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

Sin( a+b )=sinacosb+sinbcosa Sin( a – b )=sinacosb-sinbcosa Cos( a+b )=cosacosb – sinasinb Cos( a – b )=cosacosb+sinasinb

2008


GñkshkarN_RtYtBinitübec©keTs elak lwm qun elak Esn Bisidæ elak Titü em¨g elakRsI Tuy rINa elak RBwm suxnit elak pl b‘unqay

GñkrcnaRkb nig bec©keTskMBüÚT&r kBaØa lI KuNÑaka

GñkRtYtBinitüGkçaraviruTæ elak lwm miK:sir © rkßasiTæi lwm pl:ún 2008


GarmÖkfa esovePA GnuKmn_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


emeronsegçb

GnuKmn_RtIekaNmaRt 1¿ TMnak´TMngsMxan´@ enAelIrgVg´RtIekaNmaRteyIgtag è CargVas´énmMu ( ox , OM ) . eKán OP = cos è , OQ = sin è . 



y

M

Q

x'

o

 P

x

1


eKánTMnak´TMngsMxan´@énGnuKmn_rgVg´RtIekaNmaRtdUcxag eRkam ½ 1¿ sin è + cos è = 1 4¿ tan è . cot è = 1 sin è 1 2¿ tan è = cos è 5¿ 1+ tan è = cos è cos è 1 3¿ cot è = sin è 6¿ 1+ cot è = sin è 2

2

2

2

2

2

2¿ rUbmnþplbUk nig pldk 1¿ sin(a + b) = sin a cos b + sin b cos a 2¿ cos(a + b) = cos a cos b - sin a sin b tan a + tan b 3¿ tan(a + b) = 1 - tan a tan b 4¿ sin(a - b) = sin a cos b - sin b cos a 5¿ cos(a - b) = cos a cos b + sin a sin b a - tan b 6¿ tan(a - b) = 1tan + tan a tan b 2


3¿ rUmnþmMuDub 1¿ sin 2a = 2 sin a cos a 2¿ cos 2a = cos a - sin a = 2 cos a 3¿ tan 2a = 12- tan tan a 2

2

2

a - 1 = 1 - sin 2 a

2

4¿ rUbmnþknø¼mMu a 1 - cos a 1¿ sin 2 = 2 a 1 + cos a 2¿ cos 2 = 2 a 1 - cos a 3¿ tan 2 = 1 + cos a 5¿ kenßam sin x , cos x 1¿ sin x = 1 +2tt 1-t 2¿ cos x = 1 + t 3¿ tan x = 11+- tt 2

2

2

, tan x

CaGnuKmn_én

x t = tan 2

2

2

2

2

2

3


6¿ kenßam sin 3a , cos 3a , tan 3a 1¿ sin 3a = 3 sin a - 4 sin a 2¿ cos 3a = 4 cos a - 3 cos a 3 tan a - tan a 3¿ tan 3a = 1 - 3 tan a 3

3

3

2

7¿ rUbmnþbMElgBIplKuNeTAplbUk 1 1¿ cos a cos b = 2 [ cos(a + b) + cos(a - b) ] 1 2¿ sin a sin b = 2 [ cos(a - b) - cos(a + b) ] 3¿ sin a cos b = 12 [ sin(a + b) + sin(a - b) ] 4¿ sin b cos a = 12 [ sin(a + b) - sin(a - b) ] 6¿ rUbmnþbMElgBIplbUkeTAplKuN p+q p-q 1¿ cos p + cos q = 2 cos 2 cos 2 p+q p-q 2¿ cos p - cos q = -2 sin 2 sin 2 p+q p-q 3¿ sin p + sin q = 2 sin 2 cos 2 4


4¿ 5¿ 6¿ 7¿

p-q p+q sin p - sin q = 2 sin cos 2 2 sin(p + q) tan p + tan q = cos p cos q sin(p - q) tan p - tan q = cos p cos q sin(p + q) cot p + cot q = sin p sin q sin(q - p) cot p - cot q = sin p sin q

8¿ 7¿ smIkarRtIekaNmaRt 1¿ smIkar sin u = sin v mancemøIy u = v + 2kð u = ð - v + 2kð , k ∈ Z

2¿ smIkar

cos u = cos v

mancemøIy

u = v + 2kð u = - v + 2kð , k ∈ Z

3¿ smIkar

tan u = tan v

mancemøIy

u = v + kð

5


8¿ rUbmnþbEmøgFñÚEdlKYkt´sMKal´ 1¿

ð sin ( - è) = cos è 2 ð cos ( - è) = sin è 2 ð tan ( - è) = cot è 2

sin (ð - è) = sin è

2¿

cos (ð - è) = - cos è tan (ð - è) = - tan è

3¿

ð sin ( + è) = cos è 2 ð cos ( + è) = - sin è 2 ð tan ( + è) = - cot è 2 sin (ð + è) = - sin è

4¿

cos (ð + è) = - cos è tan (ð + è) = tan è sin (è + 2kð) = sin è

5¿

cos (è + 2kð) = cos è tan (è + kð) = tan è

, ∀k∈ Z 6


9¿ RkahVikGnuKmn_RtIekaNmaRt 1¿ ExßekagGnuKmn_ y = sin x y

( C ) : y = sinx 1

0

2¿ ExßekagGnuKmn_

x

1

y = cos x

y

1

0

( C ) : y = cosx

1

x

7


y = tan x

3¿ ExßekagGnuKmn_

y

1

0

4¿ ExßekagGnuKmn_

( C ) : y = tanx

1

x

y = cot x

y

1

0

1

x

8


lMhat´ nig dMeNa¼Rsay lMhat´TI1 eK[ cos á = b +a c , cos â = c b+ a , cos ã = a +c b cUrRsayfa tan á2 + tan â2 + tan ã2 = 1 . dMeNa¼Rsay Rsayfa tan á2 + tan â2 + tan ã2 = 1 á 1 - cos á á 1 + cos á eyIgman sin 2 = 2 , cos = 2 2 2

2

2

2

2

2

2

2

a á 1 - cos á b+c = b+c-a tan 2 = = a 2 1 + cos á b+c+a 1+ b+c b 1 â 1 - cos â c+a = c+a -b tan 2 = = b 2 1 + cos â c+a+b 1+ c+a 1-

eKán

c 1 - cos ã 1 - a + b a + b - c 2 ã = tan = = c a+b+c 2 1 + cos ã 1+ a+b

9


tan2

á â ã b+c - a c+a -b a+b - c + tan2 + tan2 = + + 2 2 2 a+b+c a+b+c a+b+c

tan2

á â ã b+c -a+c+a -b+a+b -c + tan2 + tan2 = 2 2 2 a+b+c á â ã a+b+c tan2 + tan2 + tan2 = =1 2 2 2 a+b+c á â ã tan2 + tan2 + tan 2 = 1 2 2 2

.

.

dUcen¼ lMhat´TI2 eK[ tan è =

cUrRsayfa dMeNa¼Rsay Rsayfa cosa

b a cos 8 è sin 8 è 1 = + b3 (a + b) 3 a3

.

8

.

sin 8 è 1 = + 3 b3 (a + b) 3 b b 2 tan è = tan è = a a sin è tan è = cos è è

eKman naM[ eday sin è b eKán cos = è a è 1 smmUl sinb è = cosa è = sin èb ++ cos = a a+b sin è b sin è 1 eKTaj b = a + b naM[ b = (a + bI 2

2

2

2

2

2

2

8

3

4

(1) 10


1 cos 2 è = a a+b

ehIy bUkTMnak´TMng

(1)

cos 8 è a = (2) 3 4 a (a + bI

naM[ nig (2) Gg:nwgGg:eKán ½

cos 8 è sin8 è a+b 1 = + = (a + b) 4 (a + b) 3 a3 b3 cos 8 è sin 8 è 1 = + b3 (a + b) 3 a3

. dUcen¼ lMhat´TI3 eK[RtIekaN ABC mYyman a , b , c CargVas´RCugQm erogKñaénmMu A , B , C . tag p Caknø¼brimaRténRtIekaN . )(p - c) k¿cUrRsayfa sin A2 = (p - bbc rYcTajrkTMnak´TMngBIreTotEdlRsedogKñaen¼ . x¿cUrbgHajfa sin A2 sin B2 sin C2 ≤ 18 3 nig cos A + cos B + cos C ≤ 2 .

11


dMeNa¼Rsay A

C

B

Rsayfa

sin

A (p - b)(p - c) = 2 bc

A 1 - cos A sin = 2 2 2

eKman tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman b +c -a a = b + c - 2bc cos A eKTaj cos A = 2bc 2

2

eKán

2

2

2

2

b2 + c 2 - a2 12bc - b2 - c 2 + a 2 2 A 2 bc sin = = 2 2 4bc 2 2 A a - (b - 0c) (a + b - c)(a - b + c) sin2 = = 2 4bc 4bc

eday a + b + c = 2p ena¼ a + b - c = 2(p - c) , a - b + c = 2(p - b) 12


A 2(p - c).2(p - b) sin = 2 4bc A (p - b)(p - c) sin = 2 bc A (p - b)(p - c) sin = 2 bc 2

eKán naM[

.

dUcen¼ . eKGacTajTMnak´TMngRsedogKñaen¼dUcxageRkam ½ B (p - a)(p - c) C (p - a)(p - b) = , sin = 2 ac 2 ab A B C 1 sin sin sin ≤ 2 2 2 8 A (p - b)(p - c) sin = 2 bc B (p - a)(p - c) C (p - a)(p - b) sin = , sin = 2 ac 2 ab A B C (p - a)(p - b)(p - c) sin sin sin = (1) 2 2 2 abc á + â ≥ 2 á .â sin

.

x¿bgHajfa

tamsRmayxagelIeKman

.

eKán tamvismPaBkUsIu eKán (p - a) + (p - b) ≥ 2 2p - a - b ≥ 2

(p - a)(p - b)

(p - a)(p - c)

c ≥ 2 (p - a)(p - b)

eKTaj

(p - a)(p - b) 1 ≤ (2) c 2 13


(p - b)(p - c) 1 ≤ a 2 (p - a)(p - c) 1 ≤ (4) b 2 (2) , (3) ,(4)

dUcKñaEdr

nig KuNTMnak´TMng

(3)

Gg:nwgGg:eKán ½

(p - a)(p - b)(p - c) 1 ≤ (5) abc 8 (1) (5)

tamTMnak´TMng nig A B C 1 eKTaj sin 2 sin 2 sin 2 ≤ 8 . 3 bgHajfa cos A + cos B + cos C ≤ 2 eyIgman A B+C B-C + 2 cos cos 2 2 2 A A B-C = 1 - 2 sin2 + 2 sin cos 2 2 2 A A B-C = 1 - 2 sin ( sin - cos ) 2 2 2 A B+C B-C = 1 - 2 sin ( cos ) - cos 2 2 2 A B C = 1+ 4 sin sin sin 2 2 2

cos A + cos B + cos C = 1 - 2 sin2

eKTaj

A B C cos A + cos B + cos C = 1 + 4 sin sin sin 2 2 2 14


A B C 1 sin sin sin ≤ 2 2 2 8 1 3 cos A + cos B + cos C ≤ 1 + 4 ( ) = 2 2

tamsRmayxagelIeKman ehtuen¼ lMhat´TI4 eK[RtIekaN ABC mYyman a , b , c CargVas´RCugQm erogKñaénmMu A , B , C . tag p Caknø¼brimaRténRtIekaN . A p(p - a) k¿cUrRsayfa cos   bc rYcTajrkTMnak´TMng BIreTotEdlRsedogKñaen¼ . x¿TajbBa¢ak´fa bc . cos A  ac. cos B  ab. cos C  p dMeNa¼Rsay Rsayfa cos A  p(pbc a) 

A

B

C

15


A eKman cos A    cos  tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman b c a a  b  c  2bc cos A eKTaj cos A  2bc 

2

2

2

eKán

2

b2  c2  a2 1 A 2bc  b 2  c 2  a 2 2 2 bc cos   2 2 4bc 2 2 A (b  c)  a (b  c  a)(b  c  a) cos 2   2 4bc 4bc

eday a  b  c  2p ena¼ b  c  a  2 (p  a) A 2p . 2(p  a) p(p  a) eKán cos 2  4bc  bc 2

¦

cos

2

2

p(p  a) bc A p(p  a) cos  2 bc

A  2

dUcen¼ . eKTajánTMnak´TMngRsedogKñaen¼dUcxageRkam ½ B p(p  b) C p(p  c) . cos  , cos  2 ac 2 ab 16


x¿TajbBa¢ak´fa

A 2 B 2 C bc . cos  ac. cos  ab. cos  p2 2 2 2 2

A p(p  a)  tamsRmayxagelIeKman 2 bc B p(p  b) C p(p  c) cos  , cos  2 ac 2 ab cos

A bc cos  p(p  a)  p 2  p . a 2 2

eKTaj

B ac cos  p(p  b)  p 2  p . b 2 2 C  p(p  c)  p 2  p . c ab cos 2 2

A 2 B 2 C bc . cos  ac. cos  ab. cos  3p 2  p (a  b  c) 2 2 2  3p 2  2p 2  p 2 A B C bc . cos 2  ac. cos 2  ab. cos 2  p 2 2 2 2 2

dUcen¼

.

17


lMhat´TI5 eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc . k¿cUrRsayfa tan A  tan B  tan C  tan A. tan B. tan C x¿TajbBa¢ak´fa tan A  tan B  tan C  3 3 . dMeNa¼Rsay k¿Rsayfa tan A  tan B  tan C  tan A. tan B. tan C eyIgman A  B  C  ð ¦ A  B  ð  C eKán tan(A  B)  tan(ð  C) tan A  tan B   tan C 1  tan A tan B tan A  tan B   tan C  tan A tan B tan C tan A  tan B  tan C  tan A. tan B. tan C

dUcen¼ x¿TajbBa¢ak´fa tan A  tan B  tan C  3 3 eday A , B , C CamMuRsYc ( tamsmµtikmµ ) eKTaj tan A  0 , tan B  0 , tan C  0 tamvismPaBkUsIueyIgGacsresr ½ tan A  tan B  tan C  3

eday

3

.

tan A. tan B. tan C

tan A  tan B  tan C  tan A. tan B. tan C

18


eKán

tan A  tan B  tan B  3 3 tan A  tan B  tan C

(tan A  tan B  tan C)3  27 (tan A  tan B  tan C) (tan A  tan B  tan C)2  27

dUcen¼ tan A  tan B  tan C  3 3 . lMhat´TI6 eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc . cUrRsayfa cos A  cos B  cos C  1 dMeNa¼Rsay Rsayfa cos A  cos B  cos C  1 2

2

2

2

2

2

A

B

tag

C

T  cos2 A  cos 2 B  cos2 C

19


1  cos 2 A 1  cos 2B   cos2 C 2 2 cos 2 A  cos 2B  1  cos 2 C 2  1  cos( A  B) cos( A  B)  cos2 C

 1  cos(ð  C) cos( A  B)  cos 2 C  1  cos C cos( A  B)  cos 2 C

 cos(A  B)  cos C   1  cos C  cos( A  B)  cos( A  B)   1  cos C

 1  2 cos A cos B cos C 2

2

2

eKán cos A  cos B  cos C  1  2 cos A cos B cos C eday A , B , C CamMuRsYcena¼ cosA  0 , cosB  0 , cosC  0 naM[ 1  2 cos A cos B cos C  1 dUcen¼ cos A  cos B  cos C  1 2

2

2

20


lMhat´TI7 eK[RtIekaN ABC mYymanmMukñúgCamMuRsYc . cUrRsayfa sin A  sin B  sin C  2 . dMeNa¼Rsay Rsayfa sin A  sin B  sin C  2 tag T  sin A  sin B  sin C 2

2

2

2

2

2

2

2

2

1  cos2A 1  cos2B  1  cos2 C  2 2 cos2A  cos2B 2  cos2 C 2  2  cos(A  B) cos(A  B)  cos2 C

 2  cosC cos(A  B)  cosC  2  cosC cos(A  B)  cosC

 2  cosC  cos(A  B)  cos(A  B)

 2  2 cos A cosB cosC

eday A , B , C CamMuRsYcena¼ cos A  0 , cos B  0 , cos C  0 naM[ 2  2 cos A cos B cos C  2 dUcen¼ sin A  sin B  sin C  2 2

