¢ak´fa ½ 8111

222

CBA sinsinsin

tag CBA

T222

111

sinsinsin

)(cotcotcotcotcotcot

)(cot)(cot)(cot

)cot()cot()cot(

16222222

6111

111

222

222

CCBBAA

CBA

CBA

edaymMu CBA ;; CasIVútFrNImaRtmYYyEdlmanersug

esµInwg 2q eKán ABCAB 422 ,

eday CBA 48

GnuKmn_RtIekaNmaRt


eKán AAA 42 naM[ 7

4

7

2

7

CBA ,,

tam )(1 eKán 67

4

7

8

7

4

7

22

77

22

cotcotcotcotcotcotT

eday 77

8

cotcot eKán ½

861262

67

4

72

7

4

7

22

7

2

72

)()cotcotcotcotcot(cot

cotcotcotcotcotcot

CACBBA

T

dUcen¼ 8Csin

1

Bsin

1

Asin

1222

.

49

GnuKmn_RtIekaNmaRt


lMhatTI28

1¿cUrRsaybBa¢ak´rUbmnþ ½

*INn,IRx,x)1ncos()nxcos(xcos2x)1ncos(

2¿Gnuvtþn_ ½ cUrsresr x7cos CaGnuKmn_én xcos .

3¿eda¼RsaysmIkar ½

01xcos14xcos112xcos244xcos128 357 .

dMeNa¼Rsay

1¿RsaybBa¢ak´rUbmnþ ½

xnnxxxn )cos()cos(cos)cos( 121

eyIgman ½

2

11

2

11211

xnxnxnxnxnxn

)()(cos.

)()(cos)cos()cos(

smmUl )cos(.cos)cos()cos( nxxxnxn 211

dUcen¼ xnnxxxn )cos()cos(cos)cos( 121 ¿.

2¿ Gnuvtþn_ ½sresr x7cos CaGnuKmn_én xcos

eKman xnnxxxn )cos()cos(cos)cos( 121 ¿

ebI 1n 1222 xx coscos

ebI 2n xxxx coscoscoscos 223 50

GnuKmn_RtIekaNmaRt


xx

xxx

coscos

cos)cos(cos

34

122

3

2

ebI 3n xxxx 2324 coscoscoscos

188

12342

24

23

xx

xxxx

coscos

)cos()coscos(cos


xxx

xxxxx

coscoscos

)coscos()coscos(cos

52016

341882

35

324


1184832

188520162

246

2435

xxx

xxxxxx

coscoscos

)coscos()coscoscos(cos


dUcen¼ xxxxx coscoscoscoscos 756112647357 ¿

3¿ eda¼RsaysmIkar ½

)(coscoscoscos 10114112244128357 xxxx ¿

EckGg:TaMgBIrénsmIkar )(1 nwg 2 eKán ½

2

17

02

175611264

357

x

xxxx

cos

coscoscoscos

eKTaján

kx 23

7 ¦ Zkk

x

,7

2

21

dUcen¼ Z'k;k;7

'k2

21x,

7

k2

21x

.

51

GnuKmn_RtIekaNmaRt


lMhatTI29

eK[sVúIténcMnYnBit )( nU kMnt´elI n eday½

10U nig aaUUINn nn sincos:

1 Edl 2

0

a

k¿tag 2

aUV nn cot .

cUrbgHajfa )( nV CasIVútFrNImaRtmYYy .

x¿KNnalImIt )....(lim nn

VVV

10 nig nnU

lim .

dMeNa¼Rsay

k¿ bgHajfa )( nV CasIVútFrNImaRtmYYy ½

man 2

aUV nn cot naM[

211

aUV nn cot

Et aaUU nn sincos 1

eKán 2

1

aaaUV nn cotsincos

aV

aa

Uaa

aU

aa

aaaU

a

aaaa

aU

aaaaaU

aaaaU

n

nn

n

n

n

n

cos

cos)cot(coscotcos

sincos;)sin(cotcos

)cos

sincossin(cotcos

)tancossin(cotcos

cotcossincos

22

2211

22

2

1

2

2

222

2

1222

22

2222

22

52

GnuKmn_RtIekaNmaRt


eday aVV nn cos1

naM[ )( nV CasIVútFrNImaRt

mYymanersug acos nig tY 2

12

00

aaUV cotcot .

x¿ KNnalImIt ½ )....(lim nnVVV

10

nig nnU

lim

eyIgman a

aa

q

qVVVV

nn

n cos

cos).cot(...

1

1

21

1

111

010

eyIgán

a

aaVVV

n

nn

n cos

cos)cot(lim)....(lim

1

1

21

1

10

eday 2

0

a ena¼ 10 acos nig 01

an

ncoslim

dUcen¼ a

a

VVV nn cos

cot)....(lim

1

21

10 .

müa¨geTot 2

aUV nn cot naM[

2

aVU nn cot

eday aa

qVV nnn cos)cot(

21

0

eKán 22

1a

aa

U nn cotcos)cot(

nig 222

1aa

aa

U n

nnncotcotcos)cot(limlim

eRBa¼ 0

an

ncoslim .

dUcen¼ 2

aU nn

cotlim

.

53

GnuKmn_RtIekaNmaRt


lMhatTI30

eK[sIVúténcMnYnBit )( nU kMnt´eday ½

1010 UU ; nig nnn UaUUINn

cos:

122 Edl IRa

k¿ tag INnUaiaUZ nnn

,)sin(cos1

.

cUrbgHajfa nn ZaiaZ )sin(cos 1

rYcTajrk nZ CaGnuKmn_

nnig a .

x¿ Tajrk nU CaGnuKmn_én n

dMeNa¼Rsay

k¿ bgHajfa n1n Z)asinia(cosZ

eyIgman nnn UaiaUZ )sin(cos 1

eyIgán 121

nnn UaiaUZ )sin(cos

n

nn

nn

nn

nnn

Uaia

UaiaUaiaaia

UUaia

UUaia

UaiaUaU

)sin(cos

)sin(cos)sin(cos

)sincos

)(sin(cos

)sin(cos

)sin(coscos

1

1

1

112

dUcen¼ nn ZaiaZ )sin(cos 1

.

KNna nZ CaGnuKmn_én n nig a ½

eday nn ZaiaZ )sin(cos 1

naM[ )( nZ CasIVútFrNImaRt

éncMnYnkMupøicEdlmanersug aiaq sincos nig 54

GnuKmn_RtIekaNmaRt


tY 1010 UaiaUZ )sin(cos .

tamrUbmnþ )sin()cos()sin(cos nainaaiaqZZ nnn

0

dUcen¼ )sin(.)cos( nainaZn .

x¿ Tajrk nU CaGnuKmn_én n ½

eyIgman )()sin(cos 11 nnn UaiaUZ

nig )()sin(cos 21 nnn UaiaUZ

dksmIkar )(1 nig )(2 Gg:nwgGg:eKán ½

nnn UaiZZ sin2 naM[ ai

ZZU nn

n sin2

Edl 0asin .

eday )sin()cos( nainaZn nig )sin()cos( nainaZn

dUcen¼ a

naU n sin

)sin( .

lMhatTI31

eK[ 432 2cos2222,

2cos222,

2cos22

BI«TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBaØak´

rUbmnþena¼pg .

dMeNa¼Rsay

rkrUbmnþTUeTA ½

eKman

432 2cos2222,

2cos222,

2cos22

55

GnuKmn_RtIekaNmaRt


tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½

1n

)n(2

cos22.........222

.

RsaybBaØakrUbmnþen¼ ½

eyIgtag )n(

n 2......222A cMeBa¼RKb´ *INn

eyIgman 21 2cos22A

Bit

eyIg«bmafavaBitdl´tYTI p KW

1p

)p(

p 2cos22......222A

Bit

eyIgnwgRsayfavaBitdl´tYTI 1p KW 2p1p 2cos2A

Bit

eyIgman p1p A2A edaytamkar«bma 1pp 2

cos2A

eyIgán 2p2p

2

1p1p 2cos2

2cos4

2cos22A

Bit

dUcen¼ 1n

)n(2

cos22.........222

.

56

GnuKmn_RtIekaNmaRt


lMhatTI32

eK[sIVúténcMnYnkMupøic )Z( n kMnt´eday ½

INn;|Z|Z

2

1Z

2

3i1Z

nn1n

0

( |Z| n CamUDulén nZ ) .

snµtfa INn,)sin.i(cosZ nnnn

Edl IR;,0 nnn .

k-cUrrkTMnak´TMngrvag n nig 1n ehIy n nig 1n

x-rkRbePTénsIVút )( n rYcKNna n CaGnuKmn_én n .

K-cUrbgHajfa2

cos....2

cos2

coscos 1n2100n

rYcbBa¢ak´ n CaGnuKmn_én n .

dMeNa¼Rsay

k-rkTMnak´TMngrvag n nig 1n ehIy n nig 1n

eyIgman )sin.i(cosZ nnnn

naM[ )sini(cosZ 1n1n1n1n

eday )|Z|Z(2

1Z nn1n

ehIy nn |Z|

eKán nnnn1n1n1n )sin.i(cos2

1)sini(cos 57

GnuKmn_RtIekaNmaRt


)

2sin.i

2(cos

2cos)sin.i(cos

)sin.icos1(2

1)sin.i(cos

nnnn1n1n1n

nnn1n1n1n

eKTaján 2

cos nn1n

nig 2n

1n

dUcen¼ 2

n1n

nig 2

cos nn1n

.

x-RbePTénsIVút )( n nig KNna n CaGnuKmn_én n ½

tamsRmayxagelIeyIgman n1n 2

1

naM[ n CasIVútFrNImaRtmanersugesµI 2

1q .

tamrUbmnþ n0n q

eday 3sin.i

3cos

2

3i1)sini(cosZ 0000

eKTaján 3

;1 00

dUcen¼ nn 2

1.

3

.

K-bgHajfa 2

cos....2

cos2

coscos n2100n


cos nn1n

¦ 2cos n

n

1n

eKán

1nk

0k

1nk

0k

k

k

1k )2

cos(

2cos.........

2cos.

2cos.cos 1n21

0

0

n

dUcen¼ 2

cos....2

cos2

coscos 1n2100n

.

M

58

GnuKmn_RtIekaNmaRt


müa¨geToteyIgman 2

cossin22

cos2

sin2sin n1n

nnn

( eRBa¼2

n1n

) eKTaj 1n

nn

sin

sin.

2

1

2cos

ehtuen¼ n

0n

n

1n

2

1

1

0nn sin

sin.

2

1

sin

sin.....

sin

sin.

sin

sin.

2

1

dUcen¼ )

2

1.

3sin(

1.

2

3

)2

1.

