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eroberogeday lwm pl:ún
briBaØabR&tKNitviTüa nig BaNiC¢kmµ
2008
Sin( a+b )=sinacosb+sinbcosa Sin( a � b )=sinacosb-sinbcosa Cos( a+b )=cosacosb � sinasinb Cos( a � b )=cosacosb+sinasinb
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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
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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
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emeronsegçb
GnuKmn_RtIekaNmaRt
1¿ TMnakTMngsMxan@
enAelIrgVg´RtIekaNmaRteyIgtag è CargVas´énmMu
)OM,ox(
.
eKán èsin=OQ,ècos=OP .
M
x 'x o
y
P
Q
1
eKánTMnak´TMngsMxan´@énGnuKmn_rgVg´RtIekaNmaRtdUcxag
eRkam ½
1¿ 1=ècos+èsin 22 4¿ 1=ècot.ètan
2¿ ècos
èsin=ètan 5¿
ècos
1=ètan+1
2
2
3¿ èsin
ècos=ècot 6¿
èsin
1=ècot+1
2
2
2¿ rUbmnþplbUk nig pldk
1¿ acosbsin+bcosasin=)b+asin(
2¿ bsinasinbcosacos=)b+acos( -
3¿ btanatan1
btan+atan=)b+atan(
-
4¿ acosbsinbcosasin=)basin( --
5¿ bsinasin+bcosacos=)bacos( -
6¿ btanatan+1
btanatan=)batan(
--
2
3¿ rUmnþmMuDub
1¿ acosasin2=a2sin
2¿ asin=acos=asinacos=acos 2222 1122 ---
3¿ atan
atan=atan 21
22
-
4¿ rUbmnþknø¼mMu
1¿ 2
12
2 acos=
asin
-
2¿ 2
12
2 acos+=
acos
3¿ acos+
acos=
atan
11
22 -
5¿ kenßam xtan,xcos,xsin CaGnuKmn_én 2x
tan=t
1¿ 212t+
t=xsin
2¿ 2
2
11
t+
t=xcos
-
3¿ 2
2
11
t+
t=xtan
-
3
6¿ kenßam atan,acos,asin 333
1¿ asinasin=asin 3433 -
2¿ acosacos=acos 343 3 -
3¿ atan
atanatan=atan 2
3
313
3-
-
7¿ rUbmnþbMElgBIplKuNeTAplbUk
1¿ ])bacos(+)b+acos([=bcosacos -21
2¿ ])b+acos()bacos([=bsinasin --21
3¿ ])basin(+)b+asin([=bcosasin -21
4¿ ])basin()b+asin([=acosbsin --21
6¿ rUbmnþbMElgBIplbUkeTAplKuN
1¿ 22
2qp
cosq+p
cos=qcos+pcos-
2¿ 22
2qp
sinq+p
sin=qcospcos-
--
3¿ 22
2qp
cosq+p
sin=qsin+psin-
4
4¿ 22
2q+p
cosqp
sin=qsinpsin-
-
5¿ qcospcos
)q+psin(=qtan+ptan
6¿ qcospcos
)qpsin(=qtanptan
--
7¿ qsinpsin
)q+psin(=qcot+pcot
8¿ qsinpsin
)pqsin(=qcotpcot
--
7¿ smIkarRtIekaNmaRt
1¿ smIkar vsin=usin mancemøIy
Zk,ðk+vð=u
ðk+v=u
∈- 2
2
2¿ smIkar vcos=ucos mancemøIy
Zk,ðk+v=u
ðk+v=u
∈- 2
2
3¿ smIkar vtan=utan mancemøIy ðk+v=u
5
8¿ rUbmnþbEmøgFñÚEdlKYktsMKal
1¿
ècot=)èð(tan
èsin=)èð(cos
ècos=)èð(sin
-
-
-
2
2
2
2¿ètan=)èð(tan
ècos=)èð(cos
èsin=)èð(sin
--
--
-
3¿
ècot=)è+ð(tan
èsin=)è+ð(cos
ècos=)è+ð(sin
-
-
2
2
2
4¿ ètan=)è+ð(tan
ècos=)è+ð(cos
èsin=)è+ð(sin
-
-
5¿ Zk,ètan=)ðk+è(tan
ècos=)ðk+è(cos
èsin=)ðk+è(sin
∈∀
2
2
6
9¿ RkahVikGnuKmn_RtIekaNmaRt
1¿ ExßekagGnuKmn_ xsin=y
0 1
1
x
y
( C ) : y = sinx
2¿ ExßekagGnuKmn_ xcos=y
0 1
1
x
y
( C ) : y = cosx
7
3¿ ExßekagGnuKmn_ xtan=y
0 1
1
x
y
( C ) : y = tanx
4¿ ExßekagGnuKmn_ xcot=y
0 1
1
x
y
8
lMhat´ nig dMeNa¼Rsay
lMhat´TI1
eK[ b+a
c=ãcos,
a+c
b=âcos,
c+b
a=ácos
cUrRsayfa 1222
222 =ã
tan+â
tan+á
tan .
dMeNa¼Rsay
Rsayfa 1222
222 =ã
tan+â
tan+á
tan
eyIgman 2
122
12
22 ácos+=
ácos,
ácos=
ásin
-
eKán a+c+b
ac+b=
c+b
a+
c+b
a
=ácos+
ácos=
átan
---
1
1
11
22
b+a+c
ba+c=
a+c
b+
a+c
b
=âcos+
âcos=
âtan
---
1
1
11
22
c+b+a
cb+a=
b+a
c+
b+a
c
=ãcos+
ãcos=
ãtan
---
1
1
11
22
9
c+b+a
cb+a+
c+b+a
ba+c+
c+b+a
ac+b=
ãtan+
âtan+
átan
---
222222
c+b+a
cb+a+ba+c+ac+b=
ãtan+
âtan+
átan
---
222222
1222
222 =c+b+a
c+b+a=
ãtan+
âtan+
átan .
dUcen¼ 1222
222 =ã
tan+â
tan+á
tan .
lMhat´TI2
eK[ a
b=ètan .
cUrRsayfa 33
8
3
8 1)b+a(
=b
èsin+
a
ècos .
dMeNa¼Rsay
Rsayfa 33
8
3
8 1)b+a(
=b
èsin+
a
ècos
eKman a
b=ètan naM[
a
b=ètan2
eday ècos
èsin=ètan
eKán a
b=
ècos
èsin2
2
smmUl b+a
=a+b
ècos+èsin=
a
ècos=
b
èsin 12222
eKTaj b+a
=b
èsin 12
naM[ )(bI+a(
b=
b
èsin143
8
10
ehIy b+a
=a
ècos 12
naM[ )(bI+a(
a=
a
ècos243
8
bUkTMnak´TMng )1( nig )2( Gg:nwgGg:eKán ½
343
8
3
8 1)b+a(
=)b+a(
b+a=
b
èsin+
a
ècos
dUcen¼ 33
8
3
8 1)b+a(
=b
èsin+
a
ècos .
lMhat´TI3
eK[RtIekaN ABC mYyman c,b,a CargVas´RCugQm
erogKñaénmMu C,B,A .
tag p Caknø¼brimaRténRtIekaN .
k¿cUrRsayfa bc
)cp)(bp(=
Asin
--
2
rYcTajrkTMnak´TMngBIreTotEdlRsedogKñaen¼ .
x¿cUrbgHajfa 81
222≤
Csin
Bsin
Asin
nig 23≤Ccos+Bcos+Acos .
11
dMeNa¼Rsay
Rsayfa bc
)cp)(bp(=
2
Asin
--
eKman 2
Acos1=
2
Asin2 -
tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman
Acosbc2c+b=a 222 - eKTaj bc2
ac+b=Acos
222 -
eKán bc4
a+cbbc2=
2
bc2
ac+b1
=2
Asin
222
222
2 ---
-
bc4
)c+ba)(cb+a(=
bc4
)cb(a=
2
Asin
22
2 --0--
eday p2=c+b+a
ena¼ )bp(2=c+ba,)cp(2=cb+a ----
A
B C
12
eKán bc4
)bp(2).cp(2=
2
Asin2 --
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 â.á2â+á ≥
eKán )bp)(ap(2)bp(+)ap( --≥--
)bp)(ap(2c
)cp)(ap(2bap2
--≥
--≥--
eKTaj )2(2
1
c
)bp)(ap(≤
--
13
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
3Ccos+Bcos+Acos ≤
eyIgman
2
CBcos
2
C+Bcos2+
2
Asin21=Ccos+Bcos+Acos 2 -
-
2
Csin
2
Bsin
2
Asin4+1=
)2
CBcos
2
C+Bcos(
2
Asin21=
)2
CBcos
2
Asin(
2
Asin21=
2
CBcos
2
Asin2+
2
Asin21= 2
---
---
--
eKTaj 2
Csin
2
Bsin
2
Asin4+1=Ccos+Bcos+Acos
14
tamsRmayxagelIeKman 8
1
2
Csin
2
Bsin
2
Asin ≤
ehtuen¼ 2
3=)
2
1(4+1Ccos+Bcos+Acos ≤
lMhat´TI4
eK[RtIekaN ABCmYyman c,b,a CargVas´RCugQm
erogKñaénmMu C,B,A .
tag p Caknø¼brimaRténRtIekaN .
k¿cUrRsayfa bc
)ap(pAcos
-
rYcTajrkTMnak´TMng
BIreTotEdlRsedogKñaen¼ .
x¿TajbBa¢ak´fa
p
Ccos.ab
Bcos.ac
Acos.bc
dMeNa¼Rsay
Rsayfa bc
)ap(pAcos
A
B C 15
eKman
AcosAcos
tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman
Acosbc2cba 222 eKTaj
bc2
acbAcos
222
eKán bc4
acbbc2
2
bc2
acb1
2
Acos
222
222
2
bc4
)acb)(acb(
bc4
a)cb(
2
Acos
22
2
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
.
16
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 .