2

2

21


lMhat´TI8 eK[ cosa x  sinb x  a 1 b Edl a  0 , b  0 , a  b  0 . cos x sin x 1 cUrRsaybBa¢ak´fa a  b  (a  b) . dMeNa¼Rsay 1 cos x sin x RsaybBa¢ak´fa a  b  (a  b) 4

eyIgman eyIgout án

4

10

10

4

4

10

10

4

4

4

4

cos 4 x sin 4 x 1   a b ab

(a  b)(b cos 4 x  a sin 4 x)  ab

ab cos4 x  a2 sin4 x  b2 cos4 x  ab sin4 x  ab  0 a2 sin4 x  b2 cos4 x  ab (sin4 x  cos4 x  1 )  0 a2 sin2 x  b2 cos4 x  ab  (sin2 x  cos2 x)2  2 sin2 x cos2 x  1 a2 sin2 x  b2 cos4 x  2ab sin2 x cos2 x  0 (a sin2 x  b cos2 x)  0

eKTaj eKán

cos 2 x sin 2 x cos 2 x  sin 2 x 1    a b ab ab c o s10 x a cos 2 x 1  (1)  4 5 a (a  b) a ab

naM[

22


10

ehIy

sin x a sin2 x 1  (2)  4 5 b (a  b) b ab cos10 x sin10 x ab 1    a4 b4 (a  b)5 (a  b)4

naM[

bUksmIkareKán lMhat´TI9 ð 2ð 4ð S  sin  sin  sin cUrKNna 7 7 7 dMeNa¼Rsay ð 2ð 4ð S  sin  sin  sin KNna 7 7 7 2

2

2

2

2

2

2ð 4ð 8ð 1  cos 1  cos 7  7  7 S 2 2 2 3 1 2ð 4ð 8ð   ( cos  cos  cos ) 2 2 7 7 7 2ð 4ð 8ð T  cos  cos  cos 7 7 7 5ð 3ð ð  cos (ð  )  cos (ð  )  cos (ð  ) 7 7 7 5ð 3ð ð   cos  cos  cos 7 7 7 1  cos

eyIgán tag

ð 7 ð 5ð ð 3ð ð ð ð 2 T sin  2 cos sin  2 cos sin  2 cos sin 7 7 7 7 7 7 7

KuNGg:TaMgBIrnwg

2 sin

eKán ½

23


tamrUbmnþ

2 cos a sin b  sin( a  b)  sin( a  b)

ð 5ð ð 3ð ð ð ð  2 cos sin  2 cos sin  2 cos sin 7 7 7 7 7 7 7 ð 6ð 4ð 4ð 2ð 2ð 2 T sin   ( sin  sin )  (sin  sin )  sin 7 7 7 7 7 7 ð 6ð ð ð 2 T sin   sin   sin(ð  )   sin 7 7 7 7

2 T sin

eKTaj

1 T 2

3 1 1 7 S   ( )  2 2 2 4 7 2 ð 2 2ð 2 4ð S  sin  sin  sin  7 7 7 4

naM[

dUcen¼ . lMhat´TI10 2ð 4ð 8ð cUrKNna S  sin 7  sin 7  sin 7 dMeNa¼Rsay eKman sin 27ð  sin 57ð , sin 47ð  sin 37ð , sin 87ð   sin 7ð 4ð 2ð ð 8ð ehIy sin 7  sin 7   sin 7 nig sin 7  0 5ð 3ð ð S  sin  sin  sin 0 eKán 7 7 7 elIkGg:TaMBIrCakaereKán ½ S2  sin2

5ð 3ð ð 5ð 3ð 5ð ð 3ð ð  sin2  sin2  2 sin sin  2 sin sin  2 sin sin 7 7 7 7 7 7 7 7 7 24


tag

yk

5ð 3ð ð  sin2  sin 2 7 7 7 10ð 6ð 2ð  cos  cos cos 3 7 7 7   2 2 3ð ð 5ð  3 1 )  cos(ð  )  cos(ð  )     cos(ð  2 2  7 7 7  3 1 3ð ð 5ð )   (  cos  cos  cos 2 2 7 7 7 5ð 3ð ð T   cos  cos  cos 7 7 7 M  sin2

KuNGg:TaMgBIrnwg 2 T sin

ð 2 sin 7

eKán ½

ð 5ð ð 3ð ð ð ð  2 cos sin  2 cos sin  2 cos sin 7 7 7 7 7 7 7

tamrUbmnþ

2 cos a sin b  sin( a  b)  sin( a  b)

ð 5ð ð 3ð ð ð ð  2 cos sin  2 cos sin  2 cos sin 7 7 7 7 7 7 7 ð 6ð 4ð 4ð 2ð 2ð 2 T sin   ( sin  sin )  (sin  sin )  sin 7 7 7 7 7 7 ð 6ð ð ð 2 T sin   sin   sin(ð  )   sin 7 7 7 7 1 3 1 7 T M   2 2 4 4 2 T sin

eKTaj tag N

naM[

2 sin 5ð 3ð sin 7 7

2 sin

5ð ð 3ð ð sin 2 sin sin 7 7 7 7 25


2ð 8ð 4ð 6ð 2ð 4ð  cos  cos  cos  cos  cos 7 7 7 7 7 7 6ð 8ð ð  cos  cos  2 sin ð . sin( )  0 7 7 7  cos

7 7 S  MN 0 4 4 2

eKán

7 S 2

ena¼

eday

S0

.

2ð 4ð 8ð 7 S  sin  sin  sin  7 7 7 2

dUcen¼ lMhat´TI11 cUrRsayfa

.

nð nð n 2 nð 8 cos  4(1) cos  4 cos  (1)n1  0 , n  IN * 7 7 7 3

dMeNa¼Rsay eKman

4nð 3nð  nð  7 7

, n  IN

4nð 3nð sin  sin (nð  ) 7 7

eKán edayeRbIrUbmnþ ½

sin 2a  2 sin a cos a sin( x  y)  sin x cos y  sin y cos x 26


nig

sin 3a  3 sin a  4 sin3 a

2nð 2nð 3nð 3nð cos  sin(nð) cos  sin cos(nð) 7 7 7 7 nð nð nð nð nð 4 sin cos ( 2 cos 2  1 )  0  (3 sin  4 sin3 ) (1)n 7 7 7 7 7 nð nð nð nð nð 4 sin cos (2 cos 2  1 )  (1)n . sin (3  4 sin2 ) 7 7 7 7 7 nð nð nð 8 cos 3  4 cos  (1)n . [ 3  4 ( 1  cos 2 )] 7 7 7 nð nð nð 8 cos 3  4 cos  (1)n .4 cos 2  (1)n 7 7 7 nð nð nð 8 cos 3  4(1)n cos 2  4 cos  (1)n1  0 7 7 7 2 sin

dUcen¼ 8 cos3

nð nð nð  4(1)n cos2  4 cos  (1)n���1  0 7 7 7

, n  IN *

lMhat´TI12 cUrKNna S  cos ð9  cos 49ð  cos 79ð dMeNa¼Rsay KNna S  cos ð9  cos 49ð  cos 79ð tamrUbmnþ cos 3a  4 cos a  3 cos a 3 1 cos a  cos a  cos 3a ¦ 4 4 3

3

3

3

3

3

3

3

27


kenßamEdl[GacsresrCa ½ S

3 ð 4ð 7ð 1 ð 4ð 7ð (cos  cos  cos )  (cos  cos  cos ) 4 9 9 9 4 3 3 3

ð 4ð 7ð  cos  cos 9 9 9 4ð 13ð  cos  cos 9 9 ð 7ð 13ð M  cos  cos  cos 9 9 9

tag eday

M  cos

KuNnwg

ð 2 sin 3

eKán

ð ð ð 7ð ð 13ð ð  2 cos sin  2 cos sin  2 cos sin 3 3 9 3 9 3 9 3 4ð 2ð 10ð 4ð 16ð 10ð 2 .M )  sin  sin  sin(  sin  sin  sin 2 9 9 9 9 9 9 3 2ð 16ð 7ð 2 M )0  sin  sin  2 sin ð cos( 2 9 9 9

2 M sin

eKTaján M  0 tag N  cos ð3  cos 43ð  cos 73ð  21  21  21  32 3 1 3 eKán S  4 M  4 N  8

28


lMhat´TI13 eK[kenßam ð 3ð 5ð  cos 3  cos 3 7 7 7 ð 3ð 5ð T  cos 4  cos 4  cos 4 7 7 7

S  cos 3

nig k¿cUrRsayfabIcMnYn cos 7ð , cos 37ð , cos 57ð Ca¦srbs´smIkar (E) : 8 x  4 x  4 x  1  0 . x¿Tajrktémø ½ 3

2

ð 3ð 5ð  cos  cos 7 7 7 ð 3ð 3ð 5ð ð 5ð cos N  cos cos  cos  cos cos 7 7 7 7 7 7

M  cos

nig P  cos 7ð cos 37ð cos 57ð . K¿KNna Q  cos 7ð  cos 37ð  cos 57ð rYcTajrktémø S nig T . dMeNa¼Rsay k¿RsayfabIcMnYn cos 7ð , cos 37ð , cos 57ð Ca¦srbs´smIkar 2

2

2

(E) : 8 x 3  4 x 2  4 x  1  0

tag

x n  cos

2n  1 ð , n 1, 2 , 3 7

Ca¦smIkar

(E) 29


eKán ½

(2n  1)ð (2n  1)ð (2n  1)ð  4 cos 2  4 cos 1 0 7 7 7 (2n  1)ð (2n  1)ð (2n  1)ð  1 )  1  4 (1  sin 2 )0 4 cos ( 2 cos 2 4 7 7 (2n  1)ð 2(2n  1)ð (2n  1)ð  ( 3  4 sin 2 )0 4 cos cos 7 7 7 (2n  1)ð (2n  1)ð 4(2n  1)ð 2(2n  1)ð  4 sin 3 3 sin sin sin 7 7 7 7   4. (2n  1)ð 2(2n  1)ð (2n  1)ð sin 2 sin 2 sin 7 7 7 4(2n  1)ð 3(2n  1)ð sin sin 7 7  0 (*) (2n  1)ð (2n  1)ð sin sin 7 7

8 cos 3

eday n  IN * : sin (2n 7 1)ð  0 ehtusmIkar ( * ) smmUl ½ 4(2n  1)ð 3(2n  1)ð  sin 0 7 7 (2n  1)ð (2n  1)ð 2 sin cos 0 7 2 sin

00

dUcen¼

cos

epÞógpÞat´ .

ð 3ð 5ð , cos , cos 7 7 7

Ca¦srbs´smIkar

(E)

. 30


x¿Tajrktémø M , N , P snµtfa x  cos 7ð , x  cos 37ð , x tamRTwsþIbTEvütGnutþn_kñúgsmIkar 8x eKán ½ 1

2

3

3

 cos

5ð 7

 4x2  4x  1  0

ð 3ð 5ð b 1  cos  cos  x1  x 2  x 3     2 7 7 7 a ð 3ð 3ð 5ð ð 5ð  cos cos  cos cos N  cos cos 7 7 7 7 7 7 c 1  x 1x 2  x 2 x 3  x 1x 3    a 2

M  cos

P  cos

nig dUcen¼

ð 3ð 5ð d 1 cos cos  x 1x 2 x 3     7 7 7 a 8

.

ð 3ð 5ð 1  cos  cos  7 7 7 2 ð 3ð 3ð 5ð ð 5ð 1 cos N  cos cos  cos  cos cos  7 7 7 7 7 7 2 M  cos

ð 3ð 5ð 1 cos cos  7 7 7 8 2 ð 2 3ð 2 5ð Q  cos  cos  cos 7 7 7

P  cos

nig K¿KNna eyIgán Q  x

.

2 1

 x 2 2  x 32

 ( x1  x 2  x 3 )2  2( x1x 2  x 2 x 3  x1x 3 )  M 2  2N 

1 1 5  2 ( )  4 2 4

31


Q  cos 2

ð 3ð 5ð 5  cos 2  cos 2  7 7 7 4

dUcen¼ . Tajrktémø S nig T eyIgán S  cos 7ð  cos 37ð  cos 57ð ¦ Sx x x ð 3ð 5ð eday x  cos 7 , x  cos 7 , x  cos 7 Ca¦srbs´ 3

3 1

3

3

(E )

3

2

3

 8x 13  4 x12  4 x 1  1  0 (1)  3 2  8x 2  4 x 2  4 x 2  1  0 (2)  3 2  8x 3  4 x 3  4 x 3  1  0 (3)

ena¼eKán

bUksmIkar

3

2

1

3

(1) , (2) , (3)

Gg:nwgGg:eKán ½

8 ( x13  x 2 3  x 33 )  4(x12  x 2 2  x 32 )  4(x1  x 2  x 3 )  3  0 8 S  4Q  4 M  3  0

5 1  QM 3 4 2 3 7 3 1 S       2 8 2 8 8 8 2 ð 3ð 5ð 1 S  cos 3  cos 3  cos 3  7 7 7 2

eKTaj . dUcen¼ ð 3ð 5ð T  cos  cos  cos x x x müa¨eTot 7 7 7 edayKuNsmIkar (1) , (2) , (3) erogKñanwg x , x , x 4

4

4

4 1

4 2

1

4 3

2

3

32


eKán

 8 x14  4 x13  4 x12  x1  0 (1' )  4 3 2  8 x 2  4 x 2  4 x 2  x 2  0 (2' )  4 3 2  8 x 3  4 x 3  4 x 3  x 2  0 (3' )

bUksmIkar

(1' ) , (2' ) , (3' )

Gg:nwgGg:eKán ½

8 ( x14  x 2 4  x 3 4 )  4 ( x13  x 2 3  x 33 )  4 ( x12  x 22  x 3 2 )  ( x1  x 2  x 8 T 4 S 4 QM0

eKTaj T  S 2 Q  M8  78  81  34 . ð 3ð 5ð 3 T  cos  cos  cos  . dUcen¼ 7 7 7 4 lMhat´TI14 eda¼RsaysmIkar ½ 64 x  112 x  56x  7  2 1  x . dMeNa¼Rsay eda¼RsaysmIkar 4

6

4

4

2

4

2

64 x 6  112 x 4  56x 2  7  2 1  x 2

(1)

2

lk&çx&NÐ 1  x  0 ¦ x  [1 , 1 ] eyIgman cos 4a  cos(a  3a) 33


 cos a cos 3a  sin a sin 3a

 cos a(4 cos 3 a  3 cos a)  sin a(3 sin a  4 sin3 a)  4 cos 4 a  3 cos 2 a  3 sin 2 a  4 sin 4 a  4 cos 4 a  4(1  cos 2 a) 2  3(cos 2 a  sin 2 a)  8 cos 4 a  8 cos 2 a  1 cos 5a  cos(a  4a)  cos a cos 4a  sin a sin 4a  cos a(8 cos4 a  8 cos2 a  1)  2 sin a sin2a cos 2a  8 cos5 a  8 cos3 a  cos a  4 sin2 a cos a(2 cos2 a  1)  8 cos5 a  8 cos3 a  cos a  4(1  cos2 a)(2 cos3 a  cos a)  16 cos5  20 cos3 a  5 cos a cos 6a  2 cos 2 3a  1  2(4 cos 3 a  3 cos a) 2  1  32 cos 6 a  48 cos 4 a  18 cos 2 a  1 cos 7a  cos(6a  a)  cos 6a cos a  sin 6a sin a  cos a cos 6a  2 sin a sin 3a cos 3a  cos a cos 6a  2 sin a(3 sin a  4 sin 3 a)(4 cos 3 a  3 cos a)  64 cos 7 a  112 cos 5 a  56 cos 3 a  7 cos a

yk

x  cos t

Edl

t[ 0 , ð ]

smIkar (1) sresr ½

64 cos 6 t  112 cos 4 t  56 cos 2 t  7  2 1  cos 2 t 64 cos 6 t  112 cos 4 t  56 cos 2 t  7  2 sin t

KuNGg:TaMgBIrnwg

cos t  0

eKán ½

64 cos 7 t  112 cos 5 t  56 cos 3 t  7 cos t  2 sin t cos t cos 7 t  sin 2t cos 7 t  cos(

ð  2 t) 2

34


eKTaj smmUl eday t{

ð  7 t   2t  2kð  2  7 t   ð  2t  2k' ð , k; k'  Z  2 ð 2kð  t    9 18   t   ð  2k' ð , k ; k'  Z  10 9

t  [ 0 , ð]

eKTajsMNMutémø

t

dUcxageRkam ½

ð 5ð 9ð 13ð 17ð 3ð 7ð ; ; ; ; ; ; } 18 18 18 18 18 10 10 ð cos t  0 t 2

eday ena¼ dUcen¼smIkar (1) mansMNMu¦sdUcxageRkam ½ x  { cos

ð 5ð 13ð 17 ð 3ð 7ð ; cos ; cos ; cos ; cos ; cos } 18 18 18 18 10 10

35


lMhat´TI15 eda¼RsaysmIkar ½ tan6 x  (tan2 x  1)3  (tan2 x  2)3  (tan2 x  3)3

dMeNa¼Rsay eda¼RsaysmIkar ½ tan6 x  (tan2 x  1)3  (tan2 x  2)3  (tan2 x  3)3

¿

lk&çx&NÐ x  ð2  kð , k  Z . tag t  tan x , t  0 smIkarsresr ½ 2

t 3  (t  1)3  (t  2)3  (t  3) 3 t 3  (t 3  3t 2  3t  1)  (t 3  6t 2  12 t  8)  t 3  9t 2  27 t  27 3t 3  9t 2  15t  9  t 3  9t 2  27 t  27 2(t 3  6t  9)