3sin

3sin

2

1

n

1n

n

nn

.

lMhatTI33

k¿cUrKNnatémø®ákdén 8

tan

x¿cUreda¼RsaysmIkar 0xcos)12(xcos.xsin2xsin 22

K¿cUreda¼RsaysmIkar 3

1

xtan)12(1

12xtan

dMeNa¼Rsay

k¿KNnatémø®ákdén 8

tan

tamrUbmnþ atan1

atan2a2tan

2 edayyktémø

8a

eKán

8tan1

8tan2

4tan

2

8tan1

8tan2

12

naM[ 018

tan28

tan 2

tag 8

tant

Edl 0t 59

GnuKmn_RtIekaNmaRt


eKán 211';01t2t 2

eKTaj¦s 021t,21t 21 ( minyk)

dUcen¼ 128

tan .

x¿ eda¼RsaysmIkar 0xcos)12(xcos.xsin2xsin 22

EckGg:TaMgBIrnwg 0xcos2 eKánsmIkar ½

0)12(xtan2xtan 2 ¿

tag xtant eKán ½

0)12(t2t 2 eday 0cba

eKTaj¦s 12t;1t 21 .

-cMeBa¼ 1t eKán 1xtan naM[ Zk,k4

x

-cMeBa¼ 12t eKán 12xtan

naM[ Zk,k8

x

dUcen¼ Zk,k8

x,k4

x

.

K¿ eda¼RsaysmIkar ½

3

1

xtan)12(1

12xtan

eday 128

tan eKán

6tan)

8xtan(

3

3

8tan.xtan1

8tanxtan

60

GnuKmn_RtIekaNmaRt


eKTaj

k68

x ¦ Zk,k24

x

dUcen¼ Zk,k24

x

.

lMhatTI34


cos nig

12

7cos

x¿cUreda¼RsayRbB&næsmIkar

8

632ycosycosxcos3

8

632ycosxcos3xcos

32

23

dMeNa¼Rsay


cos nig

12

7cos

eyIgán )43

cos(12

cos

4

62

2

2.

2

3

2

2.

2

1

4sin

3sin

4cos

3cos

ehIy )43

cos(12

7cos

4

62

2

2.

2

3

2

2.

2

1

4sin

3sin

4cos

3cos

61

GnuKmn_RtIekaNmaRt


dUcen¼ 4

62

12

7cos,

4

62

12cos

.

x¿ eda¼RsayRbB&næsmIkar½

)2(8

632ycosycosxcos3

)1(8

632ycosxcos3xcos

32

23

bUksmIkar )1( nig )2( Gg:nigGg:eKán ½

)3(2

2ycosxcos

)2

2()ycosxcos(

8

22ycosycosxcos3ycosxcos3xcos

33

3223

dksmIkar )1( nig )2( Gg:nigGg:eKán ½

)4(2

6ycosxcos

)2

6()ycosxcos(

8

66ycosycosxcos3ycosxcos3xcos

33

3223

bUksmIkar )3( nig )4( Gg:nigGg:eKán ½

4

62xcos

2

62xcos2

12

cosxcos

naM[ Zk,k212

x

62

GnuKmn_RtIekaNmaRt


dksmIkar )3( nig )4( Gg:nigGg:eKán ½

4

62ycos

2

62ycos2

12

7cosycos

naM[ Zk,k2

12

7y

dUcen¼ Zk,k212

x

nig Zk,k212

7y

lMhatTI35

k¿cUrRsaybBa¢ak´fa x4cos8

3

8

5xcosxsin 66

x¿cUreda¼RsaysmIkar x2sin4

1

16

13)xcosx(sin 3233

dMeNa¼Rsay

k¿RsaybBa¢ak´fa x4cos8

3

8

5xcosxsin 66

eyIgman INx;1xcosxsin 22

elIkGg:TaMgBIrCaKUbeKán ½

x4cos8

3

8

5x4cos

8

3

8

31xcosxsin

1)2

x4cos1(

4

3xcosxsin

1x2sin4

3xcosxsin

1xcosxsin3xcosxsin

1)xcosxsin(xcosxsin3xcosxsin

1xcosxcosxsin3xcosxsin3xsin

1)xcosxsin(

66

66

266

2266

222266

642246

322

63

GnuKmn_RtIekaNmaRt


dUcen¼ x4cos8

3

8

5xcosxsin 66 .

x¿eda¼RsaysmIkar x2sin4

1

16

13)xcosx(sin 3233

eyIgán xcosxsin216

13xcosxcosxsin2xsin 336336

16

13x4cos

4

3

8

5 ¦

2

1x4cos

eKTaj

k23

x4 ¦ Zk,2

k

12x

.

lMhatTI36

eK[sIVúténcMnYnBit )U( n kMnt´elI IN eday ½

INn,U2U

2U

n1n

0

k¿cUrKNna nU CaGnuKmn_én n .

x¿KNnaplKuN n210n U....UUUP .

dMeNa¼Rsay

KNna nU CaGnuKmn_én n ½

eyIgman 4

cos22U0

nig

8cos2

4cos22U2U 01

«bmafavaBitdl´tYTI p KW 2pp 2

cos2U

eyIgnwgRsayfavaBitdl´tYTI )1p( KW 3p1p 2

cos2U

eyIgman p1p U2U

Ettamkar«bma 2pp 2

cos2U

eyIgán 3p3p

2

2p1p 2cos2

2cos4

2cos22U

Bit

64

GnuKmn_RtIekaNmaRt


dUcen¼ 2nn 2cos2U

.

x¿ KNnaplKuN n210n U....UUUP

tamrUbmnþ acosasin2a2sin naM[ asin

a2sinacos2

2n2n

n

0k

n

0k

2k

1kn

0k2kkn

2sin

1

2sin

2sin

)

2sin

2sin

()2

cos2()U(P

lMhatTI37

eK[sIVúténcMnYnBit )U( n kMnt´elI IN eday ½

2

2U0 nig INn,

2

U11U

2

n1n

KNna nU CaGnuKmn_én n .

dMeNa¼Rsay

KNna nU CaGnuKmn_én n ½

eyIgman 4

sin2

2U0

8

sin2

4sin11

2

U11U

22

01

«bmafavaBitdl´tYTI p KW 2pp 2

sinU

eyIgnwgRsayfavaBitdl´tYTI )1p( KW 3p1p 2

sinU

Bit

eyIgman 2

U11U

2

p

1p

Ettamkar«bma

2pp 2sinU

65

GnuKmn_RtIekaNmaRt


eyIgán 2

2sin11

U2p

2

1p

3p

3p

2

2p

2sin

22

sin2

22

cos1

Bit

dUcen¼ 2nn 2sinU

.

lMhatTI38

eK[smIkardWeRkTIBIr 02mx)mm(x:)E( 22

eKsnµtfasmIkaren¼man¦sBIrtagerogKñaeday atan

nig btan .

k¿cUrkMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3ba

.

x¿cUreda¼RsaysmIkarxagelIcMeBa¼ m EdlánrkeXIj

K¿edayeRbIlTæplxagelIcUrTajrktémøRákdén 12

tan .

dMeNa¼Rsay

k¿kMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3

ba

½

smIkarman¦skalNa 08m4)mm( 22

eday atan nig btan Ca¦srbs´smIkarena¼

tamRTwsþIbTEvüteKman

)2(2mbtan.atan

)1(mmbtanatan 2

66

GnuKmn_RtIekaNmaRt


eday btan.atan1

btanatan)batan(

)3(

btan.atan1

btanatan3

btan.atan1

btanatan

3tan

ykTMnak´TMng )1( nig )2( CYskñúgsmIkar )3(

eKán ½

2m1

mm3

2

¦ 03m)31(m2

eday 0cba eKTaj¦s 3m,1m 21

-cMeBa¼ 1m ena¼ 0491.4)11( 22

( minyk )

-cMeBa¼ 3m ena¼ 0324834)33( 2

dUcen¼ 3m .

x¿eda¼RsaysmIkarxagelIcMeBa¼ m EdlánrkeXIj ½

cMeBa¼ 3m eKán ½ 023x)33(x 2

eday 0cba eKTaj¦s 32x,1x 21 .

K¿TajrktémøRákdén 12tan

½

tamlTæplxagelIeKman 1atanx1 naM[ 4

a

ehIy 3

ba

ena¼ 1243

b

ehtuen¼ 3212

tanbtanx 2

67

GnuKmn_RtIekaNmaRt


dUcen¼ 3212

tan

.

lMhatTI39

k¿cUrRsaybBa¢ak´TMnak´TMng x2cot2xcotxtan

x¿cUrKNnaplbUkxageRkam ½

nn22n 2

atan

2

1....

2

atan

2

1

2

atan

2

1atanS

dMeNa¼Rsay

k¿RsaybBa¢ak´TMnak´TMng x2cot2xcotxtan

tag x2cot2xcotA eday

xtan2

xtan1

x2tan

1x2cot

xtan

1xcot

2

eKán xtanxtan

xtan11)

xtan2

xtan1(2

xtan

1A

22

dUcen¼ x2cot2xcotxtan .

x¿KNnaplbUkxageRkam ½

a2cot22

acot

2

1

2

acot

2

1

2

acot

2

1

)2

acot2

2

a(cot

2

1

2

atan

2

12

atan

2

1....

2

atan

2

1

2

atan

2

1atanS

nn

n

0k1k1kkk

n

0k1kkk

n

0kkk

nn22n

dUcen¼ a2cot22

acot

2

1

2

atan

2

1....

2

atan

2

1atanSnnnnn

68

GnuKmn_RtIekaNmaRt


lMhatTI40

k¿cUrRsaybBa¢ak´fa xtan2x2tanxtan.x2tan 2

x¿cUrKNnaplbUk

n

0k1k

2

k

kn 2

atan

2

atan2S

dMeNa¼Rsay

k¿RsaybBa¢ak´fa xtan2x2tanxtan.x2tan 2

tag xtan2x2tan)x(f eday xtan1

xtan2x2tan

2

eKán xtan2xtan1

xtan2)x(f

2

xtan.x2tanxtan.xtan1

xtan2

xtan1

xtan2xtan1

xtan2xtan2xtan2xtan1

)xtan1(xtan2xtan2

22

22

3

2

3

2

2

dUcen¼ xtan2x2tanxtan.x2tan 2 .

x¿KNnaplbUk

n

0k1k

2

k

kn 2

atan

2

atan2S

eyIgman xtan2x2tanxtan.x2tan 2 edayyk

1k2

ax

eKán 1kk1k

2

k 2

atan2

2

atan

2

atan

2

atan

1n

1nn

0k1k

1k

k

kn 2

atan2atan

2

atan2

2

atan2S

dUcen¼ 1n

1nn

0k1k

2

k

kn 2

atan2atan

2

atan

2

atan2S

69

GnuKmn_RtIekaNmaRt


lMhatTI41

k¿cUrRsayfa )x3sinxsin3(4

1xsin 3

x¿cUrKNna n

31n

3

32

2

33n 3

asin3....

3

asin3

3

asin3

3

asinS

dMeNa¼Rsay

k¿Rsayfa )x3sinxsin3(4

1xsin 3

eyIgman )x2xsin(x3sin

tamrUbmnþ acosbsinbcosasin)basin(

xsin4xsin3

xsin2xsin2xsin2xsin

)xsin1(xsin2)xsin21(xsin

xcosxsin2)xsin21(xsin

xcosx2sinx2cosxsin

3

33

22

22

eday xsin4xsin3x3sin 3

dUcen¼ )x3sinxsin3(4

1xsin 3

.

x¿ KNna n

31n

3

32

2

33n 3

asin3....