17
lMhat´TI5
eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc .
k¿cUrRsayfa Ctan.Btan.AtanCtanBtanAtan
x¿TajbBa¢ak´fa 33CtanBtanAtan .
dMeNa¼Rsay
k¿Rsayfa Ctan.Btan.AtanCtanBtanAtan
eyIgman ðCBA ¦ CðBA
eKán )Cðtan()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 18
eKán 3 CtanBtanAtan3BtanBtanAtan
27)CtanBtanA(tan
)CtanBtanA(tan27)CtanBtanA(tan
2
3
dUcen¼ 33CtanBtanAtan .
lMhat´TI6
eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc .
cUrRsayfa 1CcosBcosAcos 222
dMeNa¼Rsay
Rsayfa 1CcosBcosAcos 222
tag CcosBcosAcosT 222
A
B C
19
CcosBcosAcos21
)BAcos()BAcos(Ccos1
Ccos)BAcos(Ccos1
Ccos)BAcos(Ccos1
Ccos)BAcos()Cðcos(1
Ccos)BAcos()BAcos(1
Ccos2
B2cosA2cos1
Ccos2
B2cos1
2
A2cos1
2
2
2
2
2
eKán CcosBcosAcos21CcosBcosAcos 222
eday C,B,A CamMuRsYcena¼ 0Ccos,0Bcos,0Acos
naM[ 1CcosBcosAcos21
dUcen¼ 1CcosBcosAcos 222
20
lMhat´TI7
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
naM[ 2CcosBcosAcos22
dUcen¼ 2CsinBsinAsin 222
21
lMhat´TI8
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
xcos2222
eKán ba
1
a
xcos2
naM[ )1(
)ba(
a
a
xcos54
10
out
22
ehIy ba
1
b
xsin2
naM[ )2(
)ba(
a
b
xsin54
10
bUksmIkareKán454
10
4
10
)ba(
1
)ba(
ba
b
xsin
a
xcos
lMhat´TI9
cUrKNna 7
ð4sin
7
ð2sin
7
ðsinS 222
dMeNa¼Rsay
KNna 7
ð4sin
7
ð2sin
7
ðsinS 222
eyIgán 2
7
ð8cos1
2
7
ð4cos1
2
7
ð2cos1
S
)7
ð8cos
7
ð4cos
7
ð2cos(
2
1
2
3
tag 7
ð8cos
7
ð4cos
7
ð2cosT
7
ðcos
7
ð3cos
7
ð5cos
)7
ðð(cos)
7
ð3ð(cos)
7
ð5ð(cos
KuNGg:TaMgBIrnwg 7
ðsin2 eKán ½
7
ðsin
7
ðcos2
7
ðsin
7
ð3cos2
7
ðsin
7
ð5cos2
7
ðsinT2
23
tamrUbmnþ )basin()basin(bsinacos2
7
ðsin)
7
ððsin(
7
ð6sin
7
ðsinT2
7
ð2sin)
7
ð2sin
7
ð4(sin)
7
ð4sin
7
ð6sin(
7
ðsinT2
7
ðsin
7
ðcos2
7
ðsin
7
ð3cos2
7
ðsin
7
ð5cos2
7
ðsinT2
eKTaj 2
1T naM[
4
7)
2
1(
2
1
2
3S
dUcen¼ 4
7
7
ð4sin
7
ð2sin
7
ðsinS 222 .
lMhat´TI10
cUrKNna 7
ð8sin
7
ð4sin
7
ð2sinS
dMeNa¼Rsay
eKman 7
ðsin
7
ð8sin,
7
ð3sin
7
ð4sin,
7
ð5sin
7
ð2sin
ehIy 7
ð8sin
7
ðsin
7
ð2sin nig 0
7
ð4sin
eKán 07
ðsin
7
ð3sin
7
ð5sinS
elIkGg:TaMBIrCakaereKán ½
7
ðsin
7
ð3sin2
7
ðsin
7
ð5sin2
7
ð3sin
7
ð5sin2
7
ðsin
7
ð3sin
7
ð5sinS 2222
24
tag 7
ðsin
7
ð3sin
7
ð5sinM 222
)7
ð5cos
7
ðcos
7
ð3cos(
2
1
2
3
)7
ð5ðcos()
7
ððcos()
7
ð3ðcos(
2
1
2
3
2
7
ð2cos
7
ð6cos
7
ð10cos
2
3
yk 7
ðcos
7
ð3cos
7
ð5cosT
KuNGg:TaMgBIrnwg 7
ðsin2 eKán ½
7
ðsin
7
ðcos2
7
ðsin
7
ð3cos2
7
ðsin
7
ð5cos2
7
ðsinT2
tamrUbmnþ )basin()basin(bsinacos2
7
ðsin)
7
ððsin(
7
ð6sin
7
ðsinT2
7
ð2sin)
7
ð2sin
7
ð4(sin)
7
ð4sin
7
ð6sin(
7
ðsinT2
7
ðsin
7
ðcos2
7
ðsin
7
ð3cos2
7
ðsin
7
ð5cos2
7
ðsinT2
eKTaj 2
1T naM[
4
7
4
1
2
3M
tag 7
ðsin
7
ð3sin2
7
ðsin
7
ð5sin2
7
ð3sin
7
ð5sin2N
25
0)
7
ðsin(.ðsin2
7
ð8cos
7
ð6cos
7
ð4cos
7
ð2cos
7
ð6cos
7
ð4cos
7
ð8cos
7
ð2cos
eKán 4
70
4
7NMS2 eday 0S
ena¼ 2
7S .
dUcen¼ 2
7
7
ð8sin
7
ð4sin
7
ð2sinS .
lMhat´TI11
cUrRsayfa
*INn,0)1(7
ðncos4
7
ðncos)1(4
7
ðncos8
1n2n3
dMeNa¼Rsay
eKman INn,7
ðn3ðn
7
ðn4
eKán )7
ðn3ðn(sin
7
ðn4sin
edayeRbIrUbmnþ ½
acosasin2a2sin
xcosysinycosxsin)yxsin( 26
nig asin4asin3a3sin 3
0)1(7
ðncos4
7
ðncos)1(4
7
ðncos8
)1(7
ðncos4.)1(
7
ðncos4
7
ðncos8
])7
ðncos1(43[.)1(
7
ðncos4
7
ðncos8
)7
ðnsin43(
7
ðnsin.)1()1
7
ðncos2(
7
ðncos
7
ðnsin4
)1()7
ðnsin4
7
ðnsin3(0)1
7
ðncos2(
7
ðncos
7
ðnsin4
)ðncos(7
ðn3sin
7
ðn3cos)ðnsin(
7
ðn2cos
7
ðn2sin2
1n2n3
n2n3
2n3
2n2
n32
dUcen¼
*INn,0)1(7
ðncos4
7
ðncos)1(4
7
ðncos8 1n2n3
lMhat´TI12
cUrKNna 9
ð7cos
9
ð4cos
9
ðcosS
333
dMeNa¼Rsay
KNna 9
ð7cos
9
ð4cos
9
ðcosS 333
tamrUbmnþ acos3acos4a3cos 3
¦ a3cos4
1acos
4
3acos3
27
kenßamEdl[GacsresrCa ½
)3
ð7cos
3
ð4cos
3
ð(cos
4
1)
9
ð7cos
9
ð4cos
9
ð(cos
4
3S
tag 9
ð7cos
9
ð4cos
9
ðcosM
eday 9
ð13cos
9
ð4cos
9
ð13cos
9
ð7cos
9
ðcosM KuNnwg
3
ðsin2 eKán
0)9
ð7cos(ðsin2
9
ð16sin
9
ð2sin
2
3M2
9
ð10sin
9
ð16sin
9
ð4sin
9
ð10sin)
9
ð2sin(
9
ð4sin
2
3M.2
3
ðsin
9
ð13cos2
3
ðsin
9
ð7cos2
3
ðsin
9
ðcos2
3
ðsinM2
eKTaján 0M
tag 2
3
2
1
2
1
2
1
3
ð7cos
3
ð4cos
3
ðcosN
eKán 8
3N
4
1M
4
3S
28
lMhat´TI13
eK[kenßam
7
ð5cos
7
ð3cos
7
ðcosS 333
nig 7
ð5cos
7
ð3cos
7
ðcosT 444
k¿cUrRsayfabIcMnYn7
ð5cos,
7
ð3cos,
7
ðcos Ca¦srbs´smIkar
01x4x4x8:)E( 23 .
x¿Tajrktémø ½
7
ð5cos
7
ðcos
7
ð5cos
7
ð3cos
7
ð3cos
7
ðcosN
7
ð5cos
7
ð3cos
7
ðcosM
nig 7
ð5cos
7
ð3cos
7
ðcosP .
K¿KNna 7
ð5cos
7
ð3cos
7
ðcosQ 222
rYcTajrktémø S nig T .
dMeNa¼Rsay
k¿RsayfabIcMnYn 7
ð5cos,
7
ð3cos,
7
ðcos Ca¦srbs´smIkar
01x4x4x8:)E( 23
tag 3,2,1n,ð7
1n2cosxn
Ca¦smIkar )E(
29
eKán ½
)*(0
7
ð)1n2(sin
7
ð)1n2(3sin
7
ð)1n2(sin
7
ð)1n2(4sin
0
7
ð)1n2(sin
7
ð)1n2(sin4
7
ð)1n2(sin3
7
ð)1n2(2sin2
7
ð)1n2(4sin
7
ð)1n2(sin2
7
ð)1n2(2sin
.4
0)7
ð)1n2(sin43(
7
ð)1n2(2cos
7
ð)1n2(cos4
0)4
ð)1n2(sin1(41)1
7
ð)1n2(cos2(
7
ð)1n2(cos4
017
ð)1n2(cos4
7
ð)1n2(cos4
7
ð)1n2(cos8
3
2
22
23
eday 07
ð)1n2(sin:*INn
ehtusmIkar )*( smmUl ½
02
ð)1n2(cos
7
ð)1n2(sin2
07
ð)1n2(3sin
7
ð)1n2(4sin
00 epÞógpÞat´ .
dUcen¼ 7
ð5cos,
7
ð3cos,
7
ðcos Ca¦srbs´smIkar )E( .
30
x¿Tajrktémø P,N,M
snµtfa 7
ð5cosx,
7
ð3cosx,
7
ðcosx 321
tamRTwsþIbTEvütGnutþn_kñúgsmIkar 01x4x4x8 23
eKán ½
2
1
a
cxxxxxx
7
ð5cos
7
ðcos
7
ð5cos
7
ð3cos
7
ð3cos
7
ðcosN
2
1
a
bxxx
7
ð5cos
7
ð3cos
7
ðcosM
313221
321
nig 8
1
a
dxxx
7
ð5cos
7
ð3cos
7
ðcosP 321 .
dUcen¼
2
1
7
ð5cos
7
ðcos
7
ð5cos
7
ð3cos
7
ð3cos
7
ðcosN
2
1
7
ð5cos
7
ð3cos
7
ðcosM
nig 8
1
7
ð5cos
7
ð3cos
7
ðcosP .
K¿KNna 7
ð5cos
7
ð3cos
7
ðcosQ
222
eyIgán 2
3
2
2
2
1 xxxQ
4
5)
2
1(2
4
1N2M
)xxxxxx(2)xxx(
2
3132212
321
31
dUcen¼ 4
5
7
ð5cos
7
ð3cos
7
ðcosQ 222
.