0

(t 3  27)  (6t  18)  0 (t  3)( t 2  3t  9)  6(t  3)  0 (t  3)( t 2  3t  3)  0

eKTaj

t3

nig

t2  3t  3  0

  9  12  0 cMeBa¼ t  3 eKán

Kµan¦seRBa¼

tan2 x  3 36


tan2 x  3  0 (tan x  3)(tan x  3)  0

eKán tan x  3  0 ¦ tan x  3 naM[ ehIy tan x  3  0 ¦ tan x   3 naM[ x   ð3  kð , k  Z

x

ð  kð , k  Z 3

lMhat´TI16 eKman á CaFñÚrmYyEdlKitCara¨düg´ehIyepÞógpÞat´ ð 0  á  . 2 eK[Exßekag (P) smIkar y  x  2x cos á  1  sin á 1¿cUrkMnt´témø  edIm,I[Exßekag (P) b¨¼nwgGk&ßGab´sIus (x' ox) rYcsg´Exßekag (P) TaMgena¼ . 2¿bgHajfaeRkABIkrNIkñúgsMNYrTI1 Exßekag (P) kat´Gk&ß Gab´sIus (x' ox) ánBIrcMnuc M' nig M' ' Edlman Gab´sIusviC¢man . 3¿etIeKRtUv[témø á b¨unµanxø¼eTIbGab´sIus x' nig x' ' éncMnuc M' nig M' ' epÞógpÞat´TMnak´TMng x'  x' '  2 4¿rkTMnak´TMngKµanGaRs&ynwg  rvagGab´sIus x' nig x' ' . 2

2

2

37


dMeNa¼Rsay 1¿ Exßekag (P) b¨¼nwgGk&ßGab´sIus (x' ox) ½ kUGredaenkMBUléná¨ra¨bUl (P) KW x   2ba  cos á ehIy y  cos á  2 cos á  1  sin á S

2

2

S

 1  cos2 á  sin á  sin2 á  sin á

Exßekag (P) b¨¼nwgGk&ßGab´sIus (x' ox) kalNa y  0 eKán sin á  sin á  0 ¦ sin á (sin á  1)  0 naM[ sin á  0 nig sin á  1 ð ð 0  á  á0 , á eday ehtu e n¼eKTaj 2 2 sg´Exßekag (P) ½ -ebI á  0 eKán y  x  2x  1  ( x  1) ð á  -ebI 2 eKán y  x S

2

2

2

2

38


y 4

3

2

1

-2

-1

0

1

2

3

4

x

-1

2¿ Gab´sIuséncMnuc M' nig M' ' Gab´sIuséncMnuc M' nig M ' ' KWCa¦srbs´smIkar ½ x 2  2 x cos á  1  sin á  0

DIsRKImINg´bRgYménsmIkarKW

'  cos2 á  1  sin á

 sin á  sin2 á  sin á (1  sin á)

eKman

sin 

nig

ð   á 0 , 2  

1  sin 

viC¢manCanic©RKb´

. 39


eKTaján '  sin á (1  sin á)  0 naM[ (P) kat´Gk&ß Gab´sIusCanic©Rtg´BIrcMnuc M ' nig M ' ' . müa¨geTotplKuN nigplbUkén¦s P  1  sin  nig S  2 cos  .    suTæEtviC¢manRKb´    0 , 2  dUcen¼ x ' nig x ' ' suTæEtviC¢man . 3¿ lk&çx&NÐ x'  x' '  2 eKman x'  x' '  S  2P 2

2

2

2

2

 4 cos2 á  2(1  sin á)  2(2 cos2  1  sin á)  2(2  2 sin2 á  1  sin á)  2(2 sin2 á  sin á  1)

eday

x '2  x ' ' 2  2

eKTaján

2(2 sin2 á  sin á  1)  2  2 sin2  sin á  1

1

 2 sin2 á  sin á  0 sin á(1  2 sin á)  0 ð 0á 2

ð á0 , á 6

eday ehtuen¼eKTaj 4¿TMnak´TMngKµanGaRs&ynwg á rvagGab´sIus eKman S  2 cos á nig P  1  sin á

x'

nig

x' '

40


eKTaján cos á  2S nig edayRKb´cMnYnBit  eKman eKán S4  (1  P)  1

sin á  1  P

cos 2 á  sin2 á  1

2

2

S 2  4(1  P)2  4 S 2  4P2  8P  0 S 2  4P(P  2)  0 2

dUcen¼ (x' x' ' )  4 x' x' ' (x' x' '2)  0 lMhat´TI17 eKmansmIkardWeRkTIBIr ½  1 2 0    (E ) : x  (  2) x  1  0 Edl cos  2 3 eK«bmafasmIkar (E) man¦sBIrEdltageday tan a nig tan b .   k¿ kMnt´témø edIm,I[ a  b  4 . x¿ eda¼RsaysmIkar (E) cMeBa¼témø  EdlánrkeXIj 2

K¿ eRbIlTæplxagelIcUrTajrktémø®ákdén

 tan 12

. 41


dMeNa¼Rsay ab

 4

k¿ kMnt´témø  edIm,I[ eday tan a nig tan b Ca¦srbs´ (E) ena¼eKmanTMnak´TMng 2 1  1 (2) tan a  tan b  2  (1) nig tan a tan b  cos  3 tan a  tan b tamrUbmnþ tan(a  b)  1  tan a tan b (3) ykTMnak´TMng (1) nig (2) CMnYskñúg (3) eKán ½ 1 3 ( 2 cos   1) cos   tan(a  b)  2 1 (  1) (2 3  2) cos  3 2

eKTaján

ab

3( 2 cos   1)  1 (2 3  2) cos  2 3 cos   3  2 3 cos   2 cos  cos  

eday

eday

 0    2

3 2

dUcen¼eKTaj



 6

.

42

 4


x¿ eda¼RsaysmIkar (E) ½ ebI   6 ena¼ (E) Gacsresr

2 2  2) x  1 0 3 3 2 2 4 8 8 28 16 3  (  2) 2  4(  1)   4 4  3 3 3 3 3 3 3 4(7  4 3) 4(2  3)2   3 3  1 2 42 3 1 x ( 2 )       1 2 3 3 3   1 2 42 3 x  (   2  )2 3  2 2 3 3  1 x1  , x2  2  3 3  tan 12

x2  (

eKTaj¦s

dUcen¼ . K¿ eRbIlTæplxagelITajrktémø®ákdén tamsRmayxagelIeKman eKTaj eday naM[

1 tan a  3

1 tan a  3   b  a  4 12

nig

naM[

1 x1  , x2  2  3 3

tan b  2  3  a 6

dUcen¼

ehIy tan

 ab 4

  2 3 12

. 43


lMhat´TI18 eK[ f(x)  a sin x  b cos x  c  a cos x  b sin x  c ab cUrRsayfa a  c  b  c  f(x)  2 2  c rYcbBa¢ak´témøGtibrma nig Gb,brmaén f (x) . dMeNa¼Rsay Rsayfa a  c  b  c  f(x)  2 a 2 b  c eyIgman f(x)  a sin x  b cos x  c  a cos x  b sin x  c (1) eday a , b , c CabIcMnYnBitviC¢manena¼ x  IR : f(x)  0 elIkGg:TaMgBIrén (1) CakaereKán ½ f (x)   a sin x  b cos x  c  a cos x  b sin x  c  2

2

2

2

2

2

2

2

2

2

2

2

2

2

f 2 (x)  a  b  2c  2 (a sin2 x  b cos2 x  c)(a cos2 x  b sin2 x  c) (2)

tamvismPaBkUsiuRKb´cMnYnBit A , B  0 eKman A  B  2 A.B ¦ 2 A.B  A  B eKán 2 (asin x  bcos x  c)(acos x  bsin x  c)  tamTMnak´TMng 2 eKTaján ½ 2

2

2

f 2 ( x)  a  b  2c  a  b  2c  4(

2

a  b  2c

ab  c) 2 44


f( x)  2

ab c 2

naM[ yk P(x)  (a sin

2

(3)

x  b cos 2 x  c)(a cos 2 x  b sin 2 x  c)

P( x)  [ a(1  cos 2 x)  b cos 2 x  c] [a cos2 x  b(1  cos2 x)  c] P( x)  [(a  c)  (b  a) cos 2 x] [ (b  c)  (b  a) cos 2 x] P( x)  (a  c)(b  c)  (b  a)2 cos 2 x  (b  a) 2 cos 4 x P( x)  (a  c)(b  c)  (b  a)2 cos 2 x sin2 x 2

2

2

eyIgman (b  a) sin x cos x  0 , x  IR eKTaján P(x)  (a  c)(b  c) , x  IR TMnak´TMng (2) eKGacsresr ½ f 2 ( x)  a  b  2c  2 P( x)  a  b  2c  2 (a  c)(b  c) f 2 ( x)  (a  c)  (b  c)  2 (a  c)(b  c) f 2 ( x)  ( a  c  b  c ) 2

eKTaj f(x)  tamTMnak´TMng

a  c  b  c (4) (3)

nig

(4)

eKTaján ½ ab  c cMeBa¼RKb´ x  IR . a  c  b  c  f( x)  2 2 dUcGnuKmn_mantémøGtibrmaesµI M  2 a 2 b  c nigmantémøGb,brmaesµI m  a  c  b  c . 45


lMhat´TI19 rktémøGb,brmaénGnuKmn_ ½ P( x)  tan2 x  cot 2 x  2(tan x  cot x)  27 2

2

Q ( x)  tan x  cot x  8(tan x  cot x)  87

0 x 

ð 2

.

dMeNa¼Rsay rktémøGb,brmaénGnuKmn_ ½ P( x)  tan2 x  cot 2 x  2(tan x  cot x)  27 ð 0 x  2

Edl tag z  tan x  cot x Edl z  2 eKán z  (tan x  cot x)  tan x  cot x  2 eKTaj tan x  cot x  z  2 eyIgán P(z)  z  2  2z  27  (z  1)  24 eday z  2 ehtuen¼eKán P(z)  1  24  25 dUcen¼témøGb,brmaén P(x) KW m  25 . müa¨geToteday Q(x)  tan x  cot x  8(tan x  cot x)  87 eKán Q(z)  z  2  8z  87  (z  4)  69 eday z  2 ehtuen¼edIm,I[ Q Gb,brmalu¼RtaEt 2

2

2

2

2

2

2

2

2

2

2

2

2

46


z4

.dUcen¼témøGb,brmaén lMhat´TI20 eda¼RsaysmIkar ½ 4 sin ( x 

ð ð ) cos ( x  )  4 12

Q ( x)

KW

m  69

.

32 2

dMeNa¼Rsay eda¼RsaysmIkar ½ 4 sin ( x 

ð ð ) cos ( x  )  4 12

32 2

tamrUbmnþ 2 sin a cos b  sin(a  b)  sin(a  b) smIkarxagelIGacsresrCabnþbnÞab´xageRkam ½ ð ð ð ð   2  sin( x   x  )  sin( x   x  )   3  2 2 4 12 4 12   ð ð  2  sin(2 x  )  sin   (1  2 ) 2 3 6  ð 2 sin(2 x  )  1  1  2 3 ð 2 sin(2 x  )  3 2 ð ð  2 x    2kð  3 4   2 x  ð  ð  ð  2kð , kZ  3 4 ð 5ð x  kð , x   kð ; k  Z 24 24

eKTaj dUcen¼

.

47


lMhat´TI21 eKmanGnuKmn_

f( x) 

1   2  sin x   2 sin x  

2

1     cos 2 x   2 cos x  

cUrrktémøtUcbMputénGnuKmn_en¼ . dMeNa¼Rsay rktémøtUcbMputénGnuKmn_en¼ 1 1 2 2 2 )  (sin x  ) cos 2 x sin2 x 1 1 4    cos 4 x  2  sin x 2 cos 4 x sin 4 x

f( x)  (cos 2 x  

4  (cos 4 x  sin 4 x)  (

cos 4 x  sin 4 x 4  (cos x  sin x)  ( ) sin 4 x cos 4 x

4  (cos 4 x  sin 4 x)(1 

4  (cos 2 x  sin 2 x) 2  2 sin 2 x cos 2 x (1 

4  (1 

4

4

edayeKman 1

1 1  ) sin 4 x cos 4 x

1 ) sin 4 x cos 4 x

1 16 sin 2 2 x)(1  ) 4 2 sin 2 x 1 1 sin 2 2 x  1 1  sin 2 2 x  2 2

16  17 sin 4 2 x

naM[

16 ) 4 sin 2 x

nig

. 48

2


eKTaj 4  (1  21 sin 2x)(1  sin162x )  4  172  252 eyIgán f(x)  4  (1  21 sin 2x)(1  sin162x )  5 22 dUcen¼témøtUcbMputénGnuKmn_KW m  5 22 . lMhat´TI22 eK[BIrcMnYnBit a nig b . cUrRsaybBa¢k´fa x  IR : | a cos x  b sin x |  a  b Gnuvtþn_ ½ rktémøGtibrma nig Gb,brmaén 2

4

2

4

2

2

f( x)  20 cos x  21sin x  22

dMeNa¼Rsay RsaybBa¢k´fa x  IR : | a cos x  b sin x |  a  b eyIgeRCIserIsvuicTr& U ( a ; b ) nig V ( cos x ; sin x ) tamniymn&y U . V  || U||.|| V ||.cos è Edl è CamMurvag BIrviucT&ren¼ . eKán U. V  ||U||.|| V ||.cos è  ||U||.|| V ||| cos è| edayeKman è  IR : | cos è |  1 eKán U . V  || U||.|| V || 2

 

2

49


eday

   U  . V  a cos x  b sin x   2 2  || U|| a  b    || V || sin 2 x  cos 2 x  1  2

2

dUcen¼ x  IR : | a cos x  b sin x |  a  b . rktémøGtibrma nig Gb,brmaén f(x)  20 cos x  21sin x  22 tamrUbmnþxagelIeyIgman | 20 cos x  21sin x | 

20 2  212  29

eKTaj  29  20 cos x  21sin x  29 naM[  7  f(x)  51 , x  IR dUcen¼GnuKmn_mantémøGtibrma 51 nig Gb,brma

7

50


lMhat´TI23 f( x) CatémøBiténGnuKmn_ f EdlcMeBa¼RKb´ x  IR eKman f(x)  2f( x)  3 cos x  sin x . cUrRsayfa f(x)  2 cMeBa¼RKb´ x  IR dMeNa¼Rsay Rsayfa f(x)  2 cMeBa¼RKb´ x  IR eKman f(x)  2f( x)  3 cos x  sin x (1) edayCMnYs x eday  x kñúgTMnak´TMng (1) eKán f( x)  2 f( x)  3 cos x  sin x

(2)

eyIgánRbB&næsmIkar ½  f( x)  2f( x)  3 cos x  sin x 1  2f( x)  f( x)  3 cos x  sin x  2    3f( x)

eKTaján

 3 cos x  3 sin x f( x)  cos x  sin x

2 2 cos x  sin x) 2 2 ð ð  2 (sin cos x  sin x cos ) 4 4 ð  2 sin(  x) 4

f ( x)  2 (

51


eday x  IR : sin( ð4  x)  1 . dUcen¼ f(x)  2 cMeBa¼RKb´ x  IR . lMhat´TI24 eKman f(x) GnuKmn_kMnt´elI IR eday ½ f(sin x)  3f(cos x)  2  cos 2 x ¿ k-cUrkMnt´rkGnuKmn_ f(x) . x-eda¼RsaysmIkar f(1  tan t).f(1  tan t) 

f(1  tan t)  f(1  tan t) 2

( t CaGBaØténsmIkar ) . dMeNa¼Rsay k-kMnt´rkGnuKmn_ f(x) CMnYs x eday ð2  x eKán f(cos x)  3f(sin x)  2  cos 2x  f(sin x)  3f(cos x)  2  cos 2 x eyIgánRbB&næ 3f(sin x)  f(cos x)  2  cos 2x 2x bMát´ f(sin x) eKTaján f(cos x)  1  cos  cos x 2 dUcen¼ f(x)  x . 2

2

52


x-eda¼RsaysmIkar f(1  tan t).f(1  tan t) 

f(1  tan t)  f(1  tan t) 2

ð  kð ; k  Z 2 (1  tan t) 2  (1  tan t) 2 2 2 (1  tan t) (1  tan t)  2

lk&çx&NÐ

t

dMeNa¼RsaysmIkaren¼eKáncMelIy ð t  kð ; t    kð , k  Z . 3 lMhat´TI25 eK[Exßekag (P) : y  f (x)  x sin   2(1  sin )x  5  sin  Edl 0     . kMnt´témø  edIm,I[Exßekag (P) sSitenAelIGkß& Gab´sIusCanic© . dMeNa¼Rsay kMnt´témø  eKman (P) : y  f (x)  x sin   2(1  sin )x  5  sin  edIm,I[ (P) sSitenAelIGkß&Gab´sIusCanic©lu¼RtaEt 2

2

f ( x )  0 , x  IR 53


a f  0 eBalKWeKRtUv[ '  0 eKman a f  sin   0 ;  ] 0 ; [

ehIy

 '  (1  sin ) 2  sin (5  sin )

'  1  2 sin   sin 2   5 sin   sin 2   '  2 sin 2   3 sin   1

ebI

 '  (2 sin   1)(sin   1) 1  sin   1 '  0 2      6 2

smmUl

eday

eKTaján dUcen¼edIm,I[Exßekag

. (P) sSitenAelIGkß&Gab´sIus

Canic©lu¼RtaEteK[l&kçx&NÐ

     6 2

0

.