3

asin3

3

asin3

3

asinS

eyIgán

n

1kk

31kn 3

asin3S

eday )x3sinxsin3(4

1xsin 3

n

k k 1 n

n k k 1 nk 1

1 a a 1 aS 3 sin 3 sin (3 sin sin a)

4 3 3 4 3

dUcen¼ 4

asin

3

asin

4

3S

n

n

n . 70

GnuKmn_RtIekaNmaRt


lMhatTI42

k¿cUrRsayfa xsin

1

x2sin

4

xcos

1222

x¿cUrKNna n

2n

2

22n

2

acos4

1....

2

acos4

1

2

acos4

1S

dMeNa¼Rsay

k¿ Rsayfa xsin

1

x2sin

4

xcos

1222

eyIgman x2sin

4

xcosxsin

1

xcosxsin

xcosxsin

xsin

1

xcos

122222

22

22

dUcen¼ xsin

1

x2sin

4

xcos

1222

.

x¿ KNna n

2n

2

22n

2

acos4

1....

2

acos4

1

2

acos4

1S

eyIgán

n

1k

k

2kn

2

acos

1.

4

1S eday

xsin

1

x2sin

4

xcos

1222

eKán ½

n

n kk 1 2 2

k 1 k

n

k 1 k 2k 1 2 2 n 2

k 1 k n

1 4 1S ( )

a a4 sin sin2 2

1 1 1 1 1 1. .

a a a4 4 sin asin sin 4 sin2 2 2

dUcen¼ n

2n2n

2

asin4

1

asin

1S . 71

GnuKmn_RtIekaNmaRt


lMhatTI43

k¿cUrRsayfa x2cotxcotx2sin

1

x¿cUrKNna n2

n

2

asin

1....

2

asin

1

2

asin

1

asin

1S

dMeNa¼Rsay

k¿Rsayfa x2cotxcotx2sin

1

tag x2cotxcot)x(f

x2sin

1

xcosxsin2

1xcos2xcos2xcosxsin2

1xcos2

xsin

xcosx2sin

x2cos

xsin

xcos

22

2

dUcen¼ x2cotxcotx2sin

1 .

x¿KNna n2

n

2

asin

1....

2

asin

1

2

asin

1

asin

1S

eyIgán

n

0k

k

n )

2

asin

1(S eday x2cotxcot

x2sin

1

acot

2

acot

2

acot

2

acotS

1n

n

0kk1kn

dUcen¼ acot2

acotS

1nn

.

72

GnuKmn_RtIekaNmaRt


lMhatTI44

k¿cUrRsayfa xcot2

xcot

xcos

11

x¿KNnaplKuN )

2

acos

11).....(

2

acos

11)(

2

acos

11)(

acos

11(P

n2

n

dMeNa¼Rsay

k¿Rsayfa xcot2

xcot

xcos

11

eyIgtag xcos

11)x(A

xcot2

xcot

xtan2

xtan

xcos

xsin.

2

xsin

2

xcos

2

xsinxcos

xsin2

xcos

2

xsinxcos

2

xsin

2

xcos2

xcos2

xcos2

xcos

1xcos22

dUcen¼ xcot2

xcot

xcos

11 .

x¿KNnaplKuN

n

nk 0

2 n k

n k 1 n 1

n 1k 0

k

1 1 1 1 1P (1 )(1 )(1 ).....(1 ) ( 1 )

a a a acosa cos cos cos cos2 2 2 2

a atan tan a2 2( ) tan cot a

a tana 2tan2

73

GnuKmn_RtIekaNmaRt


lMhatTI45

k¿cUrRsayfa xcot.xcos

)nxcos(

xcos

x)1ncos(

xcos

)nxsin(n1nn

x¿cUrKNna xcos

)nxsin(....

xcos

x3sin

xcos

x2sin

xcos

xsinS

n32n

dMeNa¼Rsay

k¿Rsayfa xcot.xcos

)nxcos(

xcos

x)1ncos(

xcos

)nxsin(n1nn

eyIgman )xnxcos(x)1ncos(

¦ xsin)nxsin(xcos)nxcos(x)1ncos(

EckGg:TaMgBIrnwg xcos 1n eKán ½

xtan.xcos

)nxsin(

xcos

)nxcos(

xcos

xsin)nxsin(

xcos

xcos)nxcos(

xcos

x)1ncos(nn1n1n1n

naM[ xtan

1.

xcos

)nxcos(

xcos

x)1ncos(

xcos

)nxsin(n1nn

dUcen¼ xcot.xcos

)nxcos(

xcos

x)1ncos(

xcos

)nxsin(n1nn

.

x¿KNna xcos

)nxsin(....

xcos

x3sin

xcos

x2sin

xcos

xsinS

n32n

eyIgán

n

1kkn xcos

)kxsin(S

eday xcot.xcos

)kxcos(

xcos

x)1kcos(

xcos

)kxsin(k1kk

eKán

n

1k1nk1kn 1xcos

x)1ncos(xcot

xcos

)kxcos(

xcos

x)1kcos(xcotS

dUcen¼ xcot1xcos

x)1ncos(S

1nn

.

74

GnuKmn_RtIekaNmaRt


lMhatTI46

k¿cUrRsayfa )nxtan(x)1ntan(xsin

1

x)1ncos().nxcos(

1

x¿ KNnaplbUk

n

1pn x)1pcos()pxcos(

1S

dMeNa¼Rsay

k¿Rsayfa )nxtan(x)1ntan(xsin

1

x)1ncos().nxcos(

1

tamrUbmnþ qcospcos

)qpsin(qtanptan

naM[ )1(qtanptan)qpsin(

1

qcospcos

1

yk )nx(q,x)1n(p nig xqp CYskñúg )1( eKán

)nxtan(x)1ntan(xsin

1

x)1ncos().nxcos(

1

.

x¿KNnaplbUk

n

1pn x)1pcos()pxcos(

1S

tamsRmayxagelIeKman ½

)pxtan(x)1ptan(xsin

1

x)1pcos().pxcos(

1

eyIgán

n

1pn )pxtan(x)1ptan(

xsin

1S

x)1ncos(xcosxsin

)nxsin(xtanx)1ntan(

xsin

1

dUcen¼ x)1ncos(x2sin

)nxsin(2Sn

.

75

GnuKmn_RtIekaNmaRt


lMhatTI47

k¿cUrRsayfa x)1ntan()nxtan(1xtan)nxtan(x)1ntan(

x¿KNnaplbUk

x)1ntan()nxtan(.....x3tanx2tanx2tanxtanSn dMeNa¼Rsay

k¿Rsayfa x)1ntan()nxtan(1xtan)nxtan(x)1ntan(

tamrUbmnþ btanatan1

btanatan)ba(ta

naM[ )1(btanatan1)batan(btanatan

edayyk nxb,x)1n(a nig xba

CYskñúg )1( eKán ½

x)1ntan()nxtan(1xtan)nxtan(x)1ntan( .

x¿KNnaplbUk

x)1ntan()nxtan(.....x3tanx2tanx2tanxtanSn

eyIgán

n

1kn x)1ktan()kxtan(S

tamsRmayxagelIeyIgman ½

x)1ntan()nxtan(1xtan)nxtan(x)1ntan(

¦ 1xcot)nxtan(x)1ntan(x)1ntan()nxtan(

eyIgán

n

1kn 1xcot)kxtan(x)1ktan(S 76

GnuKmn_RtIekaNmaRt


nxcotxcosx)1ncos(

)nxsin(

nxcotxtanx)1ntan(

dUcen¼ nxsinx)1ncos(

)nxsin(Sn

.

lMhatTI48

k¿cUrRsaybBa¢ak´fa xcos21

x2cos211xcos2

x¿cUrKNnaplKuN ½

)12

acos2).....(1

2

acos2)(1

2

acos2)(1acos2(P

n2n

dMeNa¼Rsay

k¿ RsaybBa¢ak´fa xcos21

x2cos211xcos2

tamrUbmnþ 1xcos2x2cos 2

)1xcos2)(1xcos2(1x2cos2

1xcos41x2cos2

2xcos4x2cos22

2

dUcen¼ xcos21

x2cos211xcos2

.

x¿ KNnaplKuN ½

)12

acos2).....(1

2

acos2)(1

2

acos2)(1acos2(P

n2n

n

0kk

12

acos2

eday xcos21

x2cos211xcos2

77

GnuKmn_RtIekaNmaRt


n

n

0kk

1k

n

2

acos21

a2cos21

)2

acos(21

)2

acos(21

P

dUcen¼ n

n

2

acos21

a2cos21P

.

lMhatTI49

k¿cUrRsayfa )xtan3x3tan(8

1

xtan31

xtan2

3

x¿cUrKNnaplbUk

n

0k

k

2

k

3k

n

3

atan31

3

atan3

S

dMeNa¼Rsay

k¿ Rsayfa )xtan3x3tan(8

1

xtan31

xtan2

3

tamrUbmnþ xtan31

xtanxtan3x3tan

2

3

eyIgán xtan31

xtan8xtan3

xtan31

xtanxtan3xtan3x3tan

2

3

2

3

dUcen¼ )xtan3x3tan(8

1

xtan31

xtan2

3

.

x¿KNnaplbUk

n

0k

k

2

k

3k

n

3

atan31

3

atan3

S

eyIgman )xtan3x3tan(8

1

xtan31

xtan2

3

edayyk k3

ax

78

GnuKmn_RtIekaNmaRt


eKán )3

atan3

3

a(tan

8

1

3

atan31

3

atan

k1k

k

2

k

3

eyIgán

n

1n

k

1kn

0k1k

kn 3

atan3a3tan

8

1)

3

atan3

3

atan3(

8

1S

dUcen¼ n

1n

n 3

atan

8

3

8

a3tanS

.

lMhatTI50

k¿cUrRsayfa xtanx2tan2

1

xtan1

xtan2

3

x¿cUrKNnaplbUk

n

0kn

2

n

3k

n

2

atan1

2

atan2

S

dMeNa¼Rsay

k¿Rsayfa xtanx2tan2

1

xtan1

xtan2

3

tamrUbmnþ xtan1

xtan2x2tan

2

eyIgán xtan1

xtanxtan

xtan1

xtanxtanx2tan

2

12

3

2

dUcen¼ xtanx2tan2

1

xtan1

xtan2

3

.

x¿KNnaplbUk

n

0k

k

2

k

3k

n

2

atan1

2

atan2

S

eKman xtanx2tan2

1

xtan1

xtan2

3

yk k2

ax

79

GnuKmn_RtIekaNmaRt


eKán k1k

k

2

k

3

2

atan

2

atan

2

1

2

atan1

2

atan

eyIgán n

nn

0kk

k

1k

1kn 2

atan2a2tan

2

1

2

atan2

2

atan2S

dUcen¼ n

nn 2

atan2a2tan

2

1S .

lMhatTI51

k¿cUrRsayfa x)1n2sin(x)1n2sin(xsin2

1)nx2cos(

x¿KNnaplbUk )nx2cos(.....x6cosx4cosx2cosSn

K¿TajrkplbUk )nx(cos.....x3cosx2cosxcosT 2222n

X¿KNnaplbUk )nx(sin.....x3sinx2sinxsinU 2222n

dMeNa¼Rsay

k¿Rsayfa x)1n2sin(x)1n2sin(xsin2

1)nx2cos(

tamrUbmnþ )2

qpcos()

2

qpsin(2qsinpsin

edayyk x)1n2(q,x)1n2(p

nig nx4qp,x2qp

eKán )nx2cos(xsin2x)1n2sin(x)1n2sin(

dUcen¼ x)1n2sin(x)1n2sin(xsin2

1)nx2cos( .