Tajrktémø S nig T
eyIgán 7
ð5cos
7
ð3cos
7
ðcosS 333
¦ 3
3
3
2
3
1 xxxS
eday 7
ð5cosx,
7
ð3cosx,
7
ðcosx 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
2
2
1
4
5
8
3
2
MQS
dUcen¼ 2
1
7
ð5cos
7
ð3cos
7
ðcosS 333
.
müa¨eTot 4
3
4
2
4
1444 xxx7
ð5cos
7
ð3cos
7
ðcosT
edayKuNsmIkar )3(,)2(,)1( erogKñanwg 321 x,x,x
32
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
3
2
2
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
ð5cos
7
ð3cos
7
ðcosT
444 .
lMhat´TI14
eda¼RsaysmIkar ½ 2246 x127x56x112x64 .
dMeNa¼Rsay
eda¼RsaysmIkar
)1(x127x56x112x64 2246
lk&çx&NÐ 0x1 2 ¦ ]1,1[x
eyIgman )a3acos(a4cos
33
1acos8acos8
)asina(cos3)acos1(4acos4
asin4asin3acos3acos4
)asin4asin3(asin)acos3acos4(acos
a3sinasina3cosacos
24
22224
4224
33
)a4acos(a5cos
acos5acos20cos16
)acosacos2)(acos1(4acosacos8acos8
)1acos2(acosasin4acosacos8acos8
a2cosa2sinasin2)1acos8acos8(acos
a4sinasina4cosacos
35
3235
2235
24
1acos18acos48acos32
1)acos3acos4(21a3cos2a6cos
246
232
acos7acos56acos112acos64
)acos3acos4)(asin4asin3(asin2a6cosacos
a3cosa3sinasin2a6cosacos
asina6sinacosa6cos)aa6cos(a7cos
357
33
yk tcosx Edl ]ð,0[t smIkar )1( sresr ½
tsin27tcos56tcos112tcos64
tcos127tcos56tcos112tcos64
246
2246
KuNGg:TaMgBIrnwg 0tcos eKán ½
)t22
ðcos(t7cos
t2sint7cos
tcostsin2tcos7tcos56tcos112tcos64 357
34
eKTaj
Z'k;k,ð'k2t22
ðt7
ðk2t22
ðt7
smmUl
Z'k;k,9
ð'k2
10
ðt
9
ðk2
18
ðt
eday ]ð,0[t eKTajsMNMutémø t dUcxageRkam ½
}10
ð7;
10
ð3;
18
ð17;
18
ð13;
18
ð9;
18
ð5;
18
ð{t
eday 0tcos ena¼ 2
ðt
dUcen¼smIkar )1( mansMNMu¦sdUcxageRkam ½
}10
ð7cos;
10
ð3cos;
18
ð17cos;
18
ð13cos;
18
ð5cos;
18
ðcos{x
35
lMhat´TI15
eda¼RsaysmIkar ½
3232326 )3x(tan)2x(tan)1x(tanxtan
dMeNa¼Rsay
eda¼RsaysmIkar ½
3232326 )3x(tan)2x(tan)1x(tanxtan ¿
lk&çx&NÐ Zk,ðk2
ðx .
tag 0t,xtant 2 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 03t3t2
Kµan¦seRBa¼
0129 cMeBa¼ 3t eKán 3xtan2
36
0)3x)(tan3x(tan
03xtan2
eKán 03xtan ¦ 3xtan naM[ Zk,ðk3
ðx
ehIy 03xtan ¦ 3xtan
naM[ Zk,ðk3
ðx
lMhat´TI16
eKman á CaFñÚrmYyEdlKitCara¨düg´ehIyepÞógpÞat´
2
ðá0 .
eK[Exßekag )P( smIkar ásin1ácosx2xy 2
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 ábunµanxø¼eTIbGab´sIus 'x nig ''x
éncMnuc 'M nig ''M epÞógpÞat´TMnak´TMng 2''x'x 22
4¿rkTMnak´TMngKµanGaRs&ynwg rvagGab´sIus 'x nig ''x . 37
dMeNa¼Rsay
1¿ Exßekag )P( b¨¼nwgGk&ßGab´sIus )ox'x( ½
kUGredaenkMBUléná¨ra¨bUl )(P KW ácosa2
bxS
ehIy ásin1ácos2ácosy 22
S
ásinásin
ásinácos1
2
2
Exßekag )P( b¨¼nwgGk&ßGab´sIus )ox'x( kalNa 0Sy
eKán 0ásinásin2
¦ 0)1á(sinásin naM[ 0ásin nig 1ásin
eday 2
ðá0 ehtuen¼eKTaj
2
ðá,0á
sg´Exßekag )(P ½
-ebI 0á eKán 22 )1x(1x2xy
-ebI 2
ðá eKán 2xy
38
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¦srbssmIkar ½
0ásin1ácosx2x2 DIsRKImINg´bRgYménsmIkarKW ásin1ácos' 2
)ásin1(ásin
ásinásin 2
eKman sin nig sin1 viC¢manCanic©RKb´
2
ð,0á .
39
eKTaján 0)ásin1(ásin' 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 2''x'x 22
eKman PSxx 2222 '''
)1ásinásin2(2)ásin1ásin22(2
)ásin1cos2(2)ásin1(2ácos4
22
22
eday 2''x'x22 eKTaján
0)ásin21(ásin
0ásinásin2
11ásinsin2
2)1ásinásin2(2
2
2
2
eday 2
ðá0 ehtuen¼eKTaj
6
ðá,0á
4¿TMnak´TMngKµanGaRs&ynwg á rvagGab´sIus 'x nig ''x
eKman ácos2S nig ásin1P 40
eKTaján 2
Sácos nig P1ásin
edayRKb´cMnYnBit eKman 1ásinácos 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
lMhat´TI17
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 .
41
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
eKTaján 1cos)232(
)1cos2(3
2
3cos
cos2cos323cos32
eday 2
0
dUcen¼eKTaj 6
.
42
x¿ eda¼RsaysmIkar )(E ½
ebI 6
ena¼ )E( Gacsresr 01
3
2x)2
3
2(x2
3
)32(4
3
)347(4
3
316
3
284
3
84
3
8
3
4)1
3
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 .
43
lMhat´TI18
eK[ cxsinbxcosacxcosbxsina)x(f 2222
cUrRsayfa c2
ba2)x(fcbca
rYcbBa¢ak´témøGtibrma nig Gb,brmaén )(xf .
dMeNa¼Rsay
Rsayfa c2
ba2)x(fcbca
eyIgman )1(cxsinbxcosacxcosbxsina)x(f2222
eday c,b,a CabIcMnYnBitviC¢manena¼ 0)x(f:IRx
elIkGg:TaMgBIrén )1( CakaereKán ½
)2()cxsinbxcosa)(cxcosbxsina(2c2ba)x(f
cxsinbxcosacxcosbxsina)x(f
22222
222222
tamvismPaBkUsiuRKb´cMnYnBit 0B,A
eKman B.A2BA ¦ BAB.A2
eKán c2ba)cxsinbxcosa)(cxcosbxsina(2 2222
tamTMnak´TMng 2 eKTaján ½
)c2
ba(4c2bac2ba)x(f 2
44
naM[ )3(c2
ba2)x(f
yk )cxsinbxcosa)(cxcosbxsina()x(P 2222
xsinxcos)ab()cb)(ca()x(P
xcos)ab(xcos)ab()cb)(ca()x(P
]xcos)ab()cb([]xcos)ab()ca[()x(P
]c)xcos1(bxcosa[]cxcosb)xcos1(a[)x(P
222
4222
22
2222
eyIgman IRx,0xcosxsin)ab( 222
eKTaján IRx,)cb)(ca()x(P
TMnak´TMng )2( eKGacsresr ½
22
2
2
)cbca()x(f
)cb)(ca(2)cb()ca()x(f
)cb)(ca(2c2ba)x(P2c2ba)x(f
eKTaj )4(cbca)x(f
tamTMnak´TMng )3( nig )4( eKTaján ½
c2
ba2)x(fcbca
cMeBa¼RKb´ IRx .
dUcGnuKmn_mantémøGtibrmaesµI c2
ba2M
nigmantémøGb,brmaesµI cbcam .
45
lMhat´TI19
rktémøGb,brmaénGnuKmn_ ½
87)xcotx(tan8xcotxtan)x(Q
27)xcotx(tan2xcotxtan)x(P
22
22
2
ðx0 .
dMeNa¼Rsay
rktémøGb,brmaénGnuKmn_ ½
27)xcotx(tan2xcotxtan)x(P 22
Edl 2
ðx0
tag xcotxtanz Edl 2z
eKán 2xcotxtan)xcotx(tanz 2222
eKTaj 2zxcotxtan 222
eyIgán 24)1z(27z22z)z(P 22
eday 2z ehtuen¼eKán 25241)z(P
dUcen¼témøGb,brmaén )x(P KW 25m .
müa¨geToteday 87)xcotx(tan8xcotxtan)x(Q 22
eKán 69)4z(87z82z)z(Q 22
eday 2z ehtuen¼edIm,I[ Q Gb,brmalu¼RtaEt
46
4z .dUcen¼témøGb,brmaén )x(Q KW 69m .
lMhat´TI20
eda¼RsaysmIkar ½
223)12
ðx(cos)
4
ðx(sin4
dMeNa¼Rsay
eda¼RsaysmIkar ½
223)12
ðx(cos)
4
ðx(sin4
tamrUbmnþ )basin()basin(bcosasin2
smIkarxagelIGacsresrCabnþbnÞab´xageRkam ½
2
2)
3
ðx2sin(
211)3
ðx2sin(2
)21(6
ðsin)
3
ðx2sin(2
223)12
ðx
4
ðxsin()
12
ðx
4
ðxsin(2
2
eKTaj
Zk,ðk24
ðð
3
ðx2
ðk24
ð
3
ðx2
dUcen¼ Zk;ðk24
ð5x,ðk
24
ðx .
47
lMhat´TI21
eKmanGnuKmn_ 2
2
2
2
2
2
xcos
1xcos
xsin
1xsin)x(f
cUrrktémøtUcbMputénGnuKmn_en¼ .
dMeNa¼Rsay
rktémøtUcbMputénGnuKmn_en¼
2
2
22
2
2 )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
44
44
44
4444
44
44
4
4
4
4
edayeKman 1x2sin2 naM[ 2
1x2sin
2
11 2 nig
17x2sin
161
4 .