54


lMhat´TI26 eda¼RsaysmIkar ½ 1 1 9 3 1  cos 4 x  cos 2 x   cos 4 x  cos 2 x  16 2 16 2 2

dMeNa¼Rsay eda¼RsaysmIkar ½ 1 1 9 3 1  cos 4 x  cos 2 x   cos 4 x  cos 2 x  (1) 16 2 16 2 2

smIkar

1

Gacsresr ½ 2

2

1 3 1   2 2 cos cos x x         4 4 2     1 3 1 2 2 cos x   cos x   4 4 2 t  cos 2 x

tag t

1 4

 t

Edl

3 1  4 2

0 t 1

(2)

smIkar

(2)

Gacsresr

(3)

1 3 M( t) , A( ) , B( ) 4 4

elIGkß& (x' ox) eRCIserIscMnuc tam 3 eKán MA  MB  21 eday AB  21 eKán MA  MB  AB naM[ M enAkñúg AB eKTaj 41  t  34 smmUl 41  cos x  34 2

55


1 3  | cos x |  2 2 ð ð êð x     kð 6 3    ð  k' ð  x   ð  k ' ð , k ; k '  Z  3 6

smmUl

eKTaj

lMhat´TI27 eK[RtIekaN ABC mYy . k¿ cUrRsayfa cot A cot B  cot B cot C  cot C cot A  1 . x¿ cUrRsayfa cot A  1  2 cot 2A cot A . K¿ eKdwgfamMu A ; B ; C beg;ItánCasIVútFrNImaRt mYYyEdlmanersugesµInwg q  2 . cUrRsaybBa¢ak´fa ½ 1 1 1   8 . sin A sin B sin C 2

2

2

2

dMeNa¼Rsay k¿Rsayfa cot A cot B  cot B cot C  cot C cot A  1 eyIgman A  B  C  ð ¦ A  B  ð  C eKán tan(A  B)  tan(ð  C) 56


tan A  tan B   tan C 1  tan A tan B 1 1  cot A cot B   1 1 1 cot C 1 . cot A cot B cot A  cot B 1  cot A cot B  1 cot C cot A cot C  cot B cot C   cot A cot B  1

dUcen¼ cot A cot B  cot B cot C  cot C cot A  1 . x¿Rsayfa cot A  1  2 cot 2A cot A A eyIgman tan2A  1 2 tan tan A 2

2

2 2 cot A 1  cot A  2 1 cot 2 A cot A 1 1 2 cot A 2

dUcen¼ cot A  1  2 cot 2A cot A . K¿RsaybBa¢ak´fa ½ sin1 A  sin1 B  sin1 C  8 tag T  sin1 A  sin1 B  sin1 C 2

2

2

2

2

2

 (1  cot 2 A)  (1  cot 2 B)  (1  cot 2 C)  (cot 2 A  1)  (cot 2 B  1)  (cot 2 C  1)  6  2 cot 2 A cot A  2 cot B cot 2B  2 cot 2C cot C  6 (1)

edaymMu A ; B ; C CasIVútFrNImaRtmYYyEdlmanersug esµInwg q  2 eKán B  2A , C  2B  4A

57


eday A  B  C  ð eKán A  2A  4 A  ð naM[ A  7ð , B  27ð , C  47ð tam (1) eKán T  2 cot 27ð cot 7ð  2 cot 27ð cot 47ð  cot 87ð cot 47ð  6 eday cot 87ð  cot 7ð eKán ½ ð 2ð 2ð 4ð ð 4ð cot  2 cot cot  2 cot cot 6 7 7 7 7 7 7  2(cot A cot B  cot B cot C  cot A cot C)  6  2(1)  6  8

T  2 cot

dUcen¼

1 1 1   8 sin2 A sin2 B sin2 C

.

58


lMhat´TI28 1¿cUrRsaybBa¢ak´rUbmnþ ½ cos(n  1)x  2 cos x cos(nx)  cos(n  1) x

2¿Gnuvtþn_ ½ cUrsresr 3¿eda¼RsaysmIkar ½

cos 7 x

, x  IR , n  IN *

CaGnuKmn_én

cos x

¿

.

128 cos 7 x  244 cos 5 x  112 cos 3 x  14 cos x  1  0

.

dMeNa¼Rsay 1¿RsaybBa¢ak´rUbmnþ ½ cos(n  1)x  2 cos x cos(nx)  cos(n  1) x

eyIgman ½ cos(n  1)x  cos(n  1)x  2 cos

(n  1)x  (n  1)x (n  1)x  (n  1)x . cos 2 2

smmUl cos(n  1)x  cos(n  1)x  2 cos x.cos(nx) dUcen¼ cos(n  1)x  2 cos x cos(nx)  cos(n  1)x ¿. 2¿ Gnuvtþn_ ½sresr cos 7x CaGnuKmn_én cos x eKman cos(n  1)x  2 cos x cos(nx)  cos(n  1)x ¿ ebI n  1 cos 2x  2 cos x  1 2

59


ebI

n2

cos 3x  2 cos x cos 2 x  cos x

 2 cos x(2 cos 2 x  1)  cos x  4 cos 3 x  3 cos x

ebI

n3

cos 4 x  2 cos x cos 3x  cos 2 x

 2 cos x(4 cos 3 x  3 cos x)  (2 cos 2 x  1)  8 cos 4 x  8 cos 2 x  1

ebI

n4

cos 5x  2 cos x cos 4 x  cos 3x

 2 cos x(8 cos 4 x  8 cos 2 x  1)  (4 cos 3 x  3 cos x)  16 cos5 x  20 cos 3 x  5 cos x

ebI

n5

cos 6x  2 cos x cos 5 x  cos 4 x

 2cosx(16cos5 x  20cos3 x  5cosx)  (8cos4 x  8cos2 x  1)  32cos6 x  48cos4 x  18cos2 x 1

ebI n  6 cos 7 x  2 cos x cos 6x  cos 5x dUcen¼ cos 7x  64 cos x  112 cos x  56 cos 3¿ eda¼RsaysmIkar ½ 7

5

3

x  7 cos x

128 cos7 x  244 cos5 x  112 cos3 x  14 cos x  1  0 (1)

EckGg:TaMgBIrénsmIkar

(1)

nwg

2

¿

eKán ½ 1 0 2 1 cos 7 x  2

64 cos 7 x  112 cos 5 x  56 cos 3 x  7 cos x 

60

¿


eKTaján 7x   ð3  2kð ¦ x   21ð  27kð , k  Z dUcen¼ x  21ð  27kð , x   21ð  2k7' ð ; k;k' Z . lMhat´TI29 eK[sVúIténcMnYnBit (U ) kMnt´elI n eday½ ð U  1 nig n  IN : U  U cos a  sin a Edl 0  a  . 2 k¿tag V  U  cot 2a . cUrbgHajfa (V ) CasIVútFrNImaRtmYYy . x¿KNnalImIt lim (V  V  ....  V ) nig lim U . dMeNa¼Rsay k¿ bgHajfa (V ) CasIVútFrNImaRtmYYy ½ man V  U  cot 2a naM[ V  U  cot 2a Et U  U cos a  sin a eKán V  U cos a  sin a  cot 2a n

n 1

0

n

n

n

n

n 

0

1

n

n 

n

n

n

n

n 1

n

n 1

n 1

n 1

n

61


a a a  Un cos a  2 sin cos  cot 2 2 2 a a a a  Un cos a  cot (2 sin cos tan  1) 2 2 2 2 a sin a a a 2  1)  Un cos a  cot (2 sin cos a 2 2 2 cos 2 a a a ; cos a  1  2 sin 2  Un cos a  cot (2 sin 2  1) 2 2 2 a a  Un cos a  cot cos a  (Un  cot ) cos a 2 2  Vn cos a

eday V  V cos a naM[ (V ) CasIVútFrNImaRt mYymanersug cos a nig tY V  U  cot 2a  1  cot 2a . x¿ KNnalImIt ½ lim (V  V  ....  V ) nig lim U eyIgman V  V  ...  V  V 11qq  (1  cot 2a).11coscos aa eyIgán lim (V  V  ....  V )  lim (1  cot 2a) 11coscos aa  eday 0  a  2 ena¼ 0  cos a  1 nig lim cos a  0 n 1

n

n

0

n 

0

0

1

n

n 

n1

0

1

n

n

n 1

0

n 1

n 

0

1

n

n 

n 1

n 

a 2 lim ( V0  V1  ....  Vn )  n   1  cos a 1  cot

dUcen¼ müa¨geTot

Vn  Un  cot

a 2

naM[

Un  Vn  cot

. a 2 62


eday V  V  q  (1  cot 2a) cos a eKán U  (1  cot 2a) cos a  cot 2a nig lim U  lim (1  cot 2a) cos a  cot 2a   cot 2a eRBa¼ lim cos a  0 . dUcen¼ lim U  cot 2a . n

n

n

0

n

n

n

n

n 

n 

n

n 

n

n 

lMhat´TI30 eK[sIVúténcMnYnBit (U ) kMnt´eday ½ U  0 ; U  1 nig n  IN : U  2U cos a  U Edl a  IR k¿ tag Z  U  (cos a  i sin a) U , n  IN . cUrbgHajfa Z  (cos a  i sin a) Z rYcTajrk Z CaGnuKmn_ n nig a . x¿ Tajrk U CaGnuKmn_én n . dMeNa¼Rsay k¿ bgHajfa Z  (cos a  i sin a) Z eyIgman Z  U  (cos a  i sin a)U n

0

n 2

1

n 1

n

n 1

n 1

n

n

n

n

n

n 1

n

n1

n

n

63


Z n1  Un 2  (cos a  i sin a)Un1

eyIgán

 2Un1 cos a  Un  (cos a  i sin a)Un1  (cos a  i sin a)Un1  Un Un ) cos a  i sin a  (cos a  i sin a)Un1  (cos a  i sin a)Un  (cos a  i sin a) Un  (cos a  i sin a)(Un1 

dUcen¼ Z  (cos a  i sin a) Z . KNna Z CaGnuKmn_én n nig a ½ eday Z  (cos a  i sin a) Z naM[ (Z ) CasIVútFrNImaRt éncMnYnkMupøicEdlmanersug q  cos a  i sin a nig tY Z  U  (cos a  i sin a)U  1 . tamrUbmnþ Z  Z  q  (cos a  i sin a)  cos(na)  i sin(na) dUcen¼ Z  cos(na)  i. sin(na) . x¿ Tajrk U CaGnuKmn_én n ½ eyIgman Z  U  (cos a  i sin a)U (1) nig Z  U  (cos a  i sin a)U (2) dksmIkar (1) nig (2) Gg:nwgGg:eKán ½ Z Z Z  Z  2i sin a U naM[ U  Edl sin a  0 . 2i sin a n 1

n

n

n 1

0

n

n

1

0

n

n

n

0

n

n

n 1

n

n

n

n 1

n

n

n

n

n

n

n

64


eday Z  cos(na)  i sin(na) nig na) . dUcen¼ U  sin( sin a lMhat´TI31 eK[

Zn  cos(na)  i sin(na)

n

n

2  2 cos

ð , 2 2

2  2  2 cos

ð ð , 2  2  2  2 cos 23 24

BI«TahrN_xagelIcUrrkrUbmnþTUeTA nig RsaybBaØak´ rUbmnþena¼pg . dMeNa¼Rsay rkrUbmnþTUeTA ½ eKman 2  2 cos

ð , 2 2

2  2  2 cos

ð ð , 2  2  2  2 cos 23 24

tamlMnaM«TahrN_eyIgGacTajrkrUbmnþTUeTAdUcxageRkam ½ ð 2  2  2  .........  2  2 cos .    2 n 1

(n)

RsaybBaØak´rUbmnþen¼ ½ 65


eyIgtag

A n  2  2  2  ......  2   (n)

cMeBa¼RKb´

n  IN *

ð A1  2  2 cos 2 2

eyIgman Bit eyIg«bmafavaBitdl´tYTI p KW ð A p  2  2  2  ......  2  2 cos p 1   2

Bit

(p)

A p 1  2 cos

ð

Bit eyIgnwgRsayfavaBitdl´tYTI p  1 KW 2 ð A  2 cos  A 2  A edaytamkar«bma eyIgman 2 eyIgán A  2  2 cos 2ð  4 cos 2ð  2 cos 2ð Bit p 1

p2

p

p

p 1

2

p 1

dUcen¼

p 1

p 2

p 2

ð 2  2  2  .........  2  2 cos n  1  2

.

( n )

66


lMhat´TI32 eK[sIVúténcMnYnkMupøic

(Zn )

kMnt´eday ½

 1 i 3 Z   0 2  Z  1  Z  | Z |  ; n  IN n n  n1 2

( | Z | Cam¨UDulén Z ) . snµtfa Z  ñ (cos è  i. sin è ) , n  IN Edl ñ  0 , ñ ; è  IR k-cUrrkTMnak´TMngrvag è nig è ehIy ñ nig ñ . x-rkRbePTénsIVút (è ) rYcKNna è CaGnuKmn_én n . K-cUrbgHajfa ñ  ñ cos è cos è2 cos è2 ....cos è2 rYcbBa¢ak´ ñ CaGnuKmn_én n . n

n

n

n

n

n

n

n

n

n1

n

n

n

1

n

n 1

n

0

n 1

2

0

n

dMeNa¼Rsay k-rkTMnak´TMngrvag è nig è ehIy eyIgman Z  ñ (cos è  i. sin è ) naM[ Z  ñ (cos è  i sin è ) n1

n

n

n1

n

n1

n

n 1

ñn

nig

ñ n1

n

n 1

67


eday Z eKán ñ

n1

1 ( Zn  | Zn | ) 2

ehIy | Z |  ñ n

n

1 ñ n (cos è n  i.sin è n )  ñ n  2 1 ñ n1(cos è n1  i. sin è n1)  ñ n (1  cos è n  i. sin è n ) 2 è è è ñ n1(cos è n1  i. sin è n1)  ñ n cos n (cos n  i. sin n ) 2 2 2 è è ñ n1  ñ n cos n è n1  n 2 2 è è è n1  n ñ n1  ñ n cos n 2 2 n 1(cos è n 1  i sin è n 1 ) 

eKTaján nig nig . dUcen¼ x-RbePTénsIVút (è ) nig KNna è CaGnuKmn_én n ½ tamsRmayxagelIeyIgman è  21 è naM[ è  CasIVútFrNImaRtmanersugesµI q  21 . tamrUbmnþ è  è  q eday Z  ñ (cos è  i sin è )  1 2i 3  cos ð3  i.sin ð3 eKTaján ñ  1 ; è  ð3 dUcen¼ è  ð3 . 21 . n

n

n 1

n

n

n

n

0

0

0

0

0

0

n

0

n

K-bgHajfa ñ  ñ cos è cos è2 cos è2 ....cos è2 tamsRmayxagelIeKman ñ  ñ cos è2 ¦ ññ eKán   ññ    cos( è2 ) 1

0

n

2

n

0

n

n 1

k  n 1

n

n

 cos

èn 2

k  n 1

k 1

k0

n1

k

k

k 0

68


ñn è è è  cos è 0 . cos 1 . cos 2 ......... cos n1 ñ0 2 2 2 è è è ñ n  ñ 0 cos è 0 cos 1 cos 2 .... cos n1 2 2 2 è è è sin è n  2 sin n cos n  2 sin è n1 cos n 2 2 2 è è 1 sin è n è n1  n cos n  . 2 2 2 sin è n1 sin è n1 1 sin è 0 sin è1 1 sin è 0 ñn  n . . .....  n. sin è n 2 sin è1 sin è 2 2 sin è n ð sin 3 1 1 3 ñn  n  n1 . ð 1 2 sin ð . 1 ) 2 sin( . ) 3 2n 3 2n

dUcen¼ müa¨geToteyIgman ) eKTaj ( eRBa¼ ehtuen¼ dUcen¼

.