80

GnuKmn_RtIekaNmaRt


x¿KNnaplbUk )nx2cos(.....x6cosx4cosx2cosSn

eyIgán

n

1kn )kx2cos(S

xsin

x)1ncos()nxsin(x)1ncos()nxsin(2

xsin2

1

xsinx)1n2sin(xsin2

1

x)1k2sin(x)1k2sin(xsin2

1 n

1k

dUcen¼ xsin

x)1ncos()nxsin(Sn

.

K¿TajrkplbUk )nx(cos.....x3cosx2cosxcosT 2222n

eyIgán

n

1k

2n )kx(cosT tamrUbmnþ

2

a2cos1acos2

eKán

n

1knn S

2

1

2

n

2

)kx2cos(1T

eday xsin

x)1ncos()nxsin(Sn

dUcen¼ xsin2

x)1ncos()nxsin(

2

nTn

.

X¿KNnaplbUk )nx(sin.....x3sinx2sinxsinU 2222n

eyIgán

n

1k

2n )kx(sinU

n

1kn

2 Tn)kx(cos1

eday xsin2

x)1ncos()nxsin(

2

nTn

dUcen¼ xsin2

x)1ncos()nxsin(

2

nUn

.


81

GnuKmn_RtIekaNmaRt


lMhatTI52

eK[GnuKmn_ )1x(2

8m3mx2xy

2

2

Edl IRx nig

m Caá¨ra¨Em¨Rt

kMnt´témø m edIm,I[GnuKmn_en¼Gactag[témøkUsIunUs

énmMumYYyán¦eT?

dMeNa¼Rsay

edIm,I[GnuKmn_en¼tag[témøkUsIunUsénmMumYYylu¼RtaEtcMeBa¼

RKb´ IRx eKán 1)1x(2

8m3mx2x1

2

2

edayeKman

IRx,0)1x(2 2

eKTaj 2x28m3mx2x2x2 222

¦

)2(010m3mx2x

)1(06m3mx2x32

2

cMeBa¼ 06m3mx2x3:)1( 2

smmUl

018m9m'

03a2

eday )6m)(3m(18m9m 2

eKán 0)6m)(3m('


82

GnuKmn_RtIekaNmaRt


naM[ 6m3 ¦ ]6,3[m

cMeBa¼ 010m3mx2x:)2( 2

smmUl

010m3m'

01a2

eday )5m)(2m(10m3m 2

eKán 0)5m)(2m(' naM[ 2m5

¦ ]2,5[m

edayykcemøIy ]6,3[m RbsBVnwg ]2,5[m

ena¼eKán m .

dUcen¼eKminGackMnt´témø m edIm,I[GnuKmn_en¼Gactag

[témøkUsIunUsénmMumYYyáneT .

lMhatTI53

cUrRsaybBa¢ak´fa

3

8

2x4cos

1x4sin4x4cos2

cMeBa¼RKb´cMnYnBit

dMeNa¼Rsay

eyIgtag IRx,2x4cos

1x4sin4x4cosy

eyIgán y2x4cosy1x4sin4x4cos

¦ )1(1y2x4sin4x4cos)y1( 83

GnuKmn_RtIekaNmaRt


eyIgeRCIserIsviucT&rBIr )4,y1(U

nig )x4sin,x4cos(V

tamkenßamviPaKplKuNs;aEl )2(x4sin4x4cos)y1(V.U

tam )1( nig )2( eKTaj 1y2V.U

.

müa¨geTottamniymn&y cos.||V||.||U||V.U

eday IR,1cos1

eKTaj ||V||.||U||V.U||V||.||U||

¦ 222

||V||.||U||V.U

eday 1||V||,16)y1(||U|| 22

nig 1y2V.U

eKán 16)y1()1y2( 22 ¦ 016y2y3 2

eday )8y3)(2y(16y2y3 2

ehtuen¼ 016y2y3 2 smmUl 3

8y2 .

dUcen¼ 3

8

2x4cos

1x4sin4x4cos2

cMeBa¼RKb´cMnYnBit x .


84

GnuKmn_RtIekaNmaRt


lMhatTI54

eK[cMnYnkMupøic )xsin

1x.(sini)

xcos

1x(cosZ

2

2

2

2

Edl x CacMnYnBit.

cUrkMnt´rkmUDulGb,brmaéncMnYnkMupøicen¼ ?

dMeNa¼Rsay

rkmUDulGb,brmaéncMnYnkMupøic

eyIgán 2

222

22 )

xsin

1x(sin)

xcos

1x(cos|Z|

tag 2

222

22 )

xsin

1x(sin)

xcos

1x(cos)x(f

)x2sin

161)(x2sin

2

11(4

)x2sin

161(xcosxsin2)xsinx(cos4

)xcosxsin

11)(xsinx(cos4

)xcosxsin

xsinxcos()xsinx(cos4

)xcos

1

xsin

1()xsinx(cos4

xsin

12xsin

xcos

12xcos

4

2

4

22222

4444

44

4444

44

44

4

4

4

4

edayeKman 1x2sin 2 naM[ 2

1x2sin

2

11 2

nig 17x2sin

161

4


85

GnuKmn_RtIekaNmaRt


eKTaj 2

25

2

174)

x2sin

161)(x2sin

2

11(4

4

2

eyIgán 2

25)

x2sin

161)(x2sin

2

11(4)x(f

4

2

eday )x(f|Z| eKTaján 2

25

2

5|Z|

dUcen¼m¨UDulGb,brmaén Z KW 2

25|Z| min .

lMhatTI55

eK[ x CacMnYnBitEdl 021x71x60 2 .

cUrbgHajfa 01x3

sin

.

dMeNa¼Rsay

bgHajfa 01x3

sin

tag 21x71x60)x(f 2

ebI 021x71x600)x(f 2

150405041)21)(60(4)71( 2

eKTaj¦s 5

3

120

3971x,

12

7

120

171x 21

eyIgán 021x71x60)x(f 2 naM[ 5

3x

12

7

¦ 5

9x3

4

7


86

GnuKmn_RtIekaNmaRt


¦ 5

41x3

4

3 naM[ 3

4

1x3

1

5

4

eKTaj 3

4

1x35

4

naM[ 0

1x3sin

.

dUcen¼ ebI x CacMnYnBitEdl 021x71x60 2

ena¼eKán½ 01x3

sin

.

lMhatTI56

eK[sIVúténcMnYnBit )U( n kMnt´eday

4

nsin.2U

nn

Edl *INn

k-cUrbgHajfa 4

nsin

4

ncos

4

)1n(cos.2

x-Taj[)anfa 4

)1n(cos)2(

4

ncos)2(U 1nn

n

K-KNnaplbUk n321n U.........UUUS

CaGnuKmn_én n .

dMeNa¼Rsay

k-bgHajfa 4

nsin

4

ncos

4

)1n(cos.2

tamrUbmnþ bsinasinbcosacos)bacos(

)4

sin4

nsini

4cos

4

n(cos2)

44

ncos(2

4

)1n(cos2

4

nsin

4

ncos)

4

nsin

2

2

4

ncos

2

2(2

4

)1n(cos2


87

GnuKmn_RtIekaNmaRt


dUcen¼ 4

nsin

4

ncos

4

)1n(cos.2

.

x-Taj[ánfa 4

)1n(cos)2(

4

ncos)2(U 1nn

n

eyIgman 4

nsin

4

ncos

4

)1n(cos.2

naM[ 4

)1n(cos2

4

ncos

4

nsin

KuNGg:TaMgBIrnwg n)2( eKán

4

)1n(cos)2(

4

ncos)2(

4

nsin)2( 1nnn

dUcen¼ 4

)1n(cos)2(

4

ncos)2(U 1nn

n

.

K-KNnaplbUk n321n U.........UUUS

eyIgán

nS 4

)1n(cos)2(

4cos2

4

)1k(cos)2(

4

kcos)2( 1n

n

1k

1kk

dUcen¼ 4

)1n(cos)2(1S 1n

n

.

88

GnuKmn_RtIekaNmaRt


lMhatTI57


sin nig

10cos

x¿cUrRsayfa 10

sin)yx(4)yx(x 22222

RKb´cMnYnBit IRy,x .

dMeNa¼Rsay


sin nig

10cos

eKman 10

3

210

2

eKán )10

3

2sin(

10

2sin

)10

sin1(4310

sin2

10cos43

10sin2

10cos4

10cos3

10cos

10sin2

10

3cos

10cos

10sin2

2

2

3

¦ 0110

sin210

sin4 2

tag 0

10sint

eKán 0541',01t2t4 2

eKTaj¦s 04

51t1

( minyk )

4

51t, 2

dUcen¼ 4

51

10sin

.

eday 110

cos10

sin 22

naM[

4

5210)

4

51(1

10cos 2

89

GnuKmn_RtIekaNmaRt


dUcen¼ 4

5210

10cos

.

x¿Rsayfa 10

sin)yx(4)yx(x 22222

RKb´cMnYnBit IRy,x .

tagGnuKmn_ 10

sin)yx(4)yx(x)y;x(f 22222

eKán 222222 )4

51)(yx(4yxy2xx)y;x(f

IRy,x,0y2

15x

2

15

y2

15xy2x

2

15

y2

51xy2x

2

51

2

53)yx(yxy2x2

16

526)yx(4yxy2x2

2

22

22

2222

2222

dUcen¼ 10

sin)yx(4)yx(x 22222 RKb´cMnYnBit IRy,x

90

GnuKmn_RtIekaNmaRt


lMhatTI58

eK[RtIekaN ABC mYYy .

bgHajfaebI 3

Ctan,

3

Btan,

3

Atan Ca¦srbs´smIkar

0cbxaxx:)E( 23 ena¼eKán cb3a3 .

dMeNa¼Rsay

karbgHaj ½

eyIgman CBA ( plbUkmMukñúgRtIekaN ABC )

eyIgán ]3

C)

3

B

3

Atan[()

3

C

3

B

3

Atan(

)1(

)3

Ctan

3

Btan

3

Ctan

3

Atan

3

Btan

3

A(tan1

3

Ctan

3

Btan

3

Atan

3

Ctan

3

Btan

3

Atan

3

3

Ctan.