48
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
dUcen¼témøtUcbMputénGnuKmn_KW 2
25m .
lMhat´TI22
eK[BIrcMnYnBit a nig b .
cUrRsaybBa¢k´fa 22 ba|xsinbxcosa|:IRx
Gnuvtþn_ ½ rktémøGtibrma nig Gb,brmaén
22xsin21xcos20)x(f
dMeNa¼Rsay
RsaybBa¢k´fa 22 ba|xsinbxcosa|:IRx
eyIgeRCIserIsvuicTr& )b;a(U
nig )xsin;xcos(V
tamniymn&y ècos.||V||.||U||V.U
Edl èCamMurvag
BIrviucT&ren¼ .
eKán |ècos|||V||.||U||ècos.||V||.||U||V.U
edayeKman 1|ècos|:IRè
eKán ||V||.||U||V.U
49
eday
1xcosxsin||V||
ba||U||
xsinbxcosaV.U
22
22
dUcen¼ 22 ba|xsinbxcosa|:IRx .
rktémøGtibrma nig Gb,brmaén 22xsin21xcos20)x(f
tamrUbmnþxagelIeyIgman
292120|xsin21xcos20| 22
eKTaj 29xsin21xcos2029
naM[ IRx,51)x(f7
dUcen¼GnuKmn_mantémøGtibrma 51 nig Gb,brma 7
50
lMhat´TI23
)x(f CatémøBiténGnuKmn_ f EdlcMeBa¼RKb´ IRx
eKman xsinxcos3)x(f2)x(f .
cUrRsayfa 2)x(f cMeBa¼RKb´ IRx
dMeNa¼Rsay
Rsayfa 2)x(f cMeBa¼RKb´ IRx
eKman )1(xsinxcos3)x(f2)x(f
edayCMnYs x eday x kñúgTMnak´TMng )(1 eKán
)2(xsinxcos3)x(f2)x(f
eyIgánRbB&næsmIkar ½
xsin3xcos3)x(f3
2
1
xsinxcos3)x(f)x(f2
xsinxcos3)x(f2)x(f
eKTaján xsinxcos)x(f
)x4
ðsin(2
)4
ðcosxsinxcos
4
ð(sin2
)xsin2
2xcos
2
2(2)x(f
51
eday 1)x4
ðsin(:IRx .
dUcen¼ 2)x(f cMeBa¼RKb´ IRx .
lMhat´TI24
eKman )x(f GnuKmn_kMnt´elI IR eday ½
x2cos2)x(cosf3)x(sinf ¿
k-cUrkMnt´rkGnuKmn_ )x(f .
x-eda¼RsaysmIkar
2
)ttan1(f)ttan1(f)ttan1(f).ttan1(f
( t CaGBaØténsmIkar ) .
dMeNa¼Rsay
k-kMnt´rkGnuKmn_ )x(f
CMnYs x eday x2
ð eKán x2cos2)x(sinf3)x(cosf
eyIgánRbB&næ
x2cos2)x(cosf)x(sinf3
x2cos2)x(cosf3)x(sinf
bMát´ )x(sinf eKTaján xcos2
x2cos1)x(cosf 2
dUcen¼ 2x)x(f .
52
x-eda¼RsaysmIkar
2
)ttan1(f)ttan1(f)ttan1(f).ttan1(f
lk&çx&NÐ Zk;ðk2
ðt
2
)ttan1()ttan1()ttan1()ttan1(
2222
dMeNa¼RsaysmIkaren¼eKáncMelIy
Zk,ðk3
ðt;ðkt .
lMhat´TI25
eK[Exßekag sin)sin(sin)(:)( 5122 xxxfyP
Edl 0 .
kMnt´témø edIm,I[Exßekag )(P sSitenAelIGkß&
Gab´sIusCanic© .
dMeNa¼Rsay
kMnt´témø
eKman sin)sin(sin)(:)( 5122 xxxfyP
edIm,I[ )(P sSitenAelIGkß&Gab´sIusCanic©lu¼RtaEt
IRxxf ,)( 0
53
eBalKWeKRtUv[
0'
0af
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
.
54
lMhat´TI26
eda¼RsaysmIkar ½
2
1xcos
2
3xcos
16
9xcos
2
1xcos
16
1 2424
dMeNa¼Rsay
eda¼RsaysmIkar ½
)1(2
1xcos
2
3xcos
16
9xcos
2
1xcos
16
1 2424
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
)3(2
1
4
3t
4
1t
elIGkß& )ox'x( eRCIserIscMnuc )4
3(B,)
4
1(A,)t(M
tam 3 eKán 2
1MBMA eday
2
1AB
eKán ABMBMA naM[ M enAkñúg AB
eKTaj 4
3t
4
1 smmUl
4
3xcos
4
1 2 55
smmUl 2
3|xcos|
2
1 eKTaj
Z'k;k,ð'k6
ðxð'k
3
ð
ðk3
ðxêð
6
ð
lMhat´TI27
eK[RtIekaN ABC mYy .
k¿ cUrRsayfa 1AcotCcotCcotBcotBcotAcot .
x¿ cUrRsayfa AcotA2cot21Acot2 .
K¿ eKdwgfamMu C;B;A beg;ItánCasIVútFrNImaRt
mYYyEdlmanersugesµInwg 2q .
cUrRsaybBa¢ak´fa ½
8Csin
1
Bsin
1
Asin
1222
.
dMeNa¼Rsay
k¿Rsayfa 1AcotCcotCcotBcotBcotAcot
eyIgman ðCBA ¦ CðBA
eKán )Cðtan()BAtan(
56
1BcotAcotCcotBcotCcotAcot
Ccot
1
1BcotAcot
BcotAcot
Ccot
1
Bcot
1.
Acot
11
Bcot
1
Acot
1
CtanBtanAtan1
BtanAtan
dUcen¼ 1AcotCcotCcotBcotBcotAcot .
x¿Rsayfa AcotA2cot21Acot2
eyIgman Atan1
Atan2A2tan
2
1Acot
Acot2
Acot
11
Acot
2
A2cot
12
2
dUcen¼ AcotA2cot21Acot2 .
K¿RsaybBa¢ak´fa ½ 8Csin
1
Bsin
1
Asin
1222
tag Csin
1
Bsin
1
Asin
1T
222
)1(6CcotC2cot2B2cotBcot2AcotA2cot2
6)1C(cot)1B(cot)1A(cot
)Ccot1()Bcot1()Acot1(
222
222
edaymMu C;B;A CasIVútFrNImaRtmYYyEdlmanersug
esµInwg 2q eKán A4B2C,A2B 57
eday ðCBA
eKán ðA4A2A naM[ 7
ð4C,
7
ð2B,
7
ðA
tam )1( eKán 67
ð4cot
7
ð8cot
7
ð4cot
7
ð2cot2
7
ðcot
7
ð2cot2T
eday 7
ðcot
7
ð8cot eKán ½
86)1(26)CcotAcotCcotBcotBcotA(cot2
67
ð4cot
7
ðcot2
7
ð4cot
7
ð2cot2
7
ð2cot
7
ðcot2T
dUcen¼ 8Csin
1
Bsin
1
Asin
1222
.
58
lMhat´TI28
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
x)1ncos()nxcos(xcos2x)1ncos(
eyIgman ½
2
x)1n(x)1n(cos.
2
x)1n(x)1n(cos2x)1ncos(x)1ncos(
smmUl )nxcos(.xcos2x)1ncos(x)1ncos(
dUcen¼ x)1ncos()nxcos(xcos2x)1ncos( ¿.
2¿ Gnuvtþn_ ½sresr x7cos CaGnuKmn_én xcos
eKman x)1ncos()nxcos(xcos2x)1ncos( ¿
ebI 1n 1xcos2x2cos 2
59
ebI 2n xcosx2cosxcos2x3cos
xcos3xcos4
xcos)1xcos2(xcos2
3
2
ebI 3n x2cosx3cosxcos2x4cos
1xcos8xcos8
)1xcos2()xcos3xcos4(xcos2
24
23
ebI 4n x3cosx4cosxcos2x5cos
xcos5xcos20xcos16
)xcos3xcos4()1xcos8xcos8(xcos2
35
324
ebI 5n x4cosx5cosxcos2x6cos
1xcos18xcos48xcos32
)1xcos8xcos8()xcos5xcos20xcos16(xcos2
246
2435
ebI 6n x5cosx6cosxcos2x7cos
dUcen¼ xcos7xcos56xcos112xcos64x7cos 357 ¿
3¿ eda¼RsaysmIkar ½
)1(01xcos14xcos112xcos244xcos128 357 ¿
EckGg:TaMgBIrénsmIkar )1( nwg 2 eKán ½
2
1x7cos
02
1xcos7xcos56xcos112xcos64 357
60
eKTaján ðk23
ðx7 ¦ Zk,
7
ðk2
21
ðx
dUcen¼ Z'k;k;7
ð'k2
21
ðx,
7
ðk2
21
ðx .
lMhat´TI29
eK[sVúIténcMnYnBit )U( n kMnt´elI n eday½
1U0 nig asinacosUU:INn n1n Edl 2
ða0 .
k¿tag 2
acotUV nn .
cUrbgHajfa )V( n CasIVútFrNImaRtmYYy .
x¿KNnalImIt )V....VV(lim n10n
nig nn
Ulim
.
dMeNa¼Rsay
k¿ bgHajfa )V( n CasIVútFrNImaRtmYYy ½
man 2
acotUV nn naM[
2
acotUV 1n1n
Et asinacosUU n1n
eKán 2
acotasinacosUV n1n
61
acosV
acos)2
acotU(acos
2
acotacosU
2
asin21acos;)1
2
asin2(
2
acotacosU
)1
2
acos
2
asin
2
acos
2
asin2(
2
acotacosU
)12
atan
2
acos
2
asin2(
2
acotacosU
2
acot
2
acos
2
asin2acosU
n
nn
22
n
n
n
n
eday acosVV n1n naM[ )( nV CasIVútFrNImaRt
mYymanersug acos nig tY 2
acot1
2
acotUV 00 .
x¿ KNnalImIt ½ )V....VV(lim n10n
nig nn
Ulim
eyIgman acos1
acos1).
2
acot1(
q1
q1VV...VV
1n1n
0n10
eyIgán
acos1
acos1)
2
acot1(lim)V....VV(lim
1n
nn10
n
eday 2
0
a ena¼ 10 acos nig 0acoslim 1n
n
dUcen¼ acos1
2
acot1
)V....VV(lim n10n
.
müa¨geTot 2
acotUV nn naM[
2
acotVU nn
62
eday acos)2
acot1(qVV nn
0n
eKán 2
acotacos)
2
acot1(U n
n
nig 2
acot
2
acotacos)
2
acot1(limUlim n
nn
n
eRBa¼ 0acoslim n
n
.
dUcen¼ 2
acotUlim n
n
.
lMhat´TI30
eK[sIVúténcMnYnBit )U( n kMnt´eday ½
1U;0U 10 nig n1n2n UacosU2U:INn
Edl IRa
k¿ tag INn,U)asinia(cosUZ n1nn .
cUrbgHajfa n1n Z)asinia(cosZ rYcTajrk nZ CaGnuKmn_
nnig a .
x¿ Tajrk nU CaGnuKmn_én n .
dMeNa¼Rsay
k¿ bgHajfa n1n Z)asinia(cosZ
eyIgman n1nn U)asinia(cosUZ 63
eyIgán 1n2n1n U)asinia(cosUZ
n
n1n
n1n
n1n
1nn1n
U)asinia(cos
U)asinia(cosU)asinia(cos
)asiniacos
UU)(asinia(cos
UU)asinia(cos
U)asinia(cosUacosU2
dUcen¼ n1n Z)asinia(cosZ .
KNna nZ CaGnuKmn_én n nig a ½
eday n1n Z)asinia(cosZ naM[ )Z( n CasIVútFrNImaRt
éncMnYnkMupøicEdlmanersug asiniacosq nig
tY 1U)asinia(cosUZ 010 .
tamrUbmnþ )nasin(i)nacos()asinia(cosqZZ nn
0n
dUcen¼ )nasin(.i)nacos(Zn .
x¿ Tajrk nU CaGnuKmn_én n ½
eyIgman )1(U)asinia(cosUZ n1nn
nig )2(U)asinia(cosUZ n1nn
dksmIkar )(1 nig )(2 Gg:nwgGg:eKán ½
nnn Uasini2ZZ naM[ asini2
ZZU nn
n
Edl 0asin .
