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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
id1423296 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com
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
id3602910 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com
GarmÖkfa esovePA GnuKmn_RtIekaNmaRtEdlGñksikßakMBugkan´enAkñúgéd
en¼xMJúáTánxitxMRsavRCav nigniBnæeLIgkñúgeKalbMNgTukCaÉksar
RsavRCavsRmab´GñksikßaEdlmanbMNgcg´ec¼ cg´dwgGMBIemeronen¼
[kan´Etc,as´ . enAkñúgesovePA en¼ánRbmUlpþMúnUvRbFanlMhat´lð@
ya¨geRcIn nigmanlkçN¼xusEbøk@Kña.RbFanlMhat´nImYy@xMJúáTán
xitxMeRCIserIsya¨gsRmitsRmaMgbMputRBmTaMgeFIVdMeNa¼Rsayya¨g
ek,a¼k,ayEdlGac[Gñksikßagayyl´nigqab´cgcaMGMBIviFIsaRsþeFIV
dMeNa¼RsaylMhat´nImYy@ . b¨uEnþeTa¼Caya¨gNak¾eday kgV¼xat
bec©keTs Kruekaslü nig kMhusGkçraviruTæRákdCaekItmaneLIg
edayGectnaCaBMuxaneLIy . GaRs&yehtuen¼xMJúáTCaGñkniBnæ
rg´caMTTYlnUvmtiri¼Kn´EbbsSabnaBIsMNak´GñksikßakñúgRKb´mCÄdæan
edaykþIesamnsßrIkrayCanic©edIm,IEklMGesovePAen¼[kan´Etman
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
id8414829 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com
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
id6561394 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com
1
© 2008 Lim Phalkun - 2 -
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
© 2008 Lim Phalkun - 3 -
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
© 2008 Lim Phalkun - 4 -
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
© 2008 Lim Phalkun - 5 -
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
© 2008 Lim Phalkun - 6 -
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
© 2008 Lim Phalkun - 7 -
3¿ ExßekagGnuKmn_ xtany
0 1
1
x
y
( C ) : y = tanx
4¿ ExßekagGnuKmn_ xcoty
0 1
1
x
y
7
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 8 -
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
© 2008 Lim Phalkun - 9 -
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
© 2008 Lim Phalkun - 10 -
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
© 2008 Lim Phalkun - 11 -
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
© 2008 Lim Phalkun - 12 -
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
© 2008 Lim Phalkun - 13 -
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
© 2008 Lim Phalkun - 14 -
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
© 2008 Lim Phalkun - 15 -
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
© 2008 Lim Phalkun - 16 -
27)CtanBtanA(tan
)CtanBtanA(tan27)CtanBtanA(tan2
3
dUcen¼ 33CtanBtanAtan .
lMhatTI6
eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc .
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
© 2008 Lim Phalkun - 17 -
eKán CcosBcosAcos21CcosBcosAcos 222
eday C,B,A CamMuRsYcena¼ 0Ccos,0Bcos,0Acos
naM[ 1CcosBcosAcos21
dUcen¼ 1CcosBcosAcos 222 .
lMhatTI7
eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc .
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
© 2008 Lim Phalkun - 18 -
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
© 2008 Lim Phalkun - 19 -
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
© 2008 Lim Phalkun - 20 -
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
© 2008 Lim Phalkun - 21 -
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
© 2008 Lim Phalkun - 22 -
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
© 2008 Lim Phalkun - 23 -
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
© 2008 Lim Phalkun - 24 -
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
© 2008 Lim Phalkun - 25 -
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
© 2008 Lim Phalkun - 26 -
¦ 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
© 2008 Lim Phalkun - 27 -
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
© 2008 Lim Phalkun - 28 -
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
© 2008 Lim Phalkun - 29 -
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
© 2008 Lim Phalkun - 30 -
lMhatTI16
eKman CaFñÚrmYyEdlKitCara¨düg´ehIyepÞógpÞat´
20
.
eK[Exßekag )(P smIkar sincos 122 xxy .
1¿cUrkMnt´témø edIm,I[Exßekag )P( b¨¼nwgGk&ßGab´sIus )ox'x(
rYcsg´Exßekag )(P TaMgena¼ .
2¿bgHajfaeRkABIkrNIkñúgsMNYrTI1 Exßekag )P( kat´Gk&ß
Gab´sIus )ox'x( ánBIrcMnuc 'M nig ''M Edlman
Gab´sIusviC¢man .