3

Btan

3

Atan1

3

Btan

3

Atan

1

3

Ctan

3

Btan

3

Atan1

3

Btan

3

Atan

3tan

3

Ctan)

3

B

3

Atan(1

3

Ctan)

3

B

3

Atan(

)3

CBAtan(

eday 3

Ctan,

3

Btan,

3

Atan Ca¦srbs´smIkar )E(

ena¼tamRTwsþIbTEvüteKmanTMnak´TMng ½ 91

GnuKmn_RtIekaNmaRt


)2(a3

Ctan

3

Btan

3

Atan

)4(c2

Ctan.

2

Btan.

2

Atan

)3(b2

Ctan

2

Atan

2

Ctan

2

Btan

2

Btan

2

Atan

ykTMnak´TMng )3(,)2( nig )4( CYskñúgsmIkar )1(

eKán ½

b1

ca3

¦ cab33

dUcen¼ cb3a3 .

lMhatTI59

eK[GnuKmn_

a0,

1acosx2x

acosx2acosx)x(f

2

2

bgHajfa 1)x(f1:IRx .

dMeNa¼Rsay

bgHajfa 1)x(f1:IRx

eyIgman 1acosx2x

acosx2acosx1)x(f1

2

2

IRx,0asin)xcosx(

2

acos)1x(2

)x(f1

asinacosacosx2x

)1x2x)(acos1()x(f1

1acosx2x

)acos1(x)acos1(2x)acos1()x(f1

22

22

222

2

2

2

eKTaján )1(IRx,1)x(f 92

GnuKmn_RtIekaNmaRt


müa¨geTot 1acosx2x

acosx2acosx1)x(f1

2

2

IRx,0asin)xcosx(

2

asin)1x(2

)x(f1

asinacosacosx2x

)1x2x)(acos1()x(f1

1acosx2x

)acos1(x)acos1(2x)acos1()x(f1

22

22

222

2

2

2

eKTaján )2(IRx,1)x(f

tam )1( nig )2( eKTaján 1)x(f1:IRx

lMhatTI60

eKmansmPaB ba

1

b

xcos

a

xsin 44

Edl 0ba,0b,0a

cUrbgHajfa 33

8

3

8

)ba(

1

b

xcos

a

xsin

.

dMeNa¼Rsay

bgHajfa 33

8

3

8

)ba(

1

b

xcos

a

xsin

eKman ba

1

b

xcos

a

xsin 44

naM[ abxcos)ba(axsin)ba(b 44

0)xsinbxcosa(

0xsinbxsinxcosab2xcosa

)xcosx(sinabxcosabxcosaxsinbxsinab

222

422242

422442424

eKTaj ba

1

ba

xcosxsin

b

xcos

a

xsin 2222

93

GnuKmn_RtIekaNmaRt


naM[ 44

8

4

8

)ba(

1

b

xcos

a

xsin

eKTaj )1()ba(

a

a

xsin43

8

nig )2(

)ba(

b

b

xcos43

8

bUksmIkar )1( nig )2( Gg:nigGg: eKán ½

343

8

3

8

)ba(

1

)ba(

ba

b

xcos

a

xsin

dUcen¼ 33

8

3

8

)ba(

1

b

xcos

a

xsin

.

lMhatTI61

eK[RtIekaN ABC man 5

3Bcos nig

5

4Ccos

cUrKNna )CBsin( rYckMnt´RbePTénRtIekaN ABC .

dMeNa¼Rsay

kMnt´RbePTénRtIekaN ABC

eyIgman 5

3Bcos nig

5

4Ccos

eyIgán 5

4

25

91Bcos1Bsin 2

nig 5

3

25

161Csin1Csin 2

man BcosCsinCcosBsin)CBsin(

15

3.

5

3

5

4.

5

4)CBsin(

naM[ 2

CB

ehIy 2

A

dUcen¼ ABC CaRtIekaNEkgRtg´ A . 94

GnuKmn_RtIekaNmaRt


lMhatTI62

eK[RctuekaN ABCD mYymanRCug

dDA,cCD,bBC,aAB

eKtag S CaépÞRkLarbs´ctuekaNen¼ .

cUrRsayfa )cdab(2

1S .

dMeNa¼Rsay

Rsayfa )cdab(2

1S

eyIgman ACDABC SSS

eday Bsinab2

1Bsin.BC.AB

2

1SABC

nig Dsincd2

1Dsin.DC.AD

2

1SADC

eyIgán )DsincdBsinab(2

1S

eyIgman 1Bsin ena¼ abBsinab

B

A

C

D

95

GnuKmn_RtIekaNmaRt


nig 1Dsin ena¼ cdDsincd

dUcen¼ )cdab(2

1S .

lMhatTI63

eK[RtIekaN ABC manRCugepÞógpÞat´TMnak´TMng

222 c2ba k¿cUrbgHajfa

ab4

baCcos

22

x¿TajbBa¢ak´fa 2

1Ccos

dMeNa¼Rsay

k¿bgHajfa ab4

baCcos

22

tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABCeKman ½

Ccosab2bac 222 Ettamsmµtikmµ 222 c2ba

eKán Ccosab2ba2

ba 2222

¦ 2

ba

2

babaCcosab2

222222

dUcen¼ ab4

baCcos

22

.

x¿ TajbBa¢ak´fa 2

1Ccos

eyIgman 0)ba( 2

0bab2a 22 ¦ 2

1

ab4

ba 22

96

GnuKmn_RtIekaNmaRt


edaytamsRmayxagelI ab4

baCcos

22

dUcen¼ 2

1Ccos .

lMhatTI64

eK[RtIekaN ABCmYymanRCúg c,b,a .

ebI )c

1

b

1

a

1(

2

1

c

Ccos

b

Bcos

a

Acos

cUrkMnt´RbePTénRtIekaN ABC

dMeNa¼Rsay

RbePTénRtIekaN ABC

tamRTwsþIbTkUsIunUskñúgRtIekaN ABCeKman ½

bc2

acbAcos

222 naM[ )1(

abc2

acb

a

Acos 222

dUcKñaEdreKTaj )2(abc2

bca

b

Bcos 222

nig )3(abc2

cba

c

Ccos 222

bUkTMnak´TMng )3(,)2(,)1( Gg:nwgGg:eKán ½

abc2

cba

c

Ccos

b

Bcos

a

Acos 222

eday )c

1

b

1

a

1(

2

1

c

Ccos

b

Bcos

a

Acos eKTaján ½

97

GnuKmn_RtIekaNmaRt


0)ac()cb()ba(

0)aac2c()cbc2b()bab2a(

ac2bc2ab2c2b2a2

acbcabcbaabc2

abcabc

abc2

cba

)c

1

b

1

a

1(

2

1

abc2

cba

222

222222

222

222

222

222

eKTajánsmPaB cba .

dUcen¼ ABC CaRtIekaNsmgß.

lMhatTI65

eK[smIkar 02m3x5x)3m2(x:)E( 23

«bmafasmIkaren¼man¦bItageday tan,tan,tan .

k¿cUrKNna

coscoscos

)sin(A CaGnuKmn_én m .

x¿kMnt´ m edIm,I[ 4A .

K¿eda¼RsaysmIkar )E( cMeBa¼témø m Edlán

rkeXIjxagelI .

dMeNa¼Rsay

k¿ KNna

coscoscos

)sin(A CaGnuKmn_én m

eyIgman

coscoscos

)](sin[A

98

GnuKmn_RtIekaNmaRt


tantantantantantan

tantantantantantan

coscoscos

coscossincoscossinsinsinsincoscossin

coscoscos

cos)sin()cos(sin

eday tan,tan,tan Ca¦ssmIkar )E( ena¼tamRTwsþIbTEvüt

eyIgman

2m3tantantan

5tantantantantantan

3m2tantantan

eyIgán 5m)2m3(3m2A

dUcen¼ 5mA .

x¿kMnt´ m edIm,I[ 4A

edayeyIgman 5mA

eyIgán 45m naM[ 5m .

K¿ eda¼RsaysmIkar )E( ½

cMeBa¼ 1m eKán 01x5x5x:)E( 23

eday )1x4x)(1x(1x5x5x 223

eKTaj 0)1x4x)(1x( 2 ¦

01x4x

1x2

314' eKTaj 32x,32x 21

dUcen¼sMNMu¦ssmIkar }32,1;32{x .

99

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 100 -

lMhatTI66

eK[GnuKmn_ 22 ysinbxsinaycosbxcosa)y;x(f

( Edl 0b,0a ) .

cMeBa¼RKb´ IRy;x bgHajfa 2)ba()y;x(f .

dMeNa¼Rsay

bgHajfa 2f (x; y) (a b)

22 ysinbxsinaycosbxcosa)y;x(f

)yxcos(ab2ba

)ysinxsinycosx(cosab2)ysiny(cosb)xsinx(cosa

ysinbysinxsinab2xsinaycosbycosxcosab2xcosa

22

222222

22222222

eKán )yxcos(ab2ba)y;x(f 22

edayeKman 1)yxcos(:IRy;x

eyIgán 222 )ba(ab2ba)x(f

dUcen¼ 2)ba()y;x(f .

100

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 101 -

lMhatTI67

eK[RtIekaN ABC Edl bc2

a

2

Asin .

rkRbePTénRtIekaN ABC

dMeNa¼Rsay

rkRbePTénRtIekaN ABC

eKman bc2

a

2

Asin

tamRTwsþIbTkUsIunUseKman bc2

acbAcos

222

eday 2

Asin21Acos 2

eKán ½

cb

0)cb(

0cbc2b

abc2acbbc2

abc2

bc2

acbbc4

a21

bc2

acb

2

22

2222

2222

2222

RtIekaNABCmanRCúg cb naM[vaCaRtIekaNsmát

kMBUl A .