.

lMhat´TI33 k¿cUrKNnatémø®ákdén x¿cUreda¼RsaysmIkar

tan

ð 8

sin 2 x  2 sin x. cos x  ( 2  1) cos 2 x  0

K¿cUreda¼RsaysmIkar dMeNa¼Rsay

tan x  2  1 1  1  ( 2  1) tan x 3

69


k¿KNnatémø®ákdén tan ð8 a tamrUbmnþ tan2a  1 2 tan tan a

edayyktémø

2

eKán

tan

ð  4

2 tan

a

ð 8

ð 8

1  tan2

ð 8 1 ð 1  tan2 8 ð t  tan 8

ð 8

2 tan

naM[

tan2

ð ð  2 tan 1  0 8 8

Edl t  0 tag eKán t  2t  1  0 ; '  1  1  2 eKTaj¦s t  1  2 , t  1  dUcen¼ tan ð8  2  1 . 2

1

2

20

( minyk)

x¿ eda¼RsaysmIkar sin 2 x  2 sin x. cos x  ( 2  1) cos 2 x  0

EckGg:TaMgBIrnwg

cos 2 x  0

tan2 x  2 tan x  ( 2  1)

tag

t  tan x

eKánsmIkar ½ 0 ¿

eKán ½

t 2  2 t  ( 2  1)  0

eday

abc0 70


eKTaj¦s t  1 ; t  2  1 . -cMeBa¼ t  1 eKán tan x  1 naM[ x  ð4  kð , k  Z -cMeBa¼ t  2  1 eKán tan x  2  1 naM[ x  ð8  kð , k  Z dUcen¼ x  ð4  kð , x  ð8  kð , k  Z . K¿ eda¼RsaysmIkar ½ tan x  2  1 1 ð tan  2  1 eKán eday  8 1  ( 2  1) tan x 3 1

2

ð 8  3 ð 3 1  tan x. tan 8 ð ð tan( x  )  tan 8 6 ð ð ð x    kð x  kð , k  Z 8 6 24 ð x  kð , k  Z 24 tan x  tan

eKTaj dUcen¼

¦

.

71


lMhat´TI34 k¿cUrKNnatémø®ákdén cos 12ð nig x¿cUreda¼RsayRbB&næsmIkar

cos

7ð 12

 2 3 6 3 2 cos x 3 cos x cos y    8   3 cos 2 x cos y  cos 3 y  2  3 6  8

dMeNa¼Rsay k¿KNnatémø®ákdén cos 12ð nig eyIgán cos 12ð  cos( ð3  ð4 )

cos

7ð 12

ð ð ð ð cos  sin sin 4 3 4 3 1 2 3 2 2 6  .   . 2 2 2 2 4 7ð ð ð cos  cos(  ) 12 3 4 ð ð ð ð  cos cos  sin sin 3 4 3 4 1 2 3 2 2 6  .   . 2 2 2 2 4 ð 2 6 7ð cos   , cos 12 12 4  cos

ehIy

dUcen¼

2 6 4

. 72


x¿ eda¼RsayRbB&næsmIkar½  2 3 6 3 2 cos x 3 cos x cos y    8   3 cos 2 x cos y  cos 3 y  2  3 6  8

bUksmIkar

(1)

nig

( 2)

(1) (2)

Gg:nigGg:eKán ½ 2 2 8 2 ( cos x  cos y )3  ( )3 2 2 cos x  cos y  (3) 2

cos 3 x  3 cos 2 x cos y  3 cos x cos 2 y  cos 3 y 

dksmIkar

(1)

nig

( 2)

Gg:nigGg:eKán ½ 6 6 8 6 ( cos x  cos y )3  ( ) 3 2 6 cos x  cos y  (4) 2

cos 3 x  3 cos 2 x cos y  3 cos x cos 2 y  cos 3 y 

bUksmIkar

(3)

nig

( 4)

Gg:nigGg:eKán ½ 73


2 6 2 2 6 cos x  4 ð cos x  cos 12

2 cos x 

dksmIkar

(3)

naM[ x   12ð  2kð , k  Z nig (4) Gg:nigGg:eKán ½

2 6 2 2 6 cos y  4 7ð 7ð cos y  cos y  2kð , k  Z 12 12 ð 7ð x  2kð , k  Z y  2kð , k  Z 12 12 2 cos y 

naM[

dUcen¼ nig lMhat´TI35 k¿cUrRsaybBa¢ak´fa sin x  cos x  58  38 cos 4x 13 1 x¿cUreda¼RsaysmIkar (sin x  cos x)  16  sin 4 dMeNa¼Rsay k¿RsaybBa¢ak´fa sin x  cos x  58  38 cos 4x eyIgman sin x  cos x  1 ; x  IN elIkGg:TaMgBIrCaKUbeKán ½ 6

6

3

6

2

3

2

3

2x

6

2

74


( sin2 x  cos 2 x )3  1 sin 6 x  3 sin 4 x cos 2 x  3 sin 2 x cos 4 x  cos 6 x  1 sin6 x  cos 6 x  3 sin2 x cos 2 x ( sin2 x  cos 2 x)  1 sin 6 x  cos 6 x  3 sin2 x cos 2 x  1 3 sin 2 2 x  1 4 3 1  cos 4 x sin 6 x  cos 6 x  ( ) 1 4 2 3 3 5 3 sin 6 x  cos 6 x  1   cos 4 x   cos 4 x 8 8 8 8 5 3 sin 6 x  cos 6 x   cos 4 x 8 8 13 1  sin3 2 x (sin3 x  cos 3 x)2  16 4 13 3 3 sin 6 x  2 sin3 x cos 3 x  cos 6 x  2 sin x cos x 16 5 3 13 1  cos 4 x  cos 4 x  8 4 16 2 ð ð kð x  4 x    2kð , kZ 3 12 2 sin 6 x  cos 6 x 

dUcen¼ x¿eda¼RsaysmIkar eyIgán ¦ eKTaj

.

¦

.

75


lMhat´TI36 eK[sIVúténcMnYnBit

(U n )

 U 0  2   Un1  2  Un

, n  IN

kMnt´elI

IN

eday ½

k¿cUrKNna U CaGnuKmn_én n . x¿KNnaplKuN P  U  U  U  ....  U . dMeNa¼Rsay KNna U CaGnuKmn_én n ½ eyIgman ð ð ð U  2  2 cos nig U  2  U  2  2 cos  2 cos 4 4 8 «bmafavaBitdl´tYTI p KW U  2 cos 2ð eyIgnwgRsayfavaBitdl´tYTI (p  1) KW U  2 cos 2ð eyIgman U  2  U Ettamkar«bma U  2 cos 2ð eyIgán U  2  2 cos 2ð  4 cos 2ð  2 cos 2ð Bit . dUcen¼ U  2 cos 2ð n

n

0

1

2

n

n

0

1

0

p

p 1

p2

p

p 1

p3

p

p2

2

p 1

n

p 2

p3

p 3

n 2

76


x¿ KNnaplKuN P  U  U  U  ....  U 2a tamrUbmnþ sin 2a  2 sin a cos a naM[ 2 cos a  sin sin a n

0

1

2

n

ð ð sin n n n k 1 1 ð 2 2  Pn   (Uk )   (2 cos k 2 )   ( ) ð ð ð 2 k0 k 0 k 0 sin k 2 sin n2 sin n2 2 2 2

sin

lMhat´TI37 eK[sIVúténcMnYnBit (U ) kMnt´elI 1 1 U 2 U  ni g U  2 2 KNna U CaGnuKmn_én n . dMeNa¼Rsay KNna U CaGnuKmn_én n ½ eyIgman U  22  sin ð4

IN

n

2

n

n1

0

eday ½ , n  IN

n

n

0

U1 

1  1  U0

1  1  sin 2

2

2

2

ð 4  sin ð 8 ð

«bmafavaBitdl´tYTI p KW U  sin 2 eyIgnwgRsayfavaBitdl´tYTI (p  1) KW p

eyIgman

Up  1 

1  1  Up 2 2

p2

Up1  sin

Ettamkar«bma

ð 2

p3

Up  sin

Bit ð 2p2 77


1  1  sin2 Up1 

eyIgán

ð 2 p 2

2

1  cos 

ð

2 sin 2

2 p2 

ð

2 p 3  sin ð 2 2 p3

2

Bit

dUcen¼ U  sin 2ð . lMhat´TI38 eK[smIkardWeRkTIBIr (E) : x  (m  m) x  m  2  0 eKsnµtfasmIkaren¼man¦sBIrtagerogKñaeday tan a nig tan b . k¿cUrkMnt´témøéná¨r¨aEm¨Rt m edIm,I[ a  b  ð3 . x¿cUreda¼RsaysmIkarxagelIcMeBa¼ m EdlánrkeXIj . K¿edayeRbIlTæplxagelIcUrTajrktémøRákdén tan 12ð . n

n 2

2

2

dMeNa¼Rsay k¿kMnt´témøéná¨r¨aEm¨Rt m edIm,I[ a  b  ð3 ½ smIkarman¦skalNa   (m  m)  4m  8  0 eday tan a nig tanb Ca¦srbs´smIkarena¼ 2

2

78


 tan a  tan b  m2  m (1)   tan a . tan b  m  2 (2) tan a  tan b tan(a  b)  1  tan a. tan b ð tan a  tan b tan  3 1  tan a. tan b tan a  tan b 3  (3) 1  tan a. tan b

tamRTwsþIbTEvüteKman eday

ykTMnak´TMng eKán ½ m2  m 3 1 m  2

(1)

¦

nig

( 2)

(3)

CYskñúgsmIkar

m2  (1  3) m  3  0

eday a  b  c  0 eKTaj¦s m  1 , m  3 -cMeBa¼ m  1 ena¼   (1  1)  4.1  9  4  0 ( minyk ) -cMeBa¼ m  3 ena¼   (3  3)  4 3  8  4  2 dUcen¼ m  3 . 1

2

2

2

2

30

x¿eda¼RsaysmIkarxagelIcMeBa¼ m EdlánrkeXIj ½ cMeBa¼ m  3 eKán ½ x  (3  3)x  3  2  0 eday a  b  c  0 eKTaj¦s x  1 , x  2  3 . 2

1

2

79


tan

ð 12

K¿TajrktémøRákdén ½ tamlTæplxagelIeKman x  tan a  1 naM[ ehIy a  b  ð3 ena¼ b  ð3  ð4  12ð ehtuen¼ x  tanb  tan12ð  2  3 dUcen¼ tan 12ð  2  3 . 1

a

ð 4

2

lMhat´TI39 k¿cUrRsaybBa¢ak´TMnak´TMng tan x  cot x  2 cot 2x x¿cUrKNnaplbUkxageRkam ½ S n  tan a 

1 a 1 a 1 a tan  2 tan 2  ....  n tan n 2 2 2 2 2 2

dMeNa¼Rsay k¿RsaybBa¢ak´TMnak´TMng tag

tan x  cot x  2 cot 2 x

1  cot x   tan x A  cot x  2 cot 2 x  1 1  tan2 x cot 2 x    tan 2 x 2 tan x 1 1  tan2 x 1  1  tan2 x A 2 ( )  tan x tan x 2 tan x tan x

eKán dUcen¼

eday

tan x  cot x  2 cot 2 x

. 80


x¿KNnaplbUkxageRkam ½ 1 a 1 a 1 a tan  2 tan 2  ....  n tan n 2 2 2 2 2 2 n a   1    k tan k  2  k0  2 n a a  1    k (cot k  2 cot k 1 ) 2 2  k0  2 n a 1 a  1 a  1    k cot k  k 1 cot k 1   n cot n  2 cot 2a 2 2 2 2  2 k0  2

S n  tan a 

dUcen¼ S  tana  21 tan 2a  ....  21 tan 2a  21 cot 2a  2 cot2a . lMhat´TI40 k¿cUrRsaybBa¢ak´fa tan2x .tan x  tan2x  2 tan x x¿cUrKNnaplbUk S   2 tan 2a tan 2a  dMeNa¼Rsay k¿RsaybBa¢ak´fa tan2x .tan x  tan2x  2 tan x x tag f(x)  tan 2x  2 tan x eday tan 2x  1 2 tan tan x x eKán f(x)  1 2 tan  2 tan x tan x n

n

n

n

n

2

n

k

n

2

k

k 1

k0

2

2

2

81


2 tan x  2 tan x(1  tan2 x)  1  tan2 x 2 tan x  2 tan x  2 tan3 x  1  tan2 x 2 tan3 x 2 tan x   . tan2 x  tan 2 x. tan2 x 2 2 1  tan x 1  tan x 2

dUcen¼ tan2x .tan x  tan2x  2 tan x . x¿KNnaplbUk S   2 tan 2a tan 2a  eyIgman tan2x .tan x  tan2x  2 tan x edayyk eKán tan 2a tan 2a  tan 2a  2 tan 2a n

k

2

n

k

k 1

k0

2

x

2

k

k 1

k

k 1

n

Sn 

a a  a  k k 1 n1 2 tan  2 tan  tan a  2 tan    2 n1 2k 2 k 1  k0  n a a  a  S n   2 k tan k tan2 k 1   tan a  2 n1 tan n1 2 2 2  k0 

dUcen¼

82

a 2 k 1


lMhat´TI41 k¿cUrRsayfa x¿cUrKNna S n  sin3

sin3 x 

1 ( 3 sin x  sin 3x) 4

a a a a  3 sin3 2  32 sin3 3  ....  3n1 sin3 n 3 3 3 3

dMeNa¼Rsay k¿Rsayfa sin x  41 ( 3 sin x  sin 3x) eyIgman sin 3x  sin(x  2x) tamrUbmnþ sin(a  b)  sin a cos b  sin b cos a 3

 sin x cos 2 x  sin 2 x cos x  sin x(1  2 sin2 x)  2 sin x cos 2 x  sin x(1  2 sin2 x)  2 sin x(1  sin2 x)  sin x  2 sin3 x  2 sin x  2 sin3 x  3 sin x  4 sin3 x 3

eday sin 3x  3 sin x  4 sin x dUcen¼ sin x  41 ( 3 sin x  sin 3x) . x¿ KNna S  sin 3a  3 sin 3a  3 sin 3a  ....  3 sin 3a eyIgán S    3 sin 3a  eday sin x  41 ( 3 sin x  sin 3x) 3

3

3

n

2

n 1

3

2

3

n

k 1

n

3

3

k

k 1

1 Sn  4

a a  1 n a  k k 1 3 sin 3 sin (3 sin    sin a)   k k 1  n k 1  3 3  4 3 83 n

3

n


lMhat´TI42 k¿cUrRsayfa x¿cUrKNna S dMeNa¼Rsay k¿ Rsayfa eyIgman

1 4 1   cos 2 x sin 2 2 x sin 2 x 1 1 1    ....  n a a a 4 cos 2 4 2 cos 2 4 n cos 2 n 2 2 2

1 4 1   cos 2 x sin 2 2 x sin 2 x

1 1 sin 2 x  cos 2 x 1 4     cos 2 x sin 2 x sin2 x cos 2 x sin 2 x cos 2 x sin 2 2 x 1 4 1   cos 2 x sin 2 2 x sin 2 x 1 1 1 Sn    ....  a a a 4 cos 2 4 2 cos 2 4 n cos 2 n 2 2 2    n  1 1  Sn    k . a k 1  4 cos 2 k    2  1 4 1   cos 2 x sin 2 2 x sin 2 x

dUcen¼ x¿ KNna

.

eyIgán

eday eKán ½

84


 n  1 4 1  Sn    k ( ) a a k 1  4 sin 2 k 1 sin 2 k  2 2  n  1 1 1 1    k 1 .  k. a a k 1 4  sin 2 k 1 4 sin 2 k  2 2

dUcen¼

Sn 

       1 1    2 sin a 4n sin 2 a   2n

1 1  sin 2 a 4 n sin 2 a 2n

.

lMhat´TI43 k¿cUrRsayfa sin12x  cot x  cot 2x 1 1 1 S    x¿cUrKNna a a sin a n

sin

sin

2

22

 .... 

1 sin

a 2n

dMeNa¼Rsay k¿Rsayfa sin12x  cot x  cot 2x tag f(x)  cot x  cot 2x cos x cos 2 x  sin x sin 2 x cos x 2 cos 2 x  1   sin x 2 sin x cos x 2 cos 2 x  2 cos 2 x  1 1   2 sin x cos x sin 2 x

85


1  cot x  cot 2 x sin 2 x

dUcen¼ x¿KNna

.