64
eday )nasin(i)nacos(Zn nig )nasin(i)nacos(Zn
dUcen¼ asin
)nasin(Un .
lMhat´TI31
eK[
432 2
ðcos2222,
2
ðcos222,
2
ðcos22
BI«TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBaØak´
rUbmnþena¼pg .
dMeNa¼Rsay
rkrUbmnþTUeTA ½
eKman
432 2
ðcos2222,
2
ðcos222,
2
ðcos22
tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½
1n
)n(2
ðcos22.........222
.
RsaybBaØakrUbmnþen¼ ½
65
eyIgtag )n(
n 2......222A
cMeBa¼RKb *INn
eyIgman 212
ðcos22A Bit
eyIg«bmafavaBitdl´tYTI p KW
1p
)p(
p2
ðcos22......222A
Bit
eyIgnwgRsayfavaBitdl´tYTI 1p KW 2p1p2
ðcos2A
Bit
eyIgman p1p A2A edaytamkar«bma 1pp2
ðcos2A
eyIgán 2p2p
2
1p1p2
ðcos2
2
ðcos4
2
ðcos22A
Bit
dUcen¼ 1n
)n(2
ðcos22.........222
.
66
lMhat´TI32
eK[sIVúténcMnYnkMupøic )Z( n kMnt´eday ½
INn;|Z|Z2
1Z
2
3i1Z
nn1n
0
( |Z| n CamUDulén nZ ) .
snµtfa INn,)èsin.iè(cosñZ nnnn
Edl IRè;ñ,0ñ nnn
k-cUrrkTMnak´TMngrvag nè nig 1nè ehIy nñ nig 1nñ .
x-rkRbePTénsIVút )è( n rYcKNna nè CaGnuKmn_én n .
K-cUrbgHajfa2
ècos....
2
ècos
2
ècosècosññ 1n21
00n
rYcbBa¢ak´ nñ CaGnuKmn_én n .
dMeNa¼Rsay
k-rkTMnak´TMngrvag nè nig 1nè ehIy nñ nig 1nñ
eyIgman )èsin.iè(cosñZ nnnn
naM[ )èsiniè(cosñZ 1n1n1n1n 67
eday )|Z|Z(2
1Z nn1n ehIy nn ñ|Z|
eKán nnnn1n1n1n ñ)èsin.iè(cosñ2
1)èsiniè(cosñ
)
2
èsin.i
2
è(cos
2
ècosñ)èsin.iè(cosñ
)èsin.iècos1(ñ2
1)èsin.iè(cosñ
nnnn1n1n1n
nnn1n1n1n
eKTaján 2
ècosññ n
n1n nig 2
èè n
1n
dUcen¼ 2
èè n
1n nig 2
ècosññ n
n1n .
x-RbePTénsIVút )è( n nig KNna nè CaGnuKmn_én n ½
tamsRmayxagelIeyIgman n1n è2
1è
naM[ nè CasIVútFrNImaRtmanersugesµI 2
1q .
tamrUbmnþ n
0n qèè
eday 3
ðsin.i
3
ðcos
2
3i1)èsiniè(cosñZ 0000
eKTaján 3
ðè;1ñ 00 dUcen¼
nn2
1.
3
ðè .
K-bgHajfa 2
ècos....
2
ècos
2
ècosècosññ n21
00n
tamsRmayxagelIeKman 2
ècosññ n
n1n ¦ 2
ècos
ñ
ñ n
n
1n
eKán
1nk
0k
1nk
0k
k
k
1k )2
ècos(
ñ
ñ
68
2
ècos.........
2
ècos.
2
ècos.ècos
ñ
ñ 1n210
0
n
dUcen¼ 2
ècos....
2
ècos
2
ècosècosññ 1n21
00n .
müageToteyIgman 2
ècosèsin2
2
ècos
2
èsin2èsin n
1nnn
n
( eRBa¼2
èè n
1n ) eKTaj 1n
nn
èsin
èsin.
2
1
2
ècos
ehtuen¼ n
0
nn
1n
2
1
1
0
nnèsin
èsin.
2
1
èsin
èsin.....
èsin
èsin.
èsin
èsin.
2
1ñ
dUcen¼ )
2
1.
3
ðsin(
1.
2
3
)2
1.
3
ðsin
3
ðsin
2
1ñ
n
1n
n
nn
.
lMhat´TI33
k¿cUrKNnatémø®ákdén 8
ðtan
x¿cUreda¼RsaysmIkar
0xcos)12(xcos.xsin2xsin 22
K¿cUreda¼RsaysmIkar 3
1
xtan)12(1
12xtan
dMeNa¼Rsay
69
k¿KNnatémø®ákdén 8
ðtan
tamrUbmnþ atan1
atan2a2tan
2 edayyktémø
8
ða
eKán
8
ðtan1
8
ðtan2
4
ðtan
2
8
ðtan1
8
ðtan2
12
naM[ 018
ðtan2
8
ðtan2
tag 8
ðtant Edl 0t
eKán 211';01t2t2
eKTaj¦s 021t,21t 21 ( minyk)
dUcen¼ 128
ðtan .
x¿ eda¼RsaysmIkar
0xcos)12(xcos.xsin2xsin 22
EckGg:TaMgBIrnwg 0xcos2 eKánsmIkar ½
0)12(xtan2xtan2 ¿
tag xtant eKán ½
0)12(t2t2 eday 0cba 70
eKTaj¦s 12t;1t 21 .
-cMeBa¼ 1t eKán 1xtan naM[ Zk,ðk4
ðx
-cMeBa¼ 12t eKán 12xtan
naM[ Zk,ðk8
ðx
dUcen¼ Zk,ðk8
ðx,ðk
4
ðx .
K¿ eda¼RsaysmIkar ½
3
1
xtan)12(1
12xtan
eday 128
ðtan eKán
6
ðtan)
8
ðxtan(
3
3
8
ðtan.xtan1
8
ðtanxtan
eKTaj ðk6
ð
8
ðx ¦ Zk,ðk
24
ðx
dUcen¼ Zk,ðk24
ðx .
71
lMhat´TI34
k¿cUrKNnatémø®ákdén 12
ðcos nig
12
ð7cos
x¿cUreda¼RsayRbB&næsmIkar
8
632ycosycosxcos3
8
632ycosxcos3xcos
32
23
dMeNa¼Rsay
k¿KNnatémø®ákdén 12
ðcos nig
12
ð7cos
eyIgán )4
ð
3
ðcos(
12
ðcos
4
62
2
2.
2
3
2
2.
2
1
4
ðsin
3
ðsin
4
ðcos
3
ðcos
ehIy )4
ð
3
ðcos(
12
ð7cos
4
62
2
2.
2
3
2
2.
2
1
4
ðsin
3
ðsin
4
ðcos
3
ðcos
dUcen¼ 4
62
12
ð7cos,
4
62
12
ðcos
.
72
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
73
4
62xcos
2
62xcos2
12
ðcosxcos naM[ Zk,ðk2
12
ðx
dksmIkar )3( nig )4( Gg:nigGg:eKán ½
4
62ycos
2
62ycos2
12
ð7cosycos naM[ Zk,ðk2
12
ð7y
dUcen¼ Zk,ðk212
ðx nig Zk,ðk2
12
ð7y
lMhat´TI35
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
74
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
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 ðk23
ðx4 ¦ Zk,
2
ðk
12
ðx .
75
lMhat´TI36
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
ðcos22U0 nig
8
ðcos2
4
ðcos22U2U 01
«bmafavaBitdl´tYTI p KW 2pp
2
ðcos2U
eyIgnwgRsayfavaBitdl´tYTI )1p( KW 3p1p
2
ðcos2U
eyIgman p1p U2U Ettamkar«bma 2pp
2
ðcos2U
eyIgán 3p3p
2
2p1p2
ðcos2
2
ðcos4
2
ðcos22U
Bit
dUcen¼ 2nn
2
ðcos2U
.
76
x¿ KNnaplKuN n210n U....UUUP
tamrUbmnþ acosasin2a2sin naM[ asin
a2sinacos2
2n2n
n
0k
n
0k2k
1kn
0k2kkn
2
ðsin
1
2
ðsin
2
ðsin
)
2
ðsin
2
ðsin
()2
ðcos2()U(P
lMhat´TI37
eK[sIVúténcMnYnBit )U( n kMnt´elI IN eday ½
2
2U0 nig INn,
2
U11U
2
n
1n
KNna nU CaGnuKmn_én n .
dMeNa¼Rsay
KNna nU CaGnuKmn_én n ½
eyIgman 4
ðsin
2
2U0
8
ðsin
2
4
ðsin11
2
U11U
22
0
1
«bmafavaBitdl´tYTI p KW 2pp
2
ðsinU
eyIgnwgRsayfavaBitdl´tYTI )1p( KW 3p1p
2
ðsinU
Bit
eyIgman 2
U11U
2
p
1p
Ettamkar«bma
2pp2
ðsinU
77
eyIgán 2
2
ðsin11
U2p
2
1p
3p
3p
2
2p
2
ðsin
2
2
ðsin2
2
2
ðcos1
Bit
dUcen¼ 2nn
2
ðsinU
.
lMhat´TI38
eK[smIkardWeRkTIBIr 02mx)mm(x:)E( 22
eKsnµtfasmIkaren¼man¦sBIrtagerogKñaeday atan
nig btan .
k¿cUrkMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3
ðba .
x¿cUreda¼RsaysmIkarxagelIcMeBa¼ mEdlánrkeXIj .
K¿edayeRbIlTæplxagelIcUrTajrktémøRákdén 12
ðtan .
dMeNa¼Rsay
k¿kMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3
ðba ½
smIkarman¦skalNa 08m4)mm( 22
eday atan nig btan Ca¦srbs´smIkarena¼
78
tamRTwsþIbTEvüteKman
)2(2mbtan.atan
)1(mmbtanatan 2
eday btan.atan1
btanatan)batan(
)3(
btan.atan1
btanatan3
btan.atan1
btanatan
3
ðtan
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 mEdlánrkeXIj ½
cMeBa¼ 3m eKán ½ 023x)33(x2
eday 0cba eKTaj¦s 32x,1x 21 .
79
K¿TajrktémøRákdén 12
ðtan ½
tamlTæplxagelIeKman 1atanx1 naM[ 4
ða
ehIy 3
ðba ena¼
12
ð
4
ð
3
ðb
ehtuen¼ 3212
ðtanbtanx2
dUcen¼ 3212
ðtan .
lMhat´TI39
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 .