101

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 102 -

lMhatTI68

eK[smIkar mx3sinxsinx3cosxcos:)E( 33

Edl IRm Caá¨ra¨Em¨Rt .

k¿cUreda¼RsaysmIkaren¼kalNa 8

33m

x¿rklk&çx&NÐsRmab´ m edIm,I[smIkaren¼man¦s .

dMeNa¼Rsay

k¿eda¼RsaysmIkaren¼kalNa 8

33m

eyIgman xsin4xsin3x3sin 3 naM[ )x3sinxsin3(4

1xsin 3

ehIy xcos3xcos4x3cos 3

naM[ )x3cosxcos3(4

1xcos3

smIkar )E( Gacsrsr ½

3

3

22

22

mx2cos

m4x2cos4

m4x6cosx2cos3

m4)x3sinx3(cos)xsinx3sinxcosx3(cos3

m4x3sinx3sinxsin3x3cosx3cosxcos3

mx3sin)x3sinxsin3(4

1x3cos)x3cosxcos3(

4

1

eday 8

33m eKán

2

3x2cos

naM[ Zk,k12

x

102

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 103 -

x¿ rklk&çx&NÐsRmab´ m ½

edIm,I[smIkaren¼man¦seKRKan´Et[ 1m1 3

¦ 1,1m .

lMhatTI69

eda¼RsaysmIkar ½

0)xsin2(log)x(sinlog 3

2

2

2

dMeNa¼Rsay

eda¼RsaysmIkar ½

0)xsin2(log)x(sinlog 3

2

2

2

lkç&xN&Ð 0xsin naM[ Zk,k2xk2

smIkarGacsresr ½

02)x(sinlog3)x(sinlog

02log)x(sinlog)x(sinlog

2

2

2

2

3

2

2

2

tag )x(sinlogt2

eKánsmIkar 02t3t 2

eday cab eKTaj¦s 2t,1t 21

-cMeBa¼ 1t eKán 1)x(sinlog2

smmUl 2

1xsin

naM[

Zk,k24

3k2

4x

k24

x

103

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 104 -

-cMeBa¼ 2t eKán 2)x(sinlog2

smmUl 2

1xsin

naM[

Zk,k26

5k2

6x

k26

x

lMhatTI70

k¿cUrbgHajfa

]x)1n2sin(2[]x)1n2sin(2[

)nx2cos(xsin2

x)1n2sin(2

1

x)1n2sin(2

1

x¿KNna

n

1kn x)1k2sin(2x)1k2sin(2

)kx2cos(S .

dMeNa¼Rsay

k¿ karbgHaj

tag x)1n2sin(2

1

x)1n2sin(2

1)x(f

x)1n2sin(2x)1n2sin(2

)nx2cos(xsin2

x)1n2sin(2x)1n2sin(2

x)1n2sin(x)1n2sin(

dUcen¼ ]x)1n2sin(2[]x)1n2sin(2[

)nx2cos(xsin2

x)1n2sin(2

1

x)1n2sin(2

1

x¿KNna

n

1kn x)1k2sin(2x)1k2sin(2

)kx2cos(S

eyIgán

n

1kn x)1n2sin(2

1

x)1n2sin(2 104

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 105 -

)x)1n2sin(2(xsin2xsin

x)1ncos()nxsin(

x)1n2sin(2xsin2

xsinx)1n2sin(.

xsin2

1

x)1n2sin(2

1

xsin2

1

xsin2

1

dUcen¼ )x)1n2sin(2(xsin2xsin

x)1ncos()nxsin(Sn

.

lMhatTI71

k¿cUrbgHajfa )nxcos(xcos

x)1ncos()nxtan(xtan1

x¿KNna

n

1kn )kxtan(xtan1P .

dMeNa¼Rsay

k¿bgHajfa )nxcos(xcos


eKman xsin)nxsin(xcos)nxcos(x)1ncos(

naM[

)nxtan(xtan1xcos)nxcos(

xsin)nxsin(xcos)nxcos(

)nxcos(xcos

x)1ncos(

Bit

dUcen¼ )nxcos(xcos


.

x¿KNna

n

1kn )kxtan(xtan1P

105

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 106 -

)nxcos(xcos

1

)nxcos(

x)1ncos(....

x3cos

x2cos.

x2cos

xcos.

xcos

1.

xcos

1

)kxcos(xcos

x)1kcos(

n

n

n

1k

dUcen¼ )nxcos(xcos

1)kxtan(xtan1P

n

n

1kn

.

lMhatTI72

eK[RtIekaN ABC mYymanRCug cAB,bAC,aBC

tag S CaRkLaépÞ nig R CakaMrgVg´carikeRkA

énRtIekaNen¼ .

k¿cUrRsayfa R

S2CcoscBcosbAcosa .

x¿Tajfa S4CcotcBcotbAcota 222 .

dMeNa¼Rsay

k¿Rsayfa R

S2CcoscBcosbAcosa

106

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 107 -

tag O Cap©iténrgVg´carikeRkARtIekaN ABC ehIy

H CacMnuckNþalénRCug ]BC[ .

RkLaépÞRtIekaN ABC KW OABOCAOBC SSSS

eyIgman ROCOB naM[ OBC

CaRtIekaNsmát

kMBUl O .

ehIy BCOH naM[ OH CakMBs´énRtIekaN OBC

eyIgman OH.BC2

1SOBC

kñúgRtIekaNEkg OBH eKmanOB

OH)BOHcos(

¦ )BOHcos(.OBOH

eKán )BOHcos(.OB.BC2

1SOBC

A

B C

O

H

107

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 108 -

eyIgman ABAC2

BOCBOH

mMup©itnigmMucarikkñúgrgVg´s;at´FñÚrYmBC .

eKán AcosR.a2

1SOBC

RsaydUcKñaeKán BcosR.b2

1SOAC

nig AcosR.c2

1SOAB

ehtuen¼ CcosR.c2

1BcosR.b

2

1AcosR.a

2

1S

)CcoscBcosbAcosa(R2

1S

dUcen¼ R

S2CcoscBcosbAcosa .

x¿Tajfa S4CcotcBcotbAcota 222

eyIgman AcosaR2Acosa.Asin

a

Asin

AcosaAcota 22

dUcKñaEdr BcosaR2Bcotb 2

nig Ccos.cR2Ccotc 2

eKán )CcoscBcosbAcosa(R2CcotcBcotbAcota 222

eday R

S2CcoscBcosbAcosa ( sRmayxagelI )

dUcen¼ S4CcotcBcotbAcota 222 .

108

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 109 -

lMhatTI73

eKtag r nig R erogKñaCakaMrgVg´carikkñúg nigcarikeRkA

énRtIekaN ABCmYy .

k¿cUrRsayfa R

r1CcosBcosAcos

x¿cUrRsayfa r2R .

dMeNa¼Rsay

k¿Rsayfa R

r1CcosBcosAcos

2

CBcos

2

CBcos2

2

Asin21CcosBcosAcos 2

2

Csin

2

Bsin

2

Asin41

)2

CBcos

2

CBcos(

2

Asin21

)2

CBcos

2

Asin(

2

Asin21

2

CBcos

2

Asin2

2

Asin21 2

edayeKman ½

bc

)cp)(bp(

2

Asin

ab

)bp)(ap(

2

Csin,

ac

)cp)(ap(

2

Bsin

eKán abc

)cp)(bp)(ap(

2

Csin

2

Bsin

2

Asin

109

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 110 -

eKTaj abc

)cp)(bp)(ap(41CcosBcosAcos

R

r1

R.p

r.p1

R.p

S1

R.Sp

S1

RR4

abc.p

)cp)(bp)(ap(p1

2

dUcen¼ R

r1CcosBcosAcos .

x¿Rsayfa r2R

tamvismPaBkUsIu .2

eKán )bp)(ap(2)bp()ap(

)bp)(ap(2c

)cp)(ap(2bap2

eKTaj )1(2

1

c

)bp)(ap(

dUcKñaEdr )2(2

1

a

)cp)(bp(

nig )3(2

1

b

)cp)(ap(

KuNTMnak´TMng )3(,)2(,)1( Gg:nwgGg:eKán ½

8

1

abc

)cp)(bp)(ap(

eKTaj 8

1

2

Csin

2

Bsin

2

Asin

eday 2

Csin

2

Bsin

2

Asin41CcosBcosAcos

110

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 111 -

eKán 2

3)

8

1(41CcosBcosAcos

Et R

r1CcosBcosAcos

eKTaj 2

3

R

r1 ¦ r2R

dUcen¼ r2R .

111

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 112 -

1> eK[sIVúténcMnYnBit *INn,4

ncos)2(U n

n

k> cUrbgðajfa 4

ncos

4

nsin

4

)1n(sin.2

.

x> Tajbgðajfa 4

nsin)2(

4

)1n(sin)2(U n1n

n

.

K> KNnaplbUk n321

n

1kkn U....UUUUS

.

2>eK[sIVúténcMnYnBitkMnt;eday

INn,U4U2U

1U,0U

n1n2n

10

k>eKtag n1n1n U)3i1(UZ:INn .

bgðajfa n1n Z)3i1(Z

x> cUrbgðajfa )3

nsin.i

3

n(cos2Z n

n

.

K> TajrktYtUeTA nU CaGnuKmn_én n .

3> eKmansIVúténcMnYnBit )U( n kMnt;elI IN edayTMnak;TMng ³

1U0 nig cMeBaHRKb; 4

ncos2U2U:INn n1n

k>cUrbgðajfaeKGackMnt;cMnYnBit a nig b edIm,I[sIVút )V( n Edl

kMnt;edayTMnak;TMng 4

nsinb

4

ncosaVU nn

CasIVútFrNImaRt

x> TajrktYtUeTA nU CaGnuKmn_én n . 112

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 113 -

4> eKmansIVúténcMnYnkMupøic )Z( n kMnt;eday ³

INn,2

i32Z

2

i3Z

2Z

n1n

0

k> eKtag 1ZU:INn nn .

cUrbgðajfa n1n U.2

i3U

rYcTajrk nU CaGnuKmn_én n .

x> RsaybBa¢ak;fa )12

nsin.i

12

n(cos

12

ncos2Zn

.

5- eK[sIVúténcMnYnkMupøic )Z( n kMnt;edayTMnak;TMng ³

1Z,0Z 10 nig n1n2n Z2

i1Z

2

i21Z:INn

k> tag n1nn ZZU:INn . cUrbgðajfa n1n U2

i1U

x> RsaybBa¢ak;fa 4

nsin.i

4

ncosUn

.

K> tag

n

0kkn US . cUrRsayfa n1n SZ rYcTajrk nZ

CaGnuKmn_én n .

6- eK[sIVút )V(&)U( nn kMnt;elI *IN eday

222n

222n

n

n.....

n

2

n

1V

n

nsin....

n

2sin

n

1sinU

k> bgðajfa )V( n CasIVútcuH ehIyRKb; 2

1V:*INn n .

113

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 114 -

x> eK[GnuKmn_

xsin6

xx)x(h,xcos

2

x1)x(g,xsinx)x(f

32

cUrbgðajfa 0)x(h,0)x(g,0)x(f:IRx .

K> epÞógpÞat;fa 43333 nn...321:1n

rYcTajfa nn2n VUn

1.

6

1V .

7-eK[ )U( n CasIVútnBVnþmantY n321 U........,,U,U,U

nigplsgrYm Zk,k4d .

k> cUrRsaybBa¢ak;rUbmnþ ³

2

dsin

2

UUSin.

2

ndSin

SinU.....SinUSinUSinUSinU

n1

n321

n

1kk

2

dsin

2

UUCos.

2

ndSin

CosU.....CosUCosUCosUCosU

n1

n321

n

1kk

x> Gnuvtþn_ cUrKNnaplbUkxageRkam ³

)nasin(.....a3sina2sinasinSn

)nacos(......a3cosa2cosacosCn

8-eKBinitüsIVút )U( n kMnt;eday ³

)n(

n 22.......222U . 114

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 115 -

k> edayeFIVvicarütamkMenIncUrbgðajfa 1nn 2cos2U

x> eKman )n(

n 22........222V .

cUrRsayfa 1nn 2sin2V

.

K> KNnalImIt nn

nnnV.2limandUlim

.

9- eKmanGnuKmn_ IRxxxf ,sin)(

k> cUrRsayfa xxfx

x )(6

3

RKb; IRx .

x> eKBinitüplbUk 2222sin......