1 1 1 1 Sn     ....  a a a sin a sin sin 2 sin n 2 2 2 n

Sn 

eyIgán

( k0

1 sin

a 2k

)

eday

1  cot x  cot 2 x sin 2 x

n

a a   cot cot     2 k 1 k  2  k0 a  cot n1  cot a 2 a S n  cot n1  cot a 2

Sn 

dUcen¼

.

lMhat´TI44 x 1 2  1 cos x cot x cot

k¿cUrRsayfa x¿KNnaplKuN Pn  (1 

1 1 1 1 )(1  )(1  ).....(1  ) a a a cos a cos cos 2 cos n 2 2 2

dMeNa¼Rsay x 1 2  1 cos x cot x cot

k¿Rsayfa

86


eyIgtag

A( x)  1 

1 cos x

x x x x 2 cos 2 sin cos sin x cos x  1 2 2  2 2    x x cos x cos x cos x sin cos x sin 2 2 x x cos cot 2 . sin x  tan x tan x  2  x cos x 2 cot x sin 2 x cot 1 2  1 cos x cot x 2 cos 2

dUcen¼ x¿KNnaplKuN

.

n 1 1 1 1 1 Pn  (1  )(1  )(1  ).....(1  )  ( 1 ) a a a a cosa k 0 cos cos 2 cos n cos k 2 2 2 2 a a tan tan n k 1 n 1 a 2 2  (  tan n 1 cot a ) a tan a 2 k 0 tan k 2

87


lMhat´TI45 nx)  cos(n  1)x cos(nx)  k¿cUrRsayfa sin(  . cot x  cos x cos x   cos x sin x sin 2 x sin 3x sin(nx) x¿cUrKNna S  cos    ....  x cos x cos x cos x dMeNa¼Rsay nx)  cos(n  1)x cos(nx)  k¿Rsayfa sin(  . cot x   cos x cos x   cos x eyIgman cos(n  1)x  cos(nx  x) ¦ cos(n  1)x  cos(nx) cos x  sin(nx) sin x EckGg:TaMgBIrnwg cos x eKán ½ n 1

n

n

2

n 1

n

n

3

n

n

n1

cos(n  1)x cos(nx) cos x sin(nx) sin x cos(nx) sin(nx)     . tan x n1 n1 n1 n n cos x cos x cos x cos x cos x sin(nx)  cos(n  1)x cos(nx)  1 .    cos n1 x cos n x cos n x  tan x sin(nx)  cos(n  1)x cos(nx)   . cot x  n 1 cos n x cos n x   cos x sin x sin 2 x sin 3x sin(nx) Sn     ....  cos x cos 2 x cos 3 x cos n x n  sin(kx)  Sn    k  k 1  cos x  sin(kx)  cos(k  1)x cos(kx)  . cot x   k k 1 k  cos x  cos x cos x  S cot x n  cos(k  1)x cos(kx)   cos(n  1)x  cot x 1          n n 1 k 1 k cos x cos x cos x     k 1

naM[ dUcen¼ x¿KNna eyIgán eday

.

88


lMhat´TI46 1 1 tan(n  1)x  tan(nx) k¿cUrRsayfa cos(nx). cos(  n  1)x sin x  1 x¿ KNnaplbUk S    cos(px) cos(  p  1) x  dMeNa¼Rsay 1 1 tan(n  1)x  tan(nx) k¿Rsayfa cos(nx). cos(  n  1)x sin x sin(p  q) tamrUbmnþ tanp  tan q  cos p cos q naM[ cos p1cos q  sin(p1  q)  tanp  tan q (1) yk p  (n  1)x , q  (nx) nig p  q  x CYskñúg (1) eKán 1 1 tan(n  1)x  tan(nx)  . cos(nx). cos(n  1)x sin x  1 x¿KNnaplbUk S    cos(px) cos(  p  1) x  n

n

p 1

n

n

p 1

tamsRmayxagelIeKman ½ 1 1 tan(p  1)x  tan(px)  cos(px). cos(p  1) x sin x n 1 tan(p  1)x  tan(px) Sn   sin x p 1 1 sin(nx) tan(n  1)x  tan x    sin x sin x cos x cos(n  1) x

eyIgán

89


lMhat´TI47 k¿cUrRsayfa tan(n  1)x  tan(nx)  tan x 1  tan(nx) tan(n  1) x 

x¿KNnaplbUk S n  tan x tan 2 x  tan 2 x tan 3x  .....  tan(nx) tan(n  1) x

dMeNa¼Rsay k¿Rsayfa tan(n  1)x  tan(nx)  tan x 1  tan(nx) tan(n  1)x a  tan b tamrUbmnþ ta(a  b)  1tan  tan a tan b naM[ tan a  tanb  tan(a  b) 1  tan a tanb (1) edayyk a  (n  1)x , b  nx nig a  b  x CYskñúg (1) eKán ½ tan(n  1)x  tan(nx)  tan x 1  tan(nx) tan(n  1) x  . x¿KNnaplbUk S n  tan x tan 2 x  tan 2 x tan 3x  .....  tan(nx) tan(n  1) x n

eyIgán S   tan(kx) tan(k  1)x tamsRmayxagelIeyIgman ½ n

k 1

90


tan(n  1)x  tan(nx)  tan x 1  tan(nx) tan(n  1) x 

¦ tan(nx) tan(n  1)x  tan(n  1)x  tan(nx) cot x eyIgán S   tan(k  1)x  tan(kx) cot x  1

1

n

n

k 1

 tan(n  1) x  tan x  cot x  n 

sin(nx) cot x  n cos(n  1)x cos x

dUcen¼ S  cos(nsin( 1nx)x)sin x  n . lMhat´TI48 k¿cUrRsaybBa¢ak´fa 2 cos x  1  1 1 2 2coscos2xx x¿cUrKNnaplKuN ½ n

Pn  (2 cos a  1)(2 cos

a a a  1)(2 cos 2  1).....(2 cos n  1) 2 2 2

dMeNa¼Rsay k¿ RsaybBa¢ak´fa 2 cos x tamrUbmnþ cos 2x  2 cos

 1 2

1  2 cos 2 x 1  2 cos x

x 1

2 cos 2 x  4 cos 2 x  2

dUcen¼

2 cos 2 x  1  4 cos 2 x  1 2 cos 2 x  1  (2 cos x  1)(2 cos x  1) 1  2 cos 2 x 2 cos x  1  1  2 cos x

. 91


x¿ KNnaplKuN ½ Pn  (2 cos a  1)(2 cos

a a a  1)(2 cos 2  1).....(2 cos n  1) 2 2 2

n

a   2 cos  1   k 2  k0 2 cos x  1 

eday

1  2 cos 2 x 1  2 cos x

a    1 2 cos( ) n  k 1  1  2 cos 2a 2  Pn     a a k 0  1  2 cos( k )  1  2 cos n 2 2  

dUcen¼

Pn 

lMhat´TI49 k¿cUrRsayfa

1  2 cos 2a a 1  2 cos n 2

.

tan3 x 1  ( tan 3x  3 tan x) 1  3 tan2 x 8  k 3 a  3 tan n  3k  Sn     2 a k0  1  3 tan k   3 

x¿cUrKNnaplbUk

dMeNa¼Rsay x 1 k¿ Rsayfa 1 tan  ( tan 3x  3 tan x) 3 tan x 8 x  tan x tamrUbmnþ tan3x  3 tan 1  3 tan x 3

2

3

2

92


3 tanx  tan3 x 8 tan3 x tan3x  3 tanx   3 tanx  1  3 tan2 x 1  3 tan2 x tan3 x 1  ( tan 3x  3 tan x) 2 8 1  3 tan x

eyIgán dUcen¼

.

n

x¿KNnaplbUk eyIgman eKán

Sn 

 k0

 k 3 a 3 tan  3k  a 1  3 tan2 k  3

    

tan3 x 1  ( tan 3x  3 tan x) 1  3 tan2 x 8 a tan3 k 1 a a 3  (tan k 1  3 tan k ) a 8 3 3 1  3 tan2 k 3

edayyk

x

eyIgán 1 n k a a 1 a  S n   (3 tan k 1  3k 1 tan k )   tan 3a  3n1 tan n  8 k0 8 3  3 3 tan 3a 3n1 a Sn   tan n 8 8 3

dUcen¼

.

93

a 3k


lMhat´TI50 k¿cUrRsayfa

tan3 x 1  tan 2 x  tan x 2 2 1  tan x n

x¿cUrKNnaplbUk

Sn 

 k0

 k 3 a  2 tan n 2   1  tan2 a  2n 

     

dMeNa¼Rsay k¿Rsayfa 1 tantanx x  21 tan 2x  tan x x tan x  tan x  eyIgán 21 tan 2x  tan x  1 tan tan x 1  tan x dUcen¼ 1 tantanx x  21 tan 2x  tan x . 3

2

3

2

2

3

2

n

Sn 

x¿KNnaplbUk eKman eKán

 k0

 k 3 a  2 tan k 2   1  tan2 a   2k

     

tan3 x 1 tan 2 x  tan x  1  tan2 x 2 a tan3 k 1 a a 2  tan k 1  tan k a 2 2 2 1  tan2 k 2

yk

n

eyIgán dUcen¼ S

Sn 

a

a 

1

x

a 2k

a

  2 k 1 tan 2k 1  2k tan 2k   2 tan 2a  2n tan 2n

k0

n

1 a tan 2a  2 n tan n 2 2

. 94


lMhat´TI51 k¿cUrRsayfa cos(2nx)  2 sin1 x  sin(2n  1)x  sin(2n  1)x  x¿KNnaplbUk S  cos 2x  cos 4x  cos 6x  .....  cos(2nx) K¿TajrkplbUk n

Tn  cos 2 x  cos 2 2 x  cos 2 3x  .....  cos 2 (nx)

X¿KNnaplbUk Un  sin2 x  sin2 2 x  sin2 3x  .....  sin2 (nx)

dMeNa¼Rsay k¿Rsayfa cos(2nx)  2 sin1 x  sin(2n  1)x  sin(2n  1)x  tamrUbmnþ sin p  sin q  2 sin( p 2 q) cos( p 2 q) edayyk p  (2n  1)x , q  (2n  1)x nig p  q  2x , p  q  4nx eKán sin(2n  1)x  sin(2n  1)x  2 sin x cos(2nx) dUcen¼ cos(2nx)  2 sin1 x  sin(2n  1)x  sin(2n  1)x  . x¿KNnaplbUk S  cos 2x  cos 4x  cos 6x  .....  cos(2nx) eyIgán S   cos(2kx) n

n

n

k 1

95


n 1 sin(2k  1)x  sin(2k  1)x    2 sin x k 1 1 sin(2n  1)x  sin x   2 sin x sin(nx) cos(n  1)x 1 2 sin(nx) cos(n  1)x    2 sin x sin x

dUcen¼ S  sin(nx)sincos(x n  1)x . K¿TajrkplbUk T  cos x  cos 2x  cos 3x  ..... cos (nx) 2a eyIgán T   cos (kx) tamrUbmnþ cos a  1  cos 2 eKán T   1  cos(2 2kx)   2n  21 S eday S  sin(nx)sincos(x n  1)x n  1)x . dUcen¼ T  2n  sin(nx2) cos( sin x X¿KNnaplbUk n

2

2

2

2

n

n

2

2

n

k 1

n

n

n

k 1

n

n

Un  sin2 x  sin2 2 x  sin2 3x  .....  sin2 (nx) n

eyIgán

Un 

 sin2 (kx) k 1 n

 1  cos 2 (kx)  n  Tn k 1

n  1)x eday T  2n  sin(nx2) cos( sin x n  1)x . dUcen¼ U  n2  sin(nx2) cos( sin x n

n

96


lMhat´TI52  3m  8 eK[GnuKmn_ y  x  22mx Edl x  IR nig ( x  1) m Caá¨ra¨Em¨Rt kMnt´témø m edIm,I[GnuKmn_en¼Gactag[témøkUsIunUs énmMumYyán¦eT? dMeNa¼Rsay edIm,I[GnuKmn_en¼tag[témøkUsIunUsénmMumYylu¼RtaEtcMeBa¼  3m  8 1 RKb´ x  IR eKán  1  x  22(mx x  1) edayeKman 2

2

2

2

2( x 2  1)  0 , x  IR

eKTaj ¦ cMeBa¼

 2 x 2  2  x 2  2mx  3m  8  2 x 2  2  3x 2  2mx  3m  6  0 (1)  2  x  2mx  3m  10  0 (2) (1) : 3x 2  2mx  3m  6  0

smmUl eday m

a  3  0  2  '  m  9m  18  0 

2

 9m  18  (m  3)(m  6) 97


eKán '  (m  3)(m  6)  0 naM[ ¦ m  [3,6 ] cMeBa¼(2) : x  2mx  3m  10  0

3 m 6

2

a  1  0  2    3m  10  0 ' m 

smmUl eday m  3m  10  (m  2)(m  5) eKán '  (m  2)(m  5)  0 naM[  5  m  2 ¦ m  [ 5 , 2 ] . edayykcemøIy m  [ 3 , 6 ] RbsBVnwg m  [5 , 2 ] ena¼eKán m  ö dUcen¼eKminGackMnt´témø m edIm,I[GnuKmn_en¼Gactag [témøkUsIunUsénmMumYyáneT . 2

98


lMhat´TI53 cUrRsaybBa¢ak´fa 2 

cos 4 x  4 sin 4 x  1 8  cos 4 x  2 3

cMeBa¼RKb´cMnYnBit

dMeNa¼Rsay cos 4 x  4 sin 4 x  1 , x  IR eyIgtag y  cos 4 x  2 eyIgán cos 4x  4 sin 4x  1  y cos 4x  2y (1  y) cos 4 x  4 sin 4 x  2 y  1 (1) ¦ eyIgeRCIserIsviucT&rBIr ½ U ( 1  y , 4 ) nig V ( cos 4 x , sin 4 x ) tamkenßamviPaKplKuNs;aEl 

U . V  (1  y) cos 4 x  4 sin 4 x (2) 

tam (1) nig (2) eKTaj U . V  2y  1 . müa¨geTottamniymn&y U . V  || U ||.|| V ||. cos è eday  1  cos è  1 , è  IR eKTaj  ||U||.|| V ||  U . V  ||U||.|| V|| 

 

99


2

    U  2 2  . V   || U|| .|| V ||  

¦ eday ||U||  (1  y)  16 , || V || 1 nig U . V  2y  1 eKán (2y  1)  (1  y)  16 ¦ 3y  2y  16  0 eday 3y  2y  16  (y  2)(3y  8) 8  2  y  3 y  2 y  16  0 ehtuen¼ smmUl . 3 cos 4 x  4 sin 4 x  1 8  2   dUcen¼ cos 4 x  2 3 cMeBa¼RKb´cMnYnBit x . lMhat´TI54 1 1 eK[cMnYnkMupøic Z  (cos x  cos x )  i.(sin x  sin x ) Edl x CacMnYnBit. cUrkMnt´rkm¨UDulGb,brmaéncMnYnkMupøicen¼ ? dMeNa¼Rsay rkm¨UDulGb,brmaéncMnYnkMupøic eyIgán |Z| (cos x  cos1 x )  (sin x  sin1 x ) 1 1 f ( x )  (cos x  )  (sin x  ) tag cos x sin x 

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

100


1 1 4 sin x 2    cos 4 x sin 4 x 1 1 )  4  (cos 4 x  sin 4 x)  ( 4  4 sin x cos x 4 4 cos x sin x  )  4  (cos 4 x  sin 4 x)  ( 4 4 sin x cos x 1 )  4  (cos 4 x  sin 4 x)(1  4 4 sin x cos x  cos 4 x  2 

 4  (cos 2 x  sin2 x)2  2 sin2 x cos 2 x (1   4  (1 

16 ) 4 sin 2 x

1 16 sin2 2 x)(1  ) 4 2 sin 2 x

1 1 2 1  sin 2 x  2 2

2

edayeKman sin 2 x  1 naM[ 16 nig 1  sin 2x  17 1 16 17 25 4  ( 1  sin 2 x )( 1  )  4   eKTaj 2 2 2 sin 2 x 4

2

4

1 16 25 2 f(x)  4  (1  sin 2 x)(1  ) 4 2 2 sin 2 x 5 5 2 |Z|   | Z | f( x) 2 2

eyIgán eday

eKTaján

dUcen¼m¨UDulGb,brmaén

Z

KW

| Z |min 

5 2 2

. 101


lMhat´TI55 eK[ x CacMnYnBitEdl 60x  71x  21  0 .  ð  sin    0 . cUrbgHajfa 3 x 1    dMeNa¼Rsay  ð  sin    0 bgHajfa  3x  1  tag f(x)  60x  71x  21 ebI f(x)  0  60x 2  71x  21  0 2

2

  (71) 2  4(60)(21)  5041  5040  1

x1 

71  1 7  120 12

, x2 

71  39 3  120 5

eKTaj¦s eyIgán f(x)  60x  71x  21  0 naM[ 127  x  53 4 1 4 7 9   ¦ 4  3x  5 naM[ 5 3x  1 3 4ð ð 4ð   eKTaj 5 3x  1 3 naM[ sin  3xð 1  0 . dUcen¼ ebI x CacMnYnBitEdl 60x  71x  21  0 ena¼eKán½ sin  3xð 1  0 . 2

2

102


lMhat´TI56 eK[sIVúténcMnYnBit (U ) kMnt´eday U  2 .sin n4ð Edl n  IN * . k-cUrbgHajfa 2.cos (n 41)ð  cos n4ð  sin n4ð x-Taj[ánfa U  ( 2) cos n4ð  ( 2) cos (n 41)ð K-KNnaplbUk S  U  U  U  .........  U . dMeNa¼Rsay k-bgHajfa 2.cos (n 41)ð  cos n4ð  sin n4ð tamrUbmnþ cos(a  b)  cos a cos b  sin a sin b eyIgán n

n

n

n 1

n

n

n

2 cos

1

2

3

n

(n  1)ð nð ð nð ð nð ð  2 cos(  )  2(cos cos  i sin sin ) 4 4 4 4 4 4 4

2 cos

(n  1)ð 2 nð 2 nð nð nð cos sin )  cos  2(   sin 4 2 4 2 4 4 4

2 . cos

(n  1)ð nð nð  cos  sin 4 4 4

dUcen¼ . x-Taj[ánfa U  ( 2) cos n4ð  ( 2) cos (n 41)ð (n  1)ð nð nð 2 . cos  cos  sin eyIgman 4 4 4 nð nð (n  1)ð naM[ sin 4  cos 4  2 cos 4 n

n 1

n

103


KuNGg:TaMgBIrnwg ( 2 )n sin

( 2 )n eKán

nð nð (n  1)ð  ( 2 )n cos  ( 2 )n 1 cos 4 4 4 nð (n  1)ð  ( 2 )n 1 cos Un  ( 2 )n cos 4 4

dUcen¼ K-KNnaplbUk eyIgán

.