80
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
1
2
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
1atanS
nnnnn .
lMhat´TI40
k¿cUrRsaybBa¢ak´fa xtan2x2tanxtan.x2tan 2
x¿cUrKNnaplbUk
n
0k1k
2
k
k
n2
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
81
xtan.x2tanxtan.xtan1
xtan2
xtan1
xtan2
xtan1
xtan2xtan2xtan2
xtan1
)xtan1(xtan2xtan2
22
22
3
2
3
2
2
dUcen¼ xtan2x2tanxtan.x2tan 2 .
x¿KNnaplbUk
n
0k1k
2
k
k
n2
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
k
n2
atan2atan
2
atan2
2
atan2S
dUcen¼ 1n
1nn
0k1k
2
k
k
n2
atan2atan
2
atan
2
atan2S
82
lMhat´TI41
k¿cUrRsayfa )x3sinxsin3(4
1xsin3
x¿cUrKNna
n
31n
3
32
2
33
n3
asin3....
3
asin3
3
asin3
3
asinS
dMeNa¼Rsay
k¿Rsayfa )x3sinxsin3(4
1xsin3
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
1xsin3 .
x¿ KNna n
31n
3
32
2
33
n3
asin3....
3
asin3
3
asin3
3
asinS
eyIgán
n
1kk
31k
n3
asin3S eday )x3sinxsin3(
4
1xsin3
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
83
lMhat´TI42
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
1kk
2kn
2
acos
1.
4
1S
eday xsin
1
x2sin
4
xcos
1222
eKán ½
84
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 .
lMhat´TI43
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
1xcos2xcos2
xcosxsin2
1xcos2
xsin
xcos
x2sin
x2cos
xsin
xcos
22
2
85
dUcen¼ x2cotxcotx2sin
1 .
x¿KNna n2
n
2
asin
1....
2
asin
1
2
asin
1
asin
1S
eyIgán
n
0kk
n )
2
asin
1(S eday x2cotxcot
x2sin
1
acot
2
acot
2
acot
2
acotS
1n
n
0kk1kn
dUcen¼ acot2
acotS
1nn
.
lMhat´TI44
k¿cUrRsayfa xcot
2
xcot
xcos
11
x¿KNnaplKuN
)
2
acos
11).....(
2
acos
11)(
2
acos
11)(
acos
11(P
n2
n
dMeNa¼Rsay
k¿Rsayfa xcot
2
xcot
xcos
11
86
eyIgtag xcos
11)x(A
xcot
2
xcot
xtan2
xtan
xcos
xsin.
2
xsin
2
xcos
2
xsinxcos
xsin2
xcos
2
xsinxcos
2
xsin
2
xcos2
xcos
2
xcos2
xcos
1xcos22
dUcen¼ xcot
2
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
87
lMhat´TI45
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
1kknxcos
)kxsin(S
eday xcot.xcos
)kxcos(
xcos
x)1kcos(
xcos
)kxsin(k1kk
n
1k1nk1kn 1xcos
x)1ncos(xcot
xcos
)kxcos(
xcos
x)1kcos(xcotS
88
lMhat´TI46
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
89
lMhat´TI47
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
90
x)1ntan()nxtan(1xtan)nxtan(x)1ntan(
¦ 1xcot)nxtan(x)1ntan(x)1ntan()nxtan(
eyIgán
n
1kn 1xcot)kxtan(x)1ktan(S
nxcotxcosx)1ncos(
)nxsin(
nxcotxtanx)1ntan(
dUcen¼ nxsinx)1ncos(
)nxsin(Sn
.
lMhat´TI48
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
2xcos4x2cos2
2
2
dUcen¼ xcos21
x2cos211xcos2
.
91
x¿ KNnaplKuN ½
)12
acos2).....(1
2
acos2)(1
2
acos2)(1acos2(P
n2n
n
0kk
12
acos2
eday xcos21
x2cos211xcos2
n
n
0kk
1k
n
2
acos21
a2cos21
)2
acos(21
)2
acos(21
P
dUcen¼ n
n
2
acos21
a2cos21P
.
lMhat´TI49
k¿cUrRsayfa )xtan3x3tan(8
1
xtan31
xtan2
3
x¿cUrKNnaplbUk
n
0kk
2
k
3k
n
3
atan31
3
atan3
S
dMeNa¼Rsay
k¿ Rsayfa )xtan3x3tan(8
1
xtan31
xtan2
3
tamrUbmnþ xtan31
xtanxtan3x3tan
2
3
92
eyIgán xtan31
xtan8xtan3
xtan31
xtanxtan3xtan3x3tan
2
3
2
3
dUcen¼ )xtan3x3tan(8
1
xtan31
xtan2
3
.
x¿KNnaplbUk
n
0kk
2
k
3k
n
3
atan31
3
atan3
S
eyIgman )xtan3x3tan(8
1
xtan31
xtan2
3
edayyk k3
ax
eKán )3
atan3
3
a(tan
8
1
3
atan31
3
atan
k1k
k
2
k
3
eyIgán
n
1n
k
1kn
0k1k
k
n3
atan3a3tan
8
1)
3
atan3
3
atan3(
8
1S
dUcen¼ n
1n
n3
atan
8
3
8
a3tanS
.
93
lMhat´TI50
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
eyIgán xtan1
xtanxtan
xtan1
xtanxtanx2tan
2
12
3
2
dUcen¼ xtanx2tan2
1
xtan1
xtan2
3
.
x¿KNnaplbUk
n
0kk
2
k
3k
n
2
atan1
2
atan2
S
eKman xtanx2tan2
1
xtan1
xtan2
3
yk k2
ax
eKán k1k
k
2
k
3
2
atan
2
atan
2
1
2
atan1
2
atan
eyIgán n
nn
0kk
k
1k
1k
n2
atan2a2tan
2
1
2
atan2
2
atan2S
dUcen¼ n
n
n2
atan2a2tan
2
1S .
94
lMhat´TI51
k¿cUrRsayfa x)1n2sin(x)1n2sin(xsin2
1)nx2cos(
x¿KNnaplbUk )nx2cos(.....x6cosx4cosx2cosSn
K¿TajrkplbUk
)nx(cos.....x3cosx2cosxcosT 2222
n
X¿KNnaplbUk
)nx(sin.....x3sinx2sinxsinU 2222
n
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( .
x¿KNnaplbUk )nx2cos(.....x6cosx4cosx2cosSn
eyIgán
n
1kn )kx2cos(S
95
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 2222
n
eyIgán
n
1k
2
n )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 2222
n
eyIgán
n
1k
2
n )kx(sinU
n
1kn
2 Tn)kx(cos1
eday xsin2
x)1ncos()nxsin(
2
nTn
dUcen¼ xsin2
x)1ncos()nxsin(
2
nUn
.
96
lMhat´TI52
eK[GnuKmn_ )1x(2
8m3mx2xy
2
2
Edl IRx nig
mCaá¨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(06m3mx2x3
2
2
cMeBa¼ 06m3mx2x3:)1( 2
smmUl
018m9m'
03a
2
eday )6m)(3m(18m9m2
97
eKán 0)6m)(3m(' naM[ 6m3
¦ ]6,3[m
cMeBa¼ 010m3mx2x:)2(2
smmUl
010m3m'
01a
2
eday )5m)(2m(10m3m2
eKán 0)5m)(2m(' naM[ 2m5
¦ ]2,5[m .
edayykcemøIy ]6,3[m RbsBVnwg ]2,5[m
ena¼eKán öm
dUcen¼eKminGackMnt´témø m edIm,I[GnuKmn_en¼Gactag
[témøkUsIunUsénmMumYYyáneT .
98
lMhat´TI53
cUrRsaybBa¢ak´fa
3
8
2x4cos
1x4sin4x4cos2
cMeBa¼RKb´cMnYnBit
dMeNa¼Rsay
eyIgtag IRx,2x4cos
1x4sin4x4cosy
eyIgán y2x4cosy1x4sin4x4cos
¦ )1(1y2x4sin4x4cos)y1(
eyIgeRCIserIsviucT&rBIr ½
)4,y1(U
nig )x4sin,x4cos(V
tamkenßamviPaKplKuNs;aEl
)2(x4sin4x4cos)y1(V.U
tam )1( nig )2( eKTaj 1y2V.U
.
müa¨geTottamniymn&y ècos.||V||.||U||V.U
eday IRè,1ècos1
eKTaj ||V||.||U||V.U||V||.||U||
99
¦ 22
2
||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 .
lMhat´TI54
eK[cMnYnkMupøic )xsin
1x.(sini)
xcos
1x(cosZ
2
2
2
2
Edl x CacMnYnBit.
cUrkMnt´rkmUDulGb,brmaéncMnYnkMupøicen¼ ?
dMeNa¼Rsay
rkmUDulGb,brmaéncMnYnkMupøic
eyIgán 2
2
22
2
2 )xsin
1x(sin)
xcos
1x(cos|Z|
tag 2
2
22
2
2 )xsin
1x(sin)
xcos
1x(cos)x(f
100
)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
44
44
44
4444
44
44
4
4
4
4
edayeKman 1x2sin2 naM[
2
1x2sin
2
11 2
nig 17x2sin
161
4
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|
dUcen¼mUDulGb,brmaén Z KW 2
25|Z| min .
101
lMhat´TI55
eK[ x CacMnYnBitEdl 021x71x60 2 .
cUrbgHajfa 01x3
ðsin
.
dMeNa¼Rsay
bgHajfa 01x3
ðsin
tag 21x71x60)x(f2
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 naM[
3
4
1x3
1
5
4
eKTaj 3
ð4
1x3
ð
5
ð4
naM[ 0
1x3
ðsin
.
dUcen¼ ebI x CacMnYnBitEdl 021x71x60 2
ena¼eKán½ 01x3
ðsin
.
102
lMhat´TI56
eK[sIVúténcMnYnBit )U( n kMnt´eday 4
ðnsin.2U
n
n
Edl *INn .
k-cUrbgHajfa 4
ðnsin
4
ðncos
4
ð)1n(cos.2
x-Taj[ánfa 4
ð)1n(cos)2(
4
ðncos)2(U 1nn
n
K-KNnaplbUk n321n U.........UUUS .
dMeNa¼Rsay
k-bgHajfa 4
ðnsin
4
ðncos
4
ð)1n(cos.2
tamrUbmnþ bsinasinbcosacos)bacos(
eyIgán
)4
ðsin
4
ðnsini
4
ðcos
4
ðn(cos2)
4
ð
4
ðncos(2
4
ð)1n(cos2
4
ðnsin
4
ðncos)
4
ðnsin
2
2
4
ðncos
2
2(2
4
ð)1n(cos2
dUcen¼ 4
ðnsin
4
ðncos
4
ð)1n(cos.2
.
x-Taj[ánfa 4
ð)1n(cos)2(
4
ðncos)2(U 1nn
n
eyIgman 4
ðnsin
4
ðncos
4
ð)1n(cos.2
naM[ 4
ð)1n(cos2
4
ðncos
4
ðnsin
103
KuNGg:TaMgBIrnwg n)2( eKán
4
ð)1n(cos)2(
4
ðncos)2(
4
ðnsin)2( 1nnn
dUcen¼ 4
ð)1n(cos)2(
4
ðncos)2(U 1nn
n
.