3sin

2sin

1sin

n

n

nnnSn .

cUrrkkenSamGménplbUkenH .

K> cUrKNna nn

S

lim .

10-eKmansIVút )U( n kMnt´eday

INn,

2

nsin2

2

ncosU2U

1U

n1n

0

k¿kMnt´BIrcMnYnBit A nig B edIm,I[sIVút )V( n kMnt´eday

TMnak´TMng 2

nsinB

2

ncosAVU nn

CasIVútFrNImaRt .

x¿ cUrKNna nU CaGnuKmn_én n .

115

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 116 -

11-eK[ x2tan2.......x2tan2x2tan2xtanS nn22n

k¿ cUrbgHaj 2cot2cottan .

x¿ TajbgHajfa x2cot2xcotS 1n1nn

.

12-eK[ n

n

2

3233n 3

asin3.......

3

asin3

3

asin3asinS

k¿ cUrbgHajfa x3sin4

1xsin

4

3xsin 3

.

x¿ TajbgHajfa a3sin4

1

3

asin

4

3S

n

1n

n

.

13-eK[ a2sin

1.......

a2sin

1

a2sin

1

asin

1S

n2n .

k¿ cUrbgHajfa xsin

1xcot

2

xcot .

x¿ TajbgHajfa a2cot2

acotS n

n .

14-eK[ a2cos

4......

a2cos

4

a2cos

4

acos

1S

n2

n

22

2

22n

k¿ cUrbgHajfa xsin

1

x2sin

4

xcos

1222

.

x¿ TajbgHajfa asin

1

a2sin

4S

21n2

1n

n

.

15-eK[

1n

2

n

n

2

22n 2

xcos

2

xsin2...............

2

xcos

2

xsin2

2

xcosxsinS

k¿ cUrbgHajfa 2

acosasin2a2sinasin2 2 .

x¿ TajbgHajfa x2sin2

1

2

xsin2S

n

nn .

116

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 117 -

16-eK[ n2n 2

xcos..........

2

xcos.

2

xcos.xcosP .

cUrbgHajfa n

1nn

2

xsin

x2sin.

2

1P

.

17-eK[ )

2

acos

11).....(

2

acos

11)(

2

acos

11)(

acos

11(P

n2

n .

k¿ cUrbgHajfa 2

xtan

xtan

xcos

11 .

x¿ KNnaplKuN nP .

18-cUrRsaybBa¢k´smPaBxageRkam ½

k¿ xtan1

xtan1)x

4tan(

c¿ 2sin22sintan

x¿ btanatan1

btanatan

)bacot(

)batan(22

22

q¿ x2sin2

11

xcosxsin

xcosxsin 33

K¿ ytanxtan1

ytanxtan

)yxcos(

)yxsin(

C¿ atan1

atan1

a2cos

a2sin1

X¿ 22 sinsin)sin()sin(

g¿ )tantantantantan(tan1

tantantantantantan)tan(

19-eK[RtIekaN ABCmYy .

cUrRsaybBa¢ak´smPaBxageRkam ½

k¿ 2

Ccos

2

Bcos

2

Acos4CsinBsinAsin

x¿ 2

Csin

2

Bsin

2

Asin41CcosBcosAcos

117

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 118 -

K¿ CtanBtanAtanCtanBtanAtan

X¿ 2

Ccot

2

Bcot

2

Acot

2

Ccot

2

Bcot

2

Acot

g¿ CcosBsinAsin2CsinBsinAsin 222

c¿ CcosBcosAcos22CsinBsinAsin 222

q¿ CcosBcosAcos21CcosBcosAcos 222

C¿ CsinBsinAsin4C2sinB2sinA2sin

Q¿ CcosBcosAcos41C2cosB2cosA2cos

j¿ AcosCsinBsin2CsinBsinAsin 222

20-cUrRsaybBa¢ak´smPaBxageRkam ½

k¿ x3cosx2cosxcos4x6cosx4cosx2cos1

x¿ x3tanx2cosx4cos

x2sinx4sin

K¿ x2sin)xcos21(x3sinx2sinxsin

X¿ x2cos)xcos21(x3cosx2cosxcos

21-cUrRsaybBa¢k´sMenIxageRkam ½

k¿ ccosbcosacos

)cbasin(ctanbtanatanctanbtanatan

x¿ a2sina2cos)basin()basin(2)ba(sin)ba(sin 222

K¿ )bacos()bacos(1bcosacos 22

X¿ )bacos()bacos(

)basin(2btanatan

118

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 119 -

g¿ x3tan)x3

tan()x3

tan(xtan

22-k¿ RsaybBa¢ak´ÉklkçN¼PaB ½

0xcoszcos

)xzsin(

zcosycos

)zysin(

ycosxcos

)yxsin(

x¿ TajbgHajfa ½

zcosycosxcos

)xzsin()zysin()yxsin(3

xcoszcos

)xz(sin

zcosycos

)zy(sin

ycosxcos

)yx(sin22233

3

33

3

33

3

23-cUrRsaybBa¢ak´fa acot2

1acota2cot

2

24-cUrRsaybBa¢ak´fa xcosxsin8xcosxsin4x4sin 3

nig 1xcos8xcos8x4cos 24 .

25-cUrbgHajfa atan1

)atanx1)(xa(tan2a2cosx2a2sin)x1(

2

2

.

26-RsaybBa¢ak´smPaBxageRkam ½

k¿ x2sin

2xcotxtan

x¿ x2cot2xtanxcot

K¿ x2tan2)x4

tan()x4

tan(

X¿ 2

xtan

xcos1

xsin

g¿ x2cos

2)x

4tan()x

4tan(

27-cUrsRmYlkenßamxageRkam ½

k¿ xsinxcos1

xsinxcos1

119

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 120 -

x¿ xsinxcos1

xsinxcos1

K¿ x4cosx2cos1

x6sinx4sinx2sin

X¿ ysinxsin

)yxsin(

g¿ )vucos(1

vcosucos

28-cUrbMElgCaplKuNénkenßam

x4cosx3cosx2cosxcosS .

29-eK[ a nig b CamMuRsYcEdl 2

1asin

nig 4

26bsin

cUrKNnatémøGnuKmn_rgVg´énmMu ba nig ba

rYcTajrktémøén bCar¨adüg´.

30-eK[ a CamMuRsYcEdl 2

22acos

.

cUrKNna a2cos rYcTajrktémøénmMu a Cara¨düg´ .

31-cUrbMElgplbUk

)cbasin(csinbsinasinS CaplKuNktþa .

32-RtIekaN ABC mYyman 29

20Bcos nig

29

21Ccos .

cUrrkRbePTénRtIekaN ABC .

33-cUrkMnt´rktémøGtibrma nig Gb,brma ( ebIman )

énGnuKmn_xageRkam ½ 120

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 121 -

k¿ 7xcos4xsin3y

x¿ 17xcos12xsin5y

K¿ xcosxsiny 44

X¿ 2xcos5xcosxsin4xsin3y 22

g¿ xcosxsiny 66

c¿ 2

2

2

2

2

2

xcos

1xcos

xsin

1xsiny

q¿ 2

3

3

2

3

3

xcos

1xcos

xsin

1xsiny

C¿ xcosxsiny 88

j¿ 4xtan32xtany 2

34-sRmYl acos2....222A n Edl 2

a0

35-eK[smIkardWeRkTIBIr 02mx)mm(x:)E( 22

eKsnµtfasmIkaren¼man¦sBIrtagerogKñaeday

atan nig btan .

k¿ cUrkMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3ba

.

x¿ eda¼RsaysmIkarxagelIcMeBa¼ m EdlánrkeXIj

K¿ edayeRbIlTæplxagelIcUrTajrktémøRákdén 12

tan

36-k¿ cUrKNnatémøRákdén 8tan

.

121

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 122 -

x¿ cUreda¼RsaysmIkar 010100 )x(tanlog1 122

.

37-k¿ cUrKNnatémøRákdén 5

tan .

x¿ cUreda¼RsaysmIkar

0xcos)525(3x2sin)2

52(xsin 424

38-k¿ cUrKNnatémø®ákdén 10

sin nig 10

cos

x¿ cMeBa¼RKb´cMnYnBit IRy,x cUrbgHajfa ½

10

sin)yx(4)yx(x 22222

39-eK[sIVúténcMnYnBit )U( n kMnt´eday

INn,U2U

2a0,acos2U

n1n

0

edayeFVIvicartamkMenIncUrbgHajfa nn 2

acos2U .

40-eK[GnuKmn_ 22 )ysinbxsina()ycosbxcosa()y,x(f .

cUrRsayfa 222 )ba()y,x(f .

41-k¿ cUrRsaybBa¢ak´TMnak´TMng x2cot2xcotxtan

x¿ cUrKNnaplbUkxageRkam ½

b2tan2.....b2tan2b2tan2btanB

2

atan

2

1....

2

atan

2

1

2

atan

2

1atanA

nn22n

nn22n

122

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 123 -

42-k¿ cUrRsaybBa¢ak´fa xtan2x2tanxtan.x2tan 2

x¿ cUrKNnaplbUk

n

0k

k21k

kn a2tana2tan2

1S

43-k¿ cUrRsayfa )x3sinxsin3(4

1xsin 3

x¿ cUrKNna a3sin3

1....a3sin

3

1a3sin

3

1asinS n3

n

23

2

33n

44-k¿ cUrRsayfa xsin

1

x2sin

4

xcos

1222

x¿ cUrKNna a2cos

4....

a2cos

4

acos

1S

n2

n

22n

45-k¿ cUrRsayfa x2cotxcotx2sin

1

x¿ cUrKNna a2sin

1....

a2sin

1

a2sin

1

asin

1S

n2n

46-k¿ cUrRsayfa xcot2

xcot

xcos

11

x¿ cUrKNnaplKuN ½

)a2cos

11).....(

a2cos

11)(

a2cos

11)(

acos

11(P

n2n

47-k¿ cUrRsayfa xcot.xcos

)nxcos(

xcos

x)1ncos(

xcos

)nxsin(n1nn

x¿ cUrKNna xcos

)nxsin(....

xcos

x3sin

xcos

x2sin

xcos

xsinS

n32n

48-k¿ cUrRsayfa )nxtan(x)1ntan(xsin

1

x)1ncos().nxcos(

1

x¿ KNna

x)1ncos()nxcos(

1....

x3cosx2cos

1

x2cosxcos

1Sn

123

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 124 -

49-k¿ cUrRsayfa

x)1ntan()nxtan(1xtan)nxtan(x)1ntan(

x¿ KNnaplbUk

x)1ntan()nxtan(.....x3tanx2tanx2tanxtanSn

50-k¿ cUrRsaybBa¢ak´fa xcos21

x2cos211xcos2

x¿ cUrKNnaplKuN ½

)1a2cos2).....(1a2cos2)(1a2cos2)(1acos2(P n2n

51-KNnaplKuN n2n 2

xcos.........

2

xcos.

2

xcos.xcosP

52-KNnaplKuN

)2

xtan1().........