S n  U 1  U 2  U 3  .........  U n

n

kð (k  1)ð   k k 1 S n   ( 2 ) cos  ( 2 ) cos 4 4  k 1  ð (n  1)ð  2 cos  ( 2 )n1 cos 4 4

dUcen¼

Sn  1  ( 2)

n 1

(n  1)ð cos 4

.

104


lMhat´TI57 k¿cUrKNnatémø®ákdén sin 10ð nig cos 10ð x¿cUrRsayfa x  (x  y)  4(x  y ) sin 10ð . dMeNa¼Rsay k¿KNnatémø®ákdén sin 10ð nig cos 10ð 2ð ð 3ð eKman 10   2 10  3 2 eKán sin 10  sin(  ) 2 10 2

2

2

2

2

ð ð 3ð  cos cos 10 10 10 ð ð ð ð  3 cos  4 cos 3 2 sin cos 10 10 10 10 ð ð  3  4 cos 2 2 sin 10 10 ð ð  3  4(1  sin 2 ) 2 sin 10 10 ð ð 4 sin 2  2 sin 1 0 10 10 2 sin

¦ tag t  sin 10ð  0 eKán 4t  2t  1  0 , '  1  4  5  0 eKTaj¦s t  1 4 5  0 ( minyk ) , t  1 4 5 dUcen¼ sin 10ð  1 4 5 . 2

1

2

105


eday sin 2

ð ð  cos 2 1 10 10

cos

naM[

ð 1 5 2 10  2 5  1 ( )  10 4 4

dUcen¼ cos 10ð  10 4 2 5 . x¿Rsayfa x  (x  y)  4(x  y ) sin 10ð tagGnuKmn_ f(x ; y)  x  (x  y)  4(x  y ) sin 10ð eKán f (x ; y)  x  x  2xy  y  4(x  y )(1 4 5 ) 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

62 5 16 3 5  2 x 2  2 xy  y 2  ( x 2  y 2 ) 2 1 5 2 1 5 2  x  2 xy  y 2 2  5 1 2 5 1 2    x  2 xy  y  2 2    2 x 2  2 xy  y 2  4( x 2  y 2 )

2

    

5 1 5  1  x  y  0 , x, y  IR  2 2  ð x 2  ( x  y)2  4( x 2  y 2 ) sin2 10

dUcen¼ RKb´cMnYnBit

x, y  IR

. 106


lMhat´TI58 eK[RtIekaN ABC mYYy . bgHajfaebI tan A3 , tan B3 , tan C3 Ca¦srbs´smIkar (E) : x  ax  bx  c  0 ena¼eKán 3  a  3b  c . dMeNa¼Rsay karbgHaj ½ eyIgán tan( A3  B3  C3)  tan[( A3  B3)  C3 ] 3

2

A B C  )  tan ABC 3 3 3 tan( ) A B C 3 1  tan(  ) tan 3 3 3 A B tan  tan 3 3  tan C A B 3 1  tan tan ð 3 3 tan  A B 3 tan  tan 3 3 . tan C 1 A B 3 1  tan tan 3 3 A B C A B C tan  tan  tan  tan tan tan 3 3 3 3 3 3 3 (1) A B A C B C 1  (tan tan  tan tan  tan tan ) 3 3 3 3 3 3 A B C (E) tan , tan , tan 3 3 3 tan(

eday

Ca¦srbs´smIkar

107


ena¼tamRTwsþIbTEvüteKmanTMnak´TMng ½ A B C  tan  tan   a (2) 3 3 3 A B B C A C tan tan  tan tan  tan tan  b (3) 2 2 2 2 2 2 A B C tan . tan . tan   c (4) 2 2 2

tan

ykTMnak´TMng eKán ½ 3 

ac 1 b

(2) , (3)

¦

nig

(4)

CYskñúgsmIkar

(1)

3  3 b  a  c

108


lMhat´TI59 2 x  cos a eK[GnuKmn_ f(x)  x xcos a2x cos a 1 bgHajfa x  IR :  1  f(x)  1 dMeNa¼Rsay bgHajfa x  IR :  1  f(x)  1 2

2

, 0að

.

(1  cos a) x 2  2(1  cos a) x  (1  cos a) 1  f( x)  x 2  2 x cos a  1 (1  cos a)( x 2  2 x  1) 1  f( x)  2 x  2 x cos a  cos 2 a  sin2 a a 2( x  1)2 cos 2 2  0 , x  IR 1  f( x)  ( x  cos x)2  sin2 a

eKTaján

f( x)   1 , x  IR

(1)

(1  cos a) x 2  2(1  cos a)x  (1  cos a) 1  f( x )  x 2  2 x cos a  1 (1  cos a)( x 2  2 x  1) 1  f( x )  2 x  2 x cos a  cos 2 a  sin 2 a a 2( x  1)2 sin2 2  0 , x  IR 1  f( x )  2 ( x  cos x)  sin2 a

eKTaján f(x)  1 , x  IR tam (1) nig (2) eKTaján

(2) x  IR :

 1  f( x)  1 109


lMhat´TI60 eKmansmPaB sina x  cosb x  a 1 b Edl a  0 , b  0 , a  b  0 cUrbgHajfa sina x  cosb x  (a 1b) . dMeNa¼Rsay bgHajfa sina x  cosb x  (a 1b) eKman sina x  cosb x  a 1 b naM[ b(a  b) sin x  a(a  b) cos x  ab 4

4

8

8

3

3

8

8

3

3

4

3

3

4

4

4

ab sin 4 x  b 2 sin 4 x  a 2 cos 4 x  ab cos 4 x  ab(sin 2 x  cos 2 x) 4 a 2 cos 4 x  2ab cos 2 x sin 2 x  b 2 sin 4 x  0 (a cos 2 x  b sin 2 x)2  0 sin 2 x cos 2 x sin2 x  cos 2 x 1    a b ab ab sin 8 x cos 8 x 1   a4 b4 (a  b) 4

eKTaj naM[ eKTaj sina x  (a ab) (1) nig cosb x  (a bb) bUksmIkar (1) nig (2) Gg:nigGg: eKán ½ 8 3

8

4

3

4

(2)

sin8 x cos 8 x ab 1    a3 b3 (a  b) 4 (a  b)3 110


lMhat´TI61 eK[RtIekaN ABC man cos B  53 nig cos C  54 cUrKNna sin(B  C) rYckMnt´RbePTénRtIekaN ABC . dMeNa¼Rsay kMnt´RbePTénRtIekaN ABC eyIgman cos B  53 nig cos C  54 eyIgán sin B  1  cos B  1  259  54 3  nig sin C  1  sin C  1  16 25 5 man sin(B  C)  sin B cos C  sin C cos B 2

2

4 4 3 3 sin(B  C)  .  .  1 5 5 5 5

naM[ B  C  ð2 ehIy A  ð2 dUcen¼ ABC CaRtIekaNEkgRtg´

A

.

111


lMhat´TI62 eK[ctuekaN ABCD mYyman AB  a , BC  b , CD  c , DA  d

eKtag S CaépÞRkLarbs´ctuekaNen¼ . cUrRsayfa S  21 ( ab  cd ) . dMeNa¼Rsay Rsayfa S  21 ( ab  cd ) D A

B

C

eyIgman eday S nig S eyIgán

S  S ABC  S ACD

1 1 AB . BC . sin B  ab sin B ABC 2 2 1 1  AD . DC . sin D  cd sin D ADC 2 2 1 S  ( ab sin B  cd sin D ) 2 

112


eyIgman sin B  1 ena¼ ab sinB  ab nig sin D  1 ena¼ cd sin D  cd dUcen¼ S  21 ( ab  cd ) lMhat´TI63 eK[RtIekaN ABC manRCugepÞógpÞat´TMnak´TMng a  b  2c . k¿cUrbgHajfa cos C  a 4abb x¿TajbBa¢ak´fa cos C  21 dMeNa¼Rsay k¿bgHajfa cos C  a 4abb tamRTwsþIbTkUsIunUsGnuvtþn_kñúgRtIekaN ABC eKman ½ c  a  b  2ab cos C Ettamsmµtikmµ a  b  2c eKán a 2 b  a  b  2ab cos C ¦ 2ab cos C  a  b  a 2 b  a 2 b dUcen¼ cos C  a 4abb . 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

113


x¿ TajbBa¢ak´fa cos C eyIgman (a  b)  0

1 2

2

2

2

a  2ab  b  0

¦

edaytamsRmayxagelI dUcen¼ cos C  21 .

a2  b2 1  4ab 2

a 2  b2 cos C  4ab

lMhat´TI64 eK[RtIekaN ABC mYymanRCúg a , b , c . ebI cosa A  cosb B  cosc C  21 ( 1a  b1  1c) cUrkMnt´RbePTénRtIekaN ABC dMeNa¼Rsay RbePTénRtIekaN ABC tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman ½ b c a cos A b  c  a  cos A  naM [ (1) 2bc a 2abc c b dUcKñaEdreKTaj cosb B  a 2abc (2) nig cosc C  a 2babc c (3) bUkTMnak´TMng (1) , (2) , (3) Gg:nwgGg:eKán ½ 2

2

2

2

2

2

2

2

2

2

2

2

114


cos A cos B cos C a 2  b 2  c 2    a b c 2abc

eday

cos A cos B cos C 1 1 1 1    (   ) a b c 2 a b c a2  b2  c2 1 1 1 1  (   ) 2abc 2 a b c a 2  b 2  c 2 bc  ca  ab  2abc 2abc a 2  b 2  c 2  ab  bc  ac

eKTaján ½

2a 2  2b 2  2c 2  2ab  2bc  2ac (a 2  2ab  b 2 )  (b 2  2bc  c 2 )  (c 2  2ac  a 2 )  0 (a  b)2  (b  c) 2  (c  a)2  0

eKTajánsmPaB a  b  c . dUcen¼ ABC CaRtIekaNsm&gß.

115


lMhat´TI65 eK[smIkar (E) : x  (2m  3)x  5x  3m  2  0 «bmafasmIkaren¼man¦bItageday tan á , tan â , tan ã . k¿cUrKNna A  cossin(áácos ââ cosã)ã CaGnuKmn_én m . x¿kMnt´ m edIm,I[ A  4 . K¿eda¼RsaysmIkar (E) cMeBa¼témø m Edlán rkeXIjxagelI . dMeNa¼Rsay k¿ KNna A  cossin(áácos ââ cosã)ã CaGnuKmn_én m   (   )] eyIgman A  sin[ cos  cos  cos  3

2

sin á cos(â  ã)  sin(â  ã) cos á cos á cos â cos ã sin á cos â cos ã  sin á sin â sin ã  sin â cos ã cos á  sin ã cos â cos á  cos á cos â cos ã  tan á  tan á tanâ tan ã  tan â  tan ã

 tan á  tan â  tan ã  tan á tanâ tan ã

eday tan á , tan â , tan ã Ca¦ssmIkar eyIgman

(E)

ena¼tamRTwsþIbTEvüt

 tan á  tan â  tan ã  2m  3   tan á tan â  tan â tan ã  tan ã tan á  5  tan á tan â tan ã  3m  2 

116


eyIgán A  2m  3  (3m  2)  m  5 dUcen¼ A  m  5 . x¿kMnt´ m edIm,I[ A  4 edayeyIgman A  m  5 eyIgán  m  5  4 naM[ m  1 . K¿ eda¼RsaysmIkar (E) ½ cMeBa¼ m  1 eKán (E) : x  5x  5x  1  0 eday x  5x  5x  1  (x  1)(x  4 x  1) eKTaj (x  1)(x  4x  1)  0 ¦  xx 14 x  1  0  '  4  1  3 eKTaj x  2  3 , x  2  3 dUcen¼sMNMu¦ssmIkar x  { 2  3 ; 1 , 2  3 } . 3

3

2

2

2

2

2

1

2

117


lMhat´TI66 eK[GnuKmn_ f(x ; y )  a cos x  b cos y   a sin x  b sin y  ( Edl a  0 , b  0 ) . cMeBa¼RKb´ x ; y  IR bgHajfa f(x; y)  (a  b) . dMeNa¼Rsay bgHajfa f(x; y)  (a  b) 2

2

2

2

f( x ; y )  a cos x  b cos y 2  a sin x  b sin y 2

 a 2 cos 2 x  2ab cos x cos y  b 2 cos2 y  a 2 sin2 x  2ab sin x sin y  b 2 s  a 2 (cos2 x  sin2 x)  b 2 (cos2 y  sin2 y)  2ab(cos x cos y  sin x sin y)  a 2  b 2  2ab cos(x  y) 2

2

eKán f(x; y)  a  b  2ab cos(x  y) edayeKman  x; y  IR : cos(x  y)  1 eyIgán f(x)  a  b  2ab  (a  b) dUcen¼ f(x; y)  (a  b) . 2

2

2

2

118


lMhat´TI67 eK[RtIekaN ABC Edl sin A2  2 abc . rkRbePTénRtIekaN ABC dMeNa¼Rsay rkRbePTénRtIekaN ABC eKman sin A2  2 abc tamRTwsþIbTkUsIunUseKman cos A  b 2cbc a eday cos A  1  2 sin A2 eKán ½ 2

2

2

2

b2  c2  a2 a2  1 2 2bc 4bc b2  c2  a2 2bc  a 2  2bc 2bc b 2  c 2  a 2  2bc  a 2 b 2  2bc  c 2  0 (b  c)2  0 b  c

RtIekaN ABC manRCúg kMBUl A .

bc

naM[vaCaRtIekaNsmát

119


lMhat´TI68 eK[smIkar (E) : cos x cos 3x  sin x sin 3x  m k¿cUreda¼RsaysmIkaren¼kalNa m  383 x¿rklk&çx&NÐsRmab´ m edIm,I[smIkaren¼man¦s . dMeNa¼Rsay k¿eda¼RsaysmIkaren¼kalNa m  383 eyIgman sin 3x  3 sin x  4 sin x naM[ 3

3

3

sin3 x 

1 (3 sin x  sin 3x) 4 3

ehIy cos 3x  4 cos x  3 cos x naM[ cos x  41 (3 cos x  cos 3x) smIkar (E) Gacsrsr ½ 3

1 1 (3 cos x  cos 3x) cos 3x  (3 sin x  sin 3x) sin 3x  m 4 4 3 cos x cos 3x  cos 2 3x  3 sin x sin 3x  sin 2 3x  4m 3(cos 3x cos x  sin 3x sin x)  (cos 2 3x  sin 2 3x)  4m 3 cos 2 x  cos 6x  4m 4 cos 3 2 x  4m cos 2 x  3 m

eday

m

3 3 8

eKán

cos 2 x 

3 2

120


naM[ x   12ð  kð , k  Z x¿ rklk&çx&NÐsRmab´ m ½ edIm,I[smIkaren¼man¦seKRKan´Et[ ¦ m   1 , 1 . lMhat´TI69 eda¼RsaysmIkar ½

1 

3

m  1

log 2 2 (sin x )  log 2 ( 2 sin 3 x )  0

dMeNa¼Rsay eda¼RsaysmIkar ½ log 2 2 (sin x)  log 2 (2 sin3 x)  0 2kð  x  ð  2kð , k  Z

lkç&xN&Ð sin x  0 naM[ smIkarGacsresr ½

log 2 2 (sin x)  log 2 (sin3 x)  log

2

2 0

log 2 2 (sin x)  3 log 2 (sin x)  2  0 2

tag t  log (sin x) eKánsmIkar t  3t  2  0 eday b  a  c eKTaj¦s t  1 , t  2 -cMeBa¼ t  1eKán log (sin x)  1 2

1

2

2

121


smmUl naM[

    

sin x 

1 2

ð  2kð 4 ð 3ð x  ð   2kð   2kð , k  Z 4 4

x

-cMeBa¼ t  2 eKán smmUl sin x  21 naM[

    

log 2 (sin x)  2

ð  2kð 6 ð 5ð x  ð   2kð   2kð , k  Z 6 6

x

lMhat´TI70 k¿cUrbgHajfa 1 1 2 sin x cos(2nx)   2  sin(2n  1)x 2  sin(2n  1)x [2  sin(2n  1)x ] [2  sin(2n  1)x ] n

 2kx) x¿KNna S    2  sin(2k cos( .    1) x 2  sin(2k  1)x  dMeNa¼Rsay k¿ karbgHaj tag f(x)  2  sin(12n  1)x  2  sin(12n  1)x n

k 1

122


sin(2n  1)x  sin(2n  1)x 2  sin(2n  1)x 2  sin(2n  1)x  2 sin x cos(2nx)  2  sin(2n  1)x 2  sin(2n  1)x 

dUcen¼ 2 sin x cos( 2 nx ) 1 1   2  sin( 2 n  1) x 2  sin( 2 n  1) x [2  sin( 2 n  1) x ] [2  sin( 2 n  1) x ]

n

 2kx) x¿KNna S    2  sin(2k cos( 1) x 2  sin(2k  1)x   eyIgán S  2 sin1 x   2  sin(12n  1)x  2  sin(12n  1)x  n

k 1

n

n

k 1

  1 1   2  sin x 2  sin(2n  1)x    1 sin(2n  1)x  sin x  . 2 sin x 2  sin x 2  sin(2n  1) x  sin(nx) cos(n  1) x  sin x 2  sin x (2  sin(2n  1)x) sin(nx) cos(n  1)x Sn  sin x 2  sin x (2  sin(2n  1) x) 

dUcen¼

1 2 sin x

.