3¿etIeKRtUv[témø b¨unµanxø¼eTIbGab´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
© 2008 Lim Phalkun - 31 -
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
© 2008 Lim Phalkun - 32 -
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
Mapsoft ContentScaler Tryout
32
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 33 -
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
Mapsoft ContentScaler Tryout
33
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 34 -
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
© 2008 Lim Phalkun - 35 -
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
tamsRmayxagelIeKman 323
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
© 2008 Lim Phalkun - 36 -
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
© 2008 Lim Phalkun - 37 -
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
© 2008 Lim Phalkun - 38 -
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
© 2008 Lim Phalkun - 39 -
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
© 2008 Lim Phalkun - 40 -
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
© 2008 Lim Phalkun - 41 -
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
© 2008 Lim Phalkun - 42 -
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
© 2008 Lim Phalkun - 43 -
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
© 2008 Lim Phalkun - 44 -
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
© 2008 Lim Phalkun - 45 -
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
© 2008 Lim Phalkun - 46 -
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
© 2008 Lim Phalkun - 47 -
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
© 2008 Lim Phalkun - 48 -
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
© 2008 Lim Phalkun - 49 -
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
© 2008 Lim Phalkun - 50 -
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
© 2008 Lim Phalkun - 51 -
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
ebI 4n xxxx 3425 coscoscoscos
xxx
xxxxx
coscoscos
)coscos()coscos(cos
52016
341882
35
324
ebI 5n xxxx 4526 coscoscoscos
1184832
188520162
246
2435
xxx
xxxxxx
coscoscos
)coscos()coscoscos(cos
ebI 6n xxxx 5627 coscoscoscos
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
© 2008 Lim Phalkun - 52 -
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
© 2008 Lim Phalkun - 53 -
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
© 2008 Lim Phalkun - 54 -
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
© 2008 Lim Phalkun - 55 -
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
© 2008 Lim Phalkun - 56 -
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
© 2008 Lim Phalkun - 57 -
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
© 2008 Lim Phalkun - 58 -
)
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
tamsRmayxagelIeKman 2
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
© 2008 Lim Phalkun - 59 -
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
© 2008 Lim Phalkun - 60 -
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
© 2008 Lim Phalkun - 61 -
eKTaj
k68
x ¦ Zk,k24
x
dUcen¼ Zk,k24
x
.
lMhatTI34
k¿cUrKNnatémø®ákdén 12
cos nig
12
7cos
x¿cUreda¼RsayRbB&næsmIkar
8
632ycosycosxcos3
8
632ycosxcos3xcos
32
23
dMeNa¼Rsay
k¿KNnatémø®ákdén 12
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
© 2008 Lim Phalkun - 62 -
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
© 2008 Lim Phalkun - 63 -
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
© 2008 Lim Phalkun - 64 -
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
© 2008 Lim Phalkun - 65 -
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
© 2008 Lim Phalkun - 66 -
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
© 2008 Lim Phalkun - 67 -
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
© 2008 Lim Phalkun - 68 -
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
© 2008 Lim Phalkun - 69 -
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
© 2008 Lim Phalkun - 70 -
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
© 2008 Lim Phalkun - 71 -
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
© 2008 Lim Phalkun - 72 -
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
© 2008 Lim Phalkun - 73 -
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
© 2008 Lim Phalkun - 74 -
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
© 2008 Lim Phalkun - 75 -
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
© 2008 Lim Phalkun - 76 -
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
© 2008 Lim Phalkun - 77 -
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
© 2008 Lim Phalkun - 78 -
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
© 2008 Lim Phalkun - 79 -
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
© 2008 Lim Phalkun - 80 -
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
© 2008 Lim Phalkun - 81 -
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
.
Mapsoft ContentScaler Tryout
81
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 82 -
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('
Mapsoft ContentScaler Tryout
82
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 83 -
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
© 2008 Lim Phalkun - 84 -
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 .
Mapsoft ContentScaler Tryout
84
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 85 -
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
Mapsoft ContentScaler Tryout
85
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 86 -
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
Mapsoft ContentScaler Tryout
86
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 87 -
¦ 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
Mapsoft ContentScaler Tryout
87
GnuKmn_RtIekaNmaRt
© 2008 Lim Phalkun - 88 -
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
© 2008 Lim Phalkun - 89 -
lMhatTI57
k¿cUrKNnatémø®ákdén 10
sin nig
10cos
x¿cUrRsayfa 10
sin)yx(4)yx(x 22222
RKb´cMnYnBit IRy,x .
dMeNa¼Rsay
k¿KNnatémø®ákdén 10
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
© 2008 Lim Phalkun - 90 -
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
© 2008 Lim Phalkun - 91 -
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
© 2008 Lim Phalkun - 92 -
)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
© 2008 Lim Phalkun - 93 -
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
© 2008 Lim Phalkun - 94 -
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
© 2008 Lim Phalkun - 95 -
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
© 2008 Lim Phalkun - 96 -
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
© 2008 Lim Phalkun - 97 -
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
© 2008 Lim Phalkun - 98 -
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
© 2008 Lim Phalkun - 99 -
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
x)1ncos()nxtan(xtan1
eKman xsin)nxsin(xcos)nxcos(x)1ncos(
naM[
)nxtan(xtan1xcos)nxcos(
xsin)nxsin(xcos)nxcos(
)nxcos(xcos
x)1ncos(
Bit
dUcen¼ )nxcos(xcos
x)1ncos()nxtan(xtan1
.
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(
69-eda¼RsaysmIkarxageRkam ½
k¿ 01xsin3xsin2 2
x¿ 02xsin)12(2xsin4 2
K¿ 02
3xcos)31(xcos2 2
X¿ 03xtan)31(xtan 2
g¿ 03xtan4xtan3 2
70-eda¼RsaysmIkar ½
k¿ 03xsin)13(2xsin4 2
x¿ 0xcosxcosxsin)31(xsin3 22
K¿ 03xtan)13(xtan 2
X¿ 01xcoslog3xcoslog2 222
g¿ 02xsinlog3xsinlog2
2
2
71-eda¼RsaysmIkarxageRkam ½
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
75-eda¼RsaysmIkar ½
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
76-eda¼RsaysmIkar ½
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
.
82-eda¼RsaysmIkar ½
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´RKb´cemøIyBitsmIkar
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
cMeBa¼RKb´cMnYnKt´FmµCati n nigRKb´cMnYnBit x
Edl 0x2sin n . ( 8th IMO 1966 )
108-eda¼RsaysmIkar x2x3x 3 . 134