2

xtan1)(

2

xtan1)(xtan1(P

n

2

2

222n

53-KNnaplKuN ½

)2

bcos

2

a).....(cos

2

bcos

2

a)(cos

2

bcos

2

a)(cosbcosa(cosP

nn22n

54-KNnaplKuN

)3

xsin41).....(

3

xsin41)(

3

xsin41)(xsin41(P

n

2

2

222n

55-KNnaplKuN

)3

xsin43)......(

3

xsin43)(

3

xsin43)(xsin43(P

n

2

2

222n

124

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 125 -

56-KNnaplKuN

n

2

n

2

2

2

2

2

2

2

2

n

3

xtan31

3

xtan3

.....

3

xtan31

3

xtan3

.

3

xtan31

3

xtan3

.xtan31

xtan3P

57-k¿ cUrRsayfa )xtan3x3tan(8

1

xtan31

xtan2

3

x¿ cUrKNnaplbUk

n

0kk2

k3

k

n a3tan31

a3tan3

1

S

58-k¿ cUrRsayfa xtanx2tan2

1

xtan1

xtan2

3

x¿ cUrKNnaplbUk

n

0kk2

k3

k

n a2tan1

a2tan2

1

S

59-KNnaplKuN ½

)acotan)......(taacota)(tanacota)(tanacota(tanPnn 224422

n

60-bgHajfa asin2

a2sina2cos....a4cosa2cosacos

n

n1n .

61- eK[sIVúténcMnYnBit )a( n kMnt´eday

INn,

4

a3aa

2

1a

3 1nn1n

1

k¿ cUrRsaybBa¢ak´fa 1a0 n .

x¿ eKtag nn cosa . cUrrkRbePTénsIVút )( n ? 125

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 126 -

K¿ KNna n nig na CaGnuKmn_én n .

62-eKmansIVút )b( n kMnt´eday 2

3b0

nig INn,b411

bb

2

n

n1n

k¿ eKBinitüsIVút )( n Edl INn,2

0 n

ehIy 2

tanb n

n

.

cUrkMnt´rkRbePTénsIVút )( n ?

x¿ KNna n nig nb CaGnuKmn_én n .

63-eKmansIVút )t( n kMnt´eday 3tant 0

nig INn,t42t 2

n1n

k¿ cUrbgHajfa 2t0 n .

x¿ tag nn sin2t .

cUrkMnt´rkRbePTénsIVút )( n ?

K¿ KNna n nig nt CaGnuKmn_én n .

64-eKmansIVút 3

1u 0 nig INn,u1uu 2

nn1n

k¿ tag nn cotu Edl INn,2

0 n

.

cUrkMnt´rkRbePTénsIVút )( n 126

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 127 -

x¿ KNna n nig nu CaGnuKmn_én n .

65-eK[ )u( n CasIVútviC¢manEdl 2u 0

nig n

n2

1n u1

u2u

.

KNna nu .

66-eK[RtIekaN ABC mYYymanRCúg c,b,a .

cUrRsaybBa¢ak´fa )c

1

b

1

a

1(

2

1

c

Ccos

b

Bcos

a

Acos .

67-eda¼RsaysmIkarxageRkam ½

k¿ xcos)3

x2cos(

x¿ )x3

sin(x2sin

K¿ 2

3)x

4sin(

X¿ )x3

2sin()

3x2sin(

g¿ )4

xtan(x3tan

68-eda¼RsaysmIkar ½

k¿ )6

xcos(x3sin

x¿ )x6

cos()4

x2sin(

K¿ )x23

cot()x4

tan(

X¿ )6

x3tan()x23

tan(

127

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 128 -

g¿ 5

cot)3

x3tan(


k¿ 01xsin3xsin2 2

x¿ 02xsin)12(2xsin4 2

K¿ 02

3xcos)31(xcos2 2

X¿ 03xtan)31(xtan 2

g¿ 03xtan4xtan3 2


k¿ 03xsin)13(2xsin4 2

x¿ 0xcosxcosxsin)31(xsin3 22

K¿ 03xtan)13(xtan 2

X¿ 01xcoslog3xcoslog2 222

g¿ 02xsinlog3xsinlog2

2

2


k¿ 0xcos)13(xcosxsin32xsin)13( 22

x¿ 2xsinxcos3

K¿ 2xcosxsin

X¿ 1xsin3xcos 128

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 129 -

72-eda¼RsaynwgBiPakßasmIkar m2xsinxcosm .

73-eKmansmIkar 0cosx2x)1cos2( 2

Edl 0 .

kMnt´témø edIm,I[smIkarman¦sBIrepÞógpÞat´

0sin4''x

1

'x

1 .

74-eK[smIkar ½

20,0)cos41)(32(sinx2x 22 .

k¿ cUrRsayfasmIkaren¼man¦sCanic©RKb´ .

x¿ cUrrkTMnak´TMngrvag¦s 'x nig ''x minGaRs&ynwg


k¿ 3xtan)x2

7sin(3xsin

x¿ xsin

11xcos4xcosxcot2

2

222

K¿ xcosxsinxcos2x2cos2 2

X¿ xsin2)4

x(sin 3

g¿ )4

xtan()x4

tan(xtanx2tan


k¿ 4

2x3sinxsinx3cosxcos 33

x¿ 1x2sin1

1xcos2)23xsin2(xcos 2

129

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 130 -

K¿ 13)12

xcos()4

xsin(4

X¿ x3cos

113)x

3tan()x

3tan(xtan)31(

2

77-k¿ cUrKNnatémø®ákdén 12

tan nig

12

5tan

x¿ eda¼RsaysmIkar 01xtan5xtan5xtan 23

78-k¿ KNnatémø®ákdén 8

tan

x¿ eda¼RsaysmIkar 0xtanlog1 212

.

79-eda¼RsayRbB&næsmIkar

4

25ysinysinxsin3

8

7ysinxsin3xsin

32

23

80-eda¼RsaysmIkar 4

1xcos.xsin loglog

xcos.xsinxcosxsin .

81-eda¼RsaysmIkar xx2

2 226

xxcos2

.


4347347)b

xsin2)4

x(sin)a

xcosxcos

3

( RbLgGaharUbkrN_eTArusßI éf¶ 05 emsa qñaM 2000 )

83-eK[smIkar xcos

mxcos)1m(xsinm

k¿ kMnt´ m edIm,I[smIkaren¼man¦s .

x¿ eKtag 21 x,x Ca¦sBIrénsmIkarxagelI 130

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 131 -

ehIyepÞógpÞat´

k2

xx 21

cUrKNna )xx(2cos 21 . ( Zk )

84-RsaybBa¢ak´faebI 0y,1y1,y1y1

y1y1xtan

ena¼eKán x2siny .

85-eK[RtIekaN ABC mYYyman 1CcosBcosAcos 222 .

cUrkMnt´RbePTénRtIekaN ABC ?

86-eKmansmPaB ba

1

b

xcos

a

xsin 44

.

cUrRsayfa 44

10

4

10

)ba(

1

b

xcos

a

xsin

87-eK[ 8

3bcos,

3

1acos nig

7

5ccos .

cUrRsayfa 12

ctan

2

btan

2

atan 222

?

88-eKtag c,b,a CaRCúgrbs´RtIekaN ABC

Edl 3

1

2

Btan.

2

Atan .

k¿ cUrRsayfa 2

bac

.

x¿ cUrbgHajfa 333 c8abc6ba .

89-eK«bmafasmIkar 0cbxax 2 man¦BIrtag

eday tan nig tan .

cUrKNna )cos(c)cos()sin(b)(sinaM 22

CaGnuKmn_énelxemKuN c,b,131

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 132 -

90-cUrbgHajfacMnYn 7

cos Ca¦ssmIkar

01x4x4x8 23 .

91-cUrbgHajfacMnYn 9

13cos,

9

7cos,

9cos

Ca¦ssmIkar 01x6x8 3 .

92-cUrbgHajfa 2

7

7

3sin

7

2sin

7sin

.

93-cUrbgHajfa oo0ooo 9tan69tan63tan57tantan51tan3tan .

94-KNnatémøénplKuN 00o 70tan50tan10tanP .

95-eK[smIkardWeRkTIBIr ½

03m2x)1m(x:)E( 2

k¿ kMnt´ m edIm,I[smIkaren¼man¦sBIrepßgKña .

x¿ «bmafa atan nig btan Ca¦srbs´smIkar )E( .

kMnt´ m edIm,I[ )bacos()basin( .

96-cUrKNnatémøén 7

4sin

7

2sin

7sinA 222

.

97-cUrbgHajfa 380tan40tan20tan ooo .

98-eK[smIkardWeRkTIbI ½

03x)343(x)3m(x3:)E( 23 Edl mCaá¨ra¨Em¨Rt

eK«bmafasmIkarman¦sbItagerogKñaeday

tan,tan,tan . 132

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 133 -

k¿ kMnt´témø m edIm,I[ 4

3 .

x¿ cUrkMnt´ ,, Edl 2

0

cMeBa¼témø m

EdlánrkeXIj

xagelIen¼ .

99-eK[smIkar 093mmxx3:)E( 2

Edl mCaá¨ra¨Em¨Rt .

eK«bmafasmIkarman¦sbItagerogKñaeday

tan nig tan .

k¿ kMnt´témø m edIm,I[ 3

32

)cos(

)sin(

.

x¿ cUrkMnt´ , Edl 2

0

cMeBa¼témø m

EdlánrkeXIjxagelIen¼ .

100-KNnaplbUk ½

n

2

n2

2

2

22n 2

xtan

4

1.........

2

xtan

4

1

2

xtan

4

1xtanS

101-eK[mMubI ,,0 Edl 4

3 .

cUrRsaybBa¢ak´fa ½

k¿ 2

23sinsinsin

x¿ 2

23coscoscos

133

GnuKmn_RtIekaNmaRt

© 2008 Lim Phalkun - 134 -

102-eK[ n cMnYnBitviC¢man n321 a......,,a,a,a .

cMeBa¼RKb´ nn321 IRx.....,,x,x,x cUrRsaybBa¢ak´fa ½

2n321

2n

1kkk

2n

1kkk )a....aaa(xsinaxcosa

.

103-eda¼RsaysmIkar 1xsinxcos nn Edl n

CacMnYnKt´FmµCati . ( 3rd IMO 1961 )

104-cUrkMnt´RKbćemøIyBitsmIkar

1x3cosx2cosxcos 222 .

( 4th IMO 1962 )

105-cUrbgHajfa 2

1

7

3cos

7

2cos

7cos

( 5th IMO 1963 )

106-cUrkMnt´RKb´ x éncenøa¼ 2,0 EdlepÞógpÞat´

2|x2sin1x2sin1|xcos2 . ( 7th IMO 1965 )

107-bgHajjfa x2cotxcotx2sin

1...........

x4sin

1

x2sin

1 nn

cMeBa¼RKbćMnYnKt´FmµCati n nigRKbćMnYnBit x

Edl 0x2sin n . ( 8th IMO 1966 )

108-eda¼RsaysmIkar x2x3x 3 . 134

Traingle Function

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