K-KNnaplbUk n321n U.........UUUS
eyIgán
4
ð)1n(cos)2(
4
ðcos2
4
ð)1k(cos)2(
4
ðkcos)2(S
1n
n
1k
1kk
n
dUcen¼ 4
ð)1n(cos)2(1S
1n
n
.
104
lMhat´TI57
k¿cUrKNnatémø®ákdén 10
ðsin nig
10
ðcos
x¿cUrRsayfa 10
ðsin)yx(4)yx(x 22222 .
dMeNa¼Rsay
k¿KNnatémø®ákdén 10
ðsin nig
10
ðcos
eKman 10
ð3
2
ð
10
ð2
eKán )10
3
2sin(
10
2sin
)10
ðsin1(43
10
ðsin2
10
ðcos43
10
ðsin2
10
ðcos4
10
ðcos3
10
ðcos
10
ðsin2
10
ð3cos
10
ðcos
10
ðsin2
2
2
3
¦ 0110
ðsin2
10
ðsin4 2 tag 0
10
ðsint
eKán 0541',01t2t4 2
eKTaj¦s 04
51t1
( minyk )
4
51t, 2
dUcen¼ 4
51
10
ðsin
.
105
eday
110
ðcos
10
ðsin 22 naM[
4
5210)
4
51(1
10
ðcos
2
dUcen¼ 4
5210
10
ðcos
.
x¿Rsayfa 10
ðsin)yx(4)yx(x 22222
tagGnuKmn_ 10
ðsin)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
ðsin)yx(4)yx(x 22222
RKb´cMnYnBit IRy,x .
106
lMhat´TI58
eK[RtIekaN ABC mYYy .
bgHajfaebI 3
Ctan,
3
Btan,
3
Atan Ca¦srbs´smIkar
0cbxaxx:)E( 23 ena¼eKán cb3a3 .
dMeNa¼Rsay
karbgHaj ½
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
3
ðtan
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(
107
ena¼tamRTwsþIbTEvüteKmanTMnak´TMng ½
)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
108
lMhat´TI59
eK[GnuKmn_ ða0,1acosx2x
acosx2acosx)x(f
2
2
bgHajfa 1)x(f1:IRx .
dMeNa¼Rsay
bgHajfa 1)x(f1:IRx
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
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
109
lMhat´TI60
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
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
110
lMhat´TI61
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
ðCB ehIy
2
ðA
dUcen¼ ABC CaRtIekaNEkgRtg A .
111
lMhat´TI62
eK[ctuekaNABCDmYyman
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
B
A
C
D
112
eyIgman 1Bsin ena¼ abBsinab
nig 1Dsin ena¼ cdDsincd
dUcen¼ )cdab(2
1S
lMhat´TI63
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 ABC eKman ½
Ccosab2bac 222 Ettamsmµtikmµ 222 c2ba
eKán Ccosab2ba2
ba 2222
¦ 2
ba
2
babaCcosab2
222222
dUcen¼ ab4
baCcos
22 .
113
x¿ TajbBa¢ak´fa 2
1Ccos
eyIgman 0)ba( 2
0bab2a 22 ¦ 2
1
ab4
ba 22
edaytamsRmayxagelI ab4
baCcos
22
dUcen¼ 2
1Ccos .
lMhat´TI64
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 ABC eKman 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 114
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
0)ac()cb()ba(
0)aac2c()cbc2b()bab2a(
ac2bc2ab2c2b2a2
acbcabcba
abc2
abcabc
abc2
cba
)c
1
b
1
a
1(
2
1
abc2
cba
222
222222
222
222
222
222
eKTajánsmPaB cba .
dUcen¼ ABC CaRtIekaNsm&gß.
115
lMhat´TI65
eK[smIkar 02m3x5x)3m2(x:)E( 23
«bmafasmIkaren¼man¦bItageday ãtan,âtan,átan .
k¿cUrKNna ãcosâcosácos
)ãâásin(A
CaGnuKmn_én m .
x¿kMnt´ m edIm,I[ 4A .
K¿eda¼RsaysmIkar )E( cMeBa¼témø mEdlán
rkeXIjxagelI .
dMeNa¼Rsay
k¿ KNna ãcosâcosácos
)ãâásin(A
CaGnuKmn_én m
eyIgman
coscoscos
)](sin[A
ãtanâtanátanãtanâtanátan
ãtanâtanãtanâtanátanátan
ãcosâcosácos
ácosâcosãsinácosãcosâsinãsinâsinásinãcosâcosásin
ãcosâcosácos
ácos)ãâsin()ãâcos(ásin
eday ãtan,âtan,átan Ca¦ssmIkar )E( ena¼tamRTwsþIbTEvüt
eyIgman
2m3ãtanâtanátan
5átanãtanãtanâtanâtanátan
3m2ãtanâtanátan
116
eyIgán 5m)2m3(3m2A
dUcen¼ 5mA .
x¿kMnt´ m edIm,I[ 4A
edayeyIgman 5mA
eyIgán 45m naM[ 1m .
K¿ eda¼RsaysmIkar )E( ½
cMeBa¼ 1m eKán 01x5x5x:)E( 23
eday )1x4x)(1x(1x5x5x 223
eKTaj 0)1x4x)(1x( 2 ¦
01x4x
1x
2
314' eKTaj 32x,32x 21
dUcen¼sMNMu¦ssmIkar }32,1;32{x .
117
lMhat´TI66
eK[GnuKmn_ 22ysinbxsinaycosbxcosa)y;x(f
( Edl 0b,0a ) .
cMeBa¼RKb´ IRy;x bgHajfa 2)ba()y;x(f .
dMeNa¼Rsay
bgHajfa 2)ba()y;x(f
22ysinbxsinaycosbxcosa)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 .
118
lMhat´TI67
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
abc2acb
bc2
abc2
bc2
acb
bc4
a21
bc2
acb
2
22
2222
2222
2222
RtIekaNABCmanRCúg cb naM[vaCaRtIekaNsmát
kMBUl A .
119
lMhat´TI68
eK[smIkar mx3sinxsinx3cosxcos:)E( 33
k¿cUreda¼RsaysmIkaren¼kalNa 8
33m
x¿rklk&çx&NÐsRmab´ m edIm,I[smIkaren¼man¦s .
dMeNa¼Rsay
k¿eda¼RsaysmIkaren¼kalNa 8
33m
eyIgman xsin4xsin3x3sin 3 naM[
)x3sinxsin3(4
1xsin3
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
120
naM[ Zk,ðk12
ðx
x¿ rklk&çx&NÐsRmab´ m ½
edIm,I[smIkaren¼man¦seKRKan´Et[ 1m1 3
¦ 1,1m .
lMhat´TI69
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,ðk2ðxðk2
smIkarGacsresr ½
02)x(sinlog3)x(sinlog
02log)x(sinlog)x(sinlog
2
2
2
2
3
2
2
2
tag )x(sinlogt2
eKánsmIkar 02t3t2
eday cab eKTaj¦s 2t,1t 21
-cMeBa¼ 1t eKán 1)x(sinlog2
121
smmUl 2
1xsin
naM[
Zk,ðk24
ð3ðk2
4
ððx
ðk24
ðx
-cMeBa¼ 2t eKán 2)x(sinlog2
smmUl 2
1xsin
naM[
Zk,ðk26
ð5ðk2
6
ððx
ðk26
ðx
lMhat´TI70
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
122
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
1
xsin2
1S
)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
.
123
lMhat´TI71
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
)nxcos(xcos
1
)nxcos(
x)1ncos(....
x3cos
x2cos.
x2cos
xcos.
xcos
1.
xcos
1
)kxcos(xcos
x)1kcos(
n
n
n
1k
124
lMhat´TI72
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
tag O Cap©iténrgVg´carikeRkARtIekaN ABC ehIy
A
B C
O
H
125
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
eyIgman ABAC2
BOCBOH
mMup©itnigmMucarikkñúgrgVg´s;at´FñÚrYm BC .
eKán AcosR.a2
1SOBC
RsaydUcKñaeKán BcosR.b2
1SOAC nig
AcosR.c2
1SOAB .
ehtuen¼ CcosR.c2
1BcosR.b
2
1AcosR.a
2
1S
126
)CcoscBcosbAcosa(R2
1S
dUcen¼ R
S2CcoscBcosbAcosa .
x¿Tajfa S4CcotcBcotbAcota 222
eyIgman AcosaR2Acosa.Asin
a
Asin
AcosaAcota 22
dUcKñaEdr BcosaR2Bcotb2 nig Ccos.cR2Ccotc2
eKán )CcoscBcosbAcosa(R2CcotcBcotbAcota 222
eday R
S2CcoscBcosbAcosa ( sRmayxagelI )
dUcen¼ S4CcotcBcotbAcota222
.
127
lMhat´TI73
eKtag r nig R erogKñaCakaMrgVg´carikkñúg nigcarikeRkA
énRtIekaN ABC mYy .
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
128
eKán abc
)cp)(bp)(ap(
2
Csin
2
Bsin
2
Asin
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 129
8
1
abc
)cp)(bp)(ap(
eKTaj 8
1
2
Csin
2
Bsin
2
Asin
eday 2
Csin
2
Bsin
2
Asin41CcosBcosAcos
eKán 2
3)
8
1(41CcosBcosAcos
Et R
r1CcosBcosAcos
eKTaj 2
3
R
r1 ¦ r2R .
130
lMhat´Gnuvtþn_ 1> eK[sIVúténcMnYnBit *INn,
4
ðncos)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 131
Edl
kMnt;edayTMnak;TMng 4
nsinb
4
ncosaVU nn
CasIVútFrNImaRt
x> TajrktYtUeTA nU CaGnuKmn_én n .
4> eKmansIVúténcMnYnkMupøic )Z( n kMnt;eday ³
INn,2
i32Z
2
i3Z
2Z
n1n
0
k> eKtag 1ZU:INn nn .
bgðajfa 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
132
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 .
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 .
133
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 .
134
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
135
kMnt´edayTMnak´TMng 2
nsinB
2
ncosAVU nn
CasIVútFrNImaRt .
x¿ cUrKNna nU CaGnuKmn_én n .
11-eK[ x2tan2.......x2tan2x2tan2xtanS nn22
n k¿ cUrbgHaj è2cot2ècotètan .
x¿ TajbgHajfa x2cot2xcotS1n1n
n
.
12-eK[ n
n
2
3233n
3
asin3.......
3
asin3
3
asin3asinS
k¿ cUrbgHajfa x3sin4
1xsin
4
3xsin3
.
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
. 136
x¿ TajbgHajfa asin
1
a2sin
4S
21n2
1n
n
.
15-eK[
1n
2
n
n
2
22
n2
xcos
2
xsin2...............
2
xcos
2
xsin2
2
xcosxsinS
k¿ cUrbgHajfa 2
acosasin2a2sinasin2 2 .
x¿ TajbgHajfa x2sin2
1
2
xsin2S
n
n
n .
16-eK[ n2n2
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 . 137
18-cUrRsaybBa¢k´smPaBxageRkam ½
k¿ xtan1
xtan1)x
4
ðtan(
c¿ èsin2è2sinètan 2
x¿ btanatan1
btanatan
)bacot(
)batan(22
22
q¿ x2sin2
11
xcosxsin
xcosxsin 33
K¿ ytanxtan1
ytanxtan
)yxcos(
)yxsin(
C¿ atan1
atan1
a2cos
a2sin1
X¿ âsinásin)âásin()âásin( 22
g¿ )átanãtanãtanâtanâtaná(tan1
ãtanâtanátanãtanâtanátan)ãâátan(
19-eK[RtIekaN ABCmYy .
cUrRsaybBa¢ak´smPaBxageRkam ½
k¿ 2
Ccos
2
Bcos
2
Acos4CsinBsinAsin
x¿ 2
Csin
2
Bsin
2
Asin41CcosBcosAcos
K¿ CtanBtanAtanCtanBtanAtan
X¿ 2
Ccot
2
Bcot
2
Acot
2
Ccot
2
Bcot
2
Acot
g¿ CcosBsinAsin2CsinBsinAsin 222
138
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
139
g¿ x3tan)x3
ðtan()x
3
ðtan(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
140
K¿ x2tan2)x4
ðtan()x
4
ðtan(
X¿ 2
xtan
xcos1
xsin
g¿ x2cos
2)x
4
ðtan()x
4
ðtan(
27-cUrsRmYlkenßamxageRkam ½
k¿ xsinxcos1
xsinxcos1
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
141
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
k¿ 7xcos4xsin3y
x¿ 17xcos12xsin5y
K¿ xcosxsiny 44 142
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....222An Edl
2
ða0 .
35-eK[smIkardWeRkTIBIr
02mx)mm(x:)E( 22
eKsnµtfasmIkaren¼man¦sBIrtagerogKñaeday
atan nig btan .
k¿ cUrkMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3
ðba
x¿ eda¼RsaysmIkarxagelIcMeBa¼m Edlán 143
rkeXIj .
K¿edayeRbIlTæplxagelIcUrTajrktémøRákdén
12
ðtan .
36-k¿ cUrKNnatémøRákdén 8
ðtan .
x¿ cUreda¼RsaysmIkar 010100 )x(tan122log1
37-k¿ cUrKNnatémøRákdén 5
ðtan .
x¿ cUreda¼RsaysmIkar
0xcos)525(3x2sin)2
52(xsin 424
38-k¿ cUrKNnatémø®ákdén 10
ðsin nig
10
ðcos
x¿ cMeBa¼RKb´cMnYnBit IRy,x cUrbgHajfa ½
10
ðsin)yx(4)yx(x 22222
39-eK[sIVúténcMnYnBit )U( n kMnt´eday 144
INn,U2U
2
ða0,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
nn22
n
nn22n
42-k¿ cUrRsaybBa¢ak´fa xtan2x2tanxtan.x2tan 2
x¿ cUrKNnaplbUk
n
0k
k21k
kn a2tana2tan2
1S
43-k¿ cUrRsayfa )x3sinxsin3(4
1xsin3
x¿ cUrKNna
a3sin3
1....a3sin
3
1a3sin
3
1asinS n3
n
23
2
33
n
44-k¿ cUrRsayfa xsin
1
x2sin
4
xcos
1222
145
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 xcot
2
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
146
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 n2
n
51-KNnaplKuN
n2n2
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
222
n 147
55-KNnaplKuN
)3
xsin43)......(
3
xsin43)(
3
xsin43)(xsin43(P
n
2
2
222n
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
na3tan31
a3tan3
1
S
58-k¿ cUrRsayfa xtanx2tan2
1
xtan1
xtan2
3
x¿ cUrKNnaplbUk
n
0kk2
k3
k
na2tan1
a2tan2
1
S
59-KNnaplKuN
)acotan)......(taacota)(tanacota)(tanacota(tanPn2n24422
n
60-bgHajfa asin2
a2sina2cos....a4cosa2cosacos
n
n1n
.
148
61- eK[sIVúténcMnYnBit )a( n kMnt´eday
INn,
4
a3aa
2
1a
3 1nn1n
1
k¿ cUrRsaybBa¢ak´fa 1a0 n .
x¿ eKtag nn ècosa . cUrrkRbePTénsIVút )( n
K¿ KNna nè nig na CaGnuKmn_én n .
62-eKmansIVút )b( n kMnt´eday 2
3b0
nig INn,b411
bb
2
n
n1n
k¿ eKBinitüsIVút )è( n Edl INn,2
ðè0 n
ehIy 2
ètanb nn .
cUrkMnt´rkRbePTénsIVút )è( n ?
x¿ KNna nè nig nb CaGnuKmn_én n .
63-eKmansIVút )t( n kMnt´eday 3
ðtant0
149
nig INn,t42t2
n1n
k¿ cUrbgHajfa 2t0 n .
x¿ tag nn èsin2t .
cUrkMnt´rkRbePTénsIVút )( n ?
K¿ KNna nè nig nt CaGnuKmn_én n .
64-eKmansIVút 3
1u0 nig
INn,u1uu2
nn1n
k¿ tag nn ècotu Edl INn,2
ðè0 n .
cUrkMnt´rkRbePTénsIVút )è( n
x¿ KNna nè nig nu CaGnuKmn_én n .
65-eK[ )u( n CasIVútviC¢manEdl 2u0
nig n
n2
1nu1
u2u
.
KNna nu .
66-eK[RtIekaN ABC mYYymanRCúg c,b,a . 150
cUrRsaybBa¢ak´fa )c
1
b
1
a
1(
2
1
c
Ccos
b
Bcos
a
Acos
67-eda¼RsaysmIkarxageRkam ½
k¿ xcos)3
ðx2cos(
x¿ )x3
ðsin(x2sin
K¿ 2
3)x
4
ðsin(
X¿ )x3
ð2sin()
3
ðx2sin(
g¿ )4
ðxtan(x3tan
68-eda¼RsaysmIkar ½
k¿ )6
ðxcos(x3sin
x¿ )x6
ðcos()
4
ðx2sin(
K¿ )x23
ðcot()x
4
ðtan(
X¿ )6
ðx3tan()x2
3
ðtan(
g¿ 5
ðcot)
3
ðx3tan(
69-eda¼RsaysmIkarxageRkam ½
k¿ 01xsin3xsin2 2 151
x¿ 02xsin)12(2xsin4 2
K¿ 02
3xcos)31(xcos2 2
X¿ 03xtan)31(xtan2
g¿ 03xtan4xtan3 2
70-eda¼RsaysmIkar ½
k¿ 03xsin)13(2xsin4 2
x¿ 0xcosxcosxsin)31(xsin3 22
K¿ 03xtan)13(xtan2
X¿ 01xcoslog3xcoslog2 2
2
2
g¿ 02xsinlog3xsinlog2
2
2
71-eda¼RsaysmIkarxageRkam ½
k¿ 0xcos)13(xcosxsin32xsin)13( 22
x¿ 2xsinxcos3
K¿ 2xcosxsin
X¿ 1xsin3xcos 152
72-eda¼RsaynwgBiPakßasmIkar m2xsinxcosm .
73-eKmansmIkar 0ácosx2x)1ácos2( 2
Edl ðá0 .
kMnt´témø á edIm,I[smIkarman¦sBIrepÞógpÞat´
0ásin4''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
ð7sin(3xsin
x¿ xsin
11xcos4xcosxcot2
2
222
K¿ xcosxsinxcos2x2cos2 2
X¿ xsin2)4
ðx(sin3
153
g¿ )4
ðxtan()x
4
ðtan(xtanx2tan
76-eda¼RsaysmIkar ½
k¿ 4
2x3sinxsinx3cosxcos 33
x¿ 1x2sin1
1xcos2)23xsin2(xcos 2
K¿ 13)12
ðxcos()
4
ðxsin(4
X¿ x3cos
113)x
3
ðtan()x
3
ðtan(xtan)31(
2
77-k¿ cUrKNnatémø®ákdén 12
ðtan nig
12
ð5tan
x¿ eda¼RsaysmIkar 01xtan5xtan5xtan 23
78-k¿ KNnatémø®ákdén 8
ðtan
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
.
154
82-eda¼RsaysmIkar ½
4347347)b
xsin2)4
ðx(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
ehIyepÞógpÞat´ ðk2
ðxx 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
.
155
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 0cbxax2 man¦BIrtag
eday tan nig tan .
cUrKNna
)cos(c)cos()sin(b)(sinaM 22
CaGnuKmn_énelxemKuN c,b,a .
90-cUrbgHajfacMnYn 7
ðcos Ca¦ssmIkar
01x4x4x8 23 .
91-cUrbgHajfacMnYn 9
ð13cos,
9
ð7cos,
9
ðcos
156
Ca¦ssmIkar 01x6x8 3 .
92-cUrbgHajfa 2
7
7
ð3sin
7
ð2sin
7
ðsin .
93-cUrbgHajfa oo0ooo9tan69tan63tan57tantan51tan3tan
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
ð4sin
7
ð2sin
7
ðsinA 222 .
97-cUrbgHajfa 380tan40tan20tan ooo .
98-eK[smIkardWeRkTIbI ½
03x)343(x)3m(x3:)E( 23 Edl
mCaá¨ra¨Em¨Rt
eK«bmafasmIkarman¦sbItagerogKñaeday 157
ãtan,âtan,átan .
k¿ kMnt´témø m edIm,I[ 4
ð3ãâá .
x¿ cUrkMnt´ ã,â,á Edl 2
ðãâá0
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
ðâá0 cMeBa¼témø m
EdlánrkeXIjxagelIen¼ .
100-KNnaplbUk ½
n
2
n2
2
2
22
n2
xtan
4
1.........
2
xtan
4
1
2
xtan
4
1xtanS
101-eK[mMubI ðã,â,á0 Edl 4
ð3ãâá .
158
cUrRsaybBa¢ak´fa ½
k¿ 2
23ãsinâsinásin
x¿ 2
23ãcosâcosácos
102-eK[ n cMnYnBitviC¢man n321 a......,,a,a,a .
cMeBa¼RKb´ n
n321 IRx.....,,x,x,x RsaybBa¢ak´fa ½
2
n321
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
ð3cos
7
ð2cos
7
ðcos
( 5th IMO 1963 )
159
106-cUrkMnt´RKb´ x éncenøa¼ 2,0
EdlepÞógpÞat´
2|x2sin1x2sin1|xcos2 . ( 7th IMO 1965 )
107-bgHajjfa
x2cotxcotx2sin
1...........
x4sin
1
x2sin
1 n
n
cMeBa¼RKb´cMnYnKt´FmµCati n nigRKb´cMnYnBit x
Edl 0x2sin n . ( 8th IMO 1966 )
160
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