123


lMhat´TI71  1) x k¿cUrbgHajfa 1  tan x tan(nx)  coscos(x ncos( nx) x¿KNna P   1  tan x tan(kx) . dMeNa¼Rsay  1)x k¿bgHajfa 1  tan x tan(nx)  coscos(x ncos( nx) eKman cos(n  1)x  cos(nx) cos x  sin(nx) sin x naM[ n

n

k 1

cos(n  1)x cos(nx) cos x  sin(nx) sin x   1  tan x tan(nx) cos x cos(nx) cos(nx) cos x cos(n  1)x 1  tan x tan(nx)  cos x cos(nx)

dUcen¼ x¿KNna

Bit

.

n

Pn 

 1  tan x tan(kx) k 1

n

 cos(k  1)x    cos x cos(kx)  k 1   1 1 cos x cos 2 x cos(n  1)x  . . . .... cos(nx) cos n x cos x cos 2 x cos 3x 1  cos n x cos(nx) 

124


lMhat´TI72 eK[RtIekaN ABC mYymanRCug BC  a , AC  b , AB  c tag S CaRkLaépÞ nig R CakaMrgVg´carikeRkA énRtIekaNen¼ . k¿cUrRsayfa a cos A  b cos B  c cos C  2RS . x¿Tajfa a cot A  b cot B  c cot C  4S . dMeNa¼Rsay 2S a cos A b cos B c cos C    k¿Rsayfa R 2

2

2

A

O

B

tag

O

H

C

Cap©iténrgVg´carikeRkARtIekaN

ABC

ehIy 125


CacMnuckNþalénRCug [BC] . RkLaépÞRtIekaN ABC KW S  S  S  S eyIgman OB  OC  R naM[ OBC CaRtIekaNsmát kMBUl O . ehIy OH  BC naM[ OH CakMBs´énRtIekaN OBC 1 eyIgman S  2 BC . OH kñúgRtIekaNEkg OBH eKman cos( BOH)  OH OB ¦ OH  OB.cos(BOH) 1 S  BC . OB . cos( BOH) eKán 2 H

OCA

OBC

OAB

OBC

OBC

 

 BOC BOH   BAC  A 2

eyIgman mMup©itnigmMucarikkñúgrgVg´s;at´FñÚrYm BC . 1 S  a .R cos A eKán 2 RsaydUcKñaeKán S  21 b .R cos B nig 1 S  c .R cos A . 2 ehtuen¼ S  21 a.R cos A  21 b.R cos B  21 c.R cos C OBC

OAC

OAB

126


1 R ( a cos A  b cos B  c cos C) 2 2S a cos A  b cos B  c cos C  R

S

. dUcen¼ x¿Tajfa a cot A  b cot B  c cot C  4S A a eyIgman a cot A  a cos  .a cos A  2R a cos A sin A sin A dUcKñaEdr b cot B  2R a cos B nig c cot C  2R c.cos C eKán a cot A  b cotB  c cotC  2R (a cos A  b cosB  c cosC) 2S a cos A  b cos B  c cos C  ( sRmayxagelI ) eday R dUcen¼ a cot A  b cot B  c cot C  4S . 2

2

2

2

2

2

2

2

2

2

2

2

2



127


lMhat´TI73 eKtag r nig R erogKñaCakaMrgVg´carikkñúg nigcarikeRkA énRtIekaN ABC mYy . r cos A  cos B  cos C  1  k¿cUrRsayfa R x¿cUrRsayfa R  2 r . dMeNa¼Rsay r k¿Rsayfa cos A  cos B  cos C  1  R cos A  cos B  cos C  1  2 sin2

A BC BC  2 cos cos 2 2 2

A A BC  2 sin cos 2 2 2 A A BC )  1  2 sin ( sin  cos 2 2 2 A BC BC )  1  2 sin ( cos  cos 2 2 2 A B C  1  4 sin sin sin 2 2 2  1  2 sin2

edayeKman ½ A (p  b)(p  c)  2 bc B (p  a)(p  c) C sin  , sin  2 ac 2 sin

(p  a)(p  b) ab 128


eKán eKTaj

sin

A B C (p  a)(p  b)(p  c) sin sin  2 2 2 abc

cos A  cos B  cos C  1  4

(p  a)(p  b)(p  c) abc

p(p  a)(p  b)(p  c) S2  1  1 abc p S .R R p. 4R S p.r r  1  1  1 p. R p.R R

r cos A  cos B  cos C  1  R

dUcen¼ . x¿Rsayfa R  2 r tamvismPaBkUsIu     2  . eKán (p  a)  (p  b)  2 (p  a)(p  b) 2p  a  b  2

(p  a)(p  c)

c  2 (p  a)(p  b)

eKTaj

(p  a)(p  b) 1 (1)  c 2 (p  b)(p  c) 1  (2) a 2 (p  a)(p  c) 1  (3) b 2

dUcKñaEdr nig KuNTMnak´TMng

(1) , ( 2) , (3)

Gg:nwgGg:eKán ½ 129


(p  a)(p  b)(p  c) 1  abc 8 A B C 1 sin sin sin  2 2 2 8

eKTaj A B C cos A  cos B  cos C  1  4 sin sin sin eday 2 2 2 eKán cos A  cos B  cos C  1  4 ( 81 )  23 Et cos A  cos B  cos C  1  Rr r 3 eKTaj 1  R  2 ¦ R  2 r . 

130


lMhat´Gnuvtþn_ nð Un  ( 2 ) cos , n  IN * 4 (n  1) n n 2 . sin  sin  cos 4 4 4 (n  1) n n 1 n U n  ( 2 ) sin  ( 2 ) sin 4 4 n

1> eK[sIVúténcMnYnBit k> cUrbgðajfa x> Tajbgðajfa

.

.

n

K> KNnaplbUk S   U n

k

  U1  U 2  U 3  ....  U n

k 1

.

U 0  0 , U 1  1  U n  2  2 U n 1  4 U n , n  IN

2>eK[sIVúténcMnYnBitkMnt;eday k>eKtag n  IN : Z n 1  U n 1  (1  i 3 )U n . bgðajfa Z  (1  i 3)Z x> cUrbgðajfa Z  2 (cos n3  i.sin n3 ) . K> TajrktYtUeTA U CaGnuKmn_én n . 3> eKmansIVúténcMnYnBit (U ) kMnt;elI IN edayTMnak;TMng ³ n U  1 nig cMeBaHRKb; n  IN : U  2U  2 cos 4 k>cUrbgðajfaeKGackMnt;cMnYnBit a nig b edIm,I[sIVút (Vn ) n 1

n

n

n

n

n

0

n 1

n

131


Edl n n U  V  a cos  b sin kMnt;edayTMnak;TMng 4 4 CasIVútFrNImaRt x> TajrktYtUeTA U n CaGnuKmn_én n . 4> eKmansIVúténcMnYnkMupøic (Z ) kMnt;eday ³ n

n

n

Z 0  2   3i 2 3 i   , n  IN Z Z  n 1 n 2 2 

k> eKtag n  IN : U  Z  1 . 3 i U  .U rYcTajrk U n CaGnuKmn_én n bgðajfa 2 n

n

n 1

n

n n n Z n  2 cos (cos  i. sin ) 12 12 12

x> RsaybBa¢ak;fa 5- eK[sIVúténcMnYnkMupøic (Z ) kMnt;edayTMnak;TMng ³ 1 2  i  n  IN : Z  Z Z  0 , Z  1 nig 2 k> tag n  IN : U  Z  Z . 1 i U  U cUrbgðajfa 2

.

n

0

n2

1

n

n 1

n 1

n 1 

1 i Zn 2

n

n

132


x> RsaybBa¢ak;fa

n n U n  cos  i. sin 4 4

.

n

K> tag S   U  . cUrRsayfa Z n 1  Sn rYcTajrk Z CaGnuKmn_én n . 6- eK[sIVút ( U n ) & (Vn ) kMnt;elI IN * eday n

k

k 0

n

1 2 n      U sin sin .... sin  n n2 n2 n2  V  1  2  .....  n  n n 2 n 2 n2 ( Vn )

k> bgðajfa CasIVútcuH ehIyRKb; x> eK[GnuKmn_

n  IN* : Vn 

1 2

.

x2 x3 f ( x )  x  sin x , g ( x )  1   cos x , h ( x )   x   sin x 2 6

cUrbgðajfa x  IR : f (x)  0 , g(x)  0 , h(x)  0 . 3 3 3 3 4  n  1 : 1  2  3  ...  n  n K> epÞógpÞat;fa 1 1 rYcTajfa V  6 . n  U  V . 

n

2

n

n

133


7-eK[ (U ) CasIVútnBVnþmantY U1 , U 2 , U 3 , ........, U n nigplsgrYm d  4k , k  Z . k> cUrRsaybBa¢ak;rUbmnþ ³ n

Sin

n

 SinU k  SinU1  SinU 2  SinU 3  .....  SinU n  k 1

Sin

n

 CosU k  CosU1  CosU2  CosU3  .....  CosUn  k 1

U  Un nd .Sin 1 2 2 d sin 2

U  Un nd . Cos 1 2 2 d sin 2

x> Gnuvtþn_ cUrKNnaplbUkxageRkam ³ S n  sin a  sin 2a  sin 3a  .....  sin( na ) C n  cos a  cos 2a  cos 3a  ......  cos( na )

8-eKBinitüsIVút (U ) kMnt;eday ³ n

U n  2  2  2  .......  2  2  (n)

. 134


k> edayeFIVvicarütamkMenIncUrbgðajfa x> eKman V

n

 U n  2 cos n 1 2

 2  2  2  ........  2  2 

.

(n)

Vn  2 sin

 2 n 1

cUrRsayfa . K> KNnalImIt lim U and lim 2 .V  . 9- eKmanGnuKmn_ f ( x)  sin x , x  IR n

n

n  

n  

n

x3 x  f ( x)  x 6

RKb; x  IR . k> cUrRsayfa 1 2 3 n x> eKBinitüplbUk S  sin n  sin n  sin n  ......  sin n . cUrrkkenSamGménplbUkenH . K> cUrKNna lim S . 10-eKmansIVút (U ) kMnt´eday n

n  

2

2

2

2

n

n

U 0  1   n n U n 1  2U n  cos 2  2 sin 2 , n  IN

k¿kMnt´BIrcMnYnBit

A

nig

B

edIm,I[sIVút

(Vn ) 135


U n  Vn  A cos

n n  B sin 2 2

kMnt´edayTMnak´TMng CasIVútFrNImaRt . x¿ cUrKNna U CaGnuKmn_én n . 11-eK[ S  tan x  2 tan2x  2 tan2 x  .......  2 tan2 x k¿ cUrbgHaj tanè  cot è  2 cot 2è . x¿ TajbgHajfa S  cot x  2 cot 2 x . 12-eK[ S  sin a  3 sin 3a  3 sin 3a  .......  3 sin 3a 3 1 sin x  sin x  sin 3x . k¿ cUrbgHajfa 4 4 3 a 1 x¿ TajbgHajfa S  4 sin 3  4 sin 3a . 1 1 1 1 13-eK[ S  sin a  sin 2a  sin 2 a  .......  sin 2 a . x 1 k¿ cUrbgHajfa cot 2  cot x  sin x . a x¿ TajbgHajfa S  cot 2  cot 2 a . 14-eK[ S  cos1 a  cos4 2a  cos4 2 a  ......  cos4 2 a 1 4 1 k¿ cUrbgHajfa cos x  sin 2x  sin x . n

2

2

n

n

n

n 1

n 1

n

3

3

2

3

n

n

2

n

3

n 1

n

n

n

2

n

n

n

2

n

2

2

2

2

2

n

2

2

n

2

136


x¿ TajbgHajfa

4 n 1 1  Sn  sin2 2n 1 a sin2 a

.

15-eK[ x x 2 x x n 2 x Sn  sinx cos  2 sin cos 2  .......... ..... 2 sin n cos n1 2 2 2 2 2 2

a 2 sin a  sin 2a  2 sin a cos 2 2

k¿ cUrbgHajfa . x 1 S  2 sin  sin 2 x . x¿ TajbgHajfa 2 2 x x x 16-eK[ P  cos x. cos 2 .cos 2 .......... cos 2 . n

n

n

n

cUrbgHajfa

2

Pn 

1

2

. n 1

sin 2 x x sin n 2

n

.

17-eK[ 1 1 1 1 Pn  (1  )(1  )(1  ).....(1  ) a a a cos a cos cos 2 cos n 2 2 2 1 tan x 1  cos x tan x 2

.

k¿ cUrbgHajfa

x¿ KNnaplKuN

Pn

. 137


18-cUrRsaybBa¢k´smPaBxageRkam ½ x c¿ tan è sin 2è  2 sin è k¿ tan(ð4  x)  11 tan tan x x¿ q¿ sin( x  y) tan x  tan y 1  sin 2a 1  tan a  C¿  K¿ cos( cos 2a 1  tan a x  y) 1  tan x tan y X¿ sin(á  â) sin(á  â)  sin á  sin â â  tan ã  tan á tan â tan ã g¿ tan(á  â  ã)  1 tan(taná átan tan â  tan â tan ã  tan ã tan á) 19-eK[RtIekaN ABC mYy . cUrRsaybBa¢ak´smPaBxageRkam ½ k¿ sin A  sin B  sin C  4 cos A2 cos B2 cos C2 x¿ cos A  cos B  cos C  1  4 sin A2 sin B2 sin C2 K¿ tan A  tan B  tan C  tan A tan B tan C X¿ cot A2  cot B2  cot C2  cot A2 cot B2 cot C2 g¿ sin A  sin B  sin C  2 sin A sin B cos C 2

sin3 x  cos3 x 1  1  sin 2 x sin x  cos x 2

tan(a  b) tan2 a  tan2 b  cot(a  b) 1  tan2 a tan2 b

2

2

2

2

2

138


2

2

2

c¿ sin A  sin B  sin C  2  2 cos A cos B cos C q¿ cos A  cos B  cos C  1  2 cos A cos B cos C C¿ sin 2A  sin 2B  sin 2C  4 sin A sin B sin C Q¿ cos 2A  cos 2B  cos 2C  1  4 cos A cos B cos C j¿ sin A  sin B  sin C  2 sin B sin C cos A 20-cUrRsaybBa¢ak´smPaBxageRkam ½ k¿ 1  cos 2x  cos 4x  cos 6x  4 cos x cos 2x cos 3x sin 4 x  sin 2 x x¿ cos  tan 3x 4 x  cos 2 x K¿ sin x  sin 2x  sin 3x  (1  2 cos x) sin 2x X¿ cos x  cos 2x  cos 3x  (1  2 cos x) cos 2x 21-cUrRsaybBa¢k´sMenIxageRkam ½  c) k¿ tan a  tanb  tan c  tan a tanb tan c  cossin(aacos bb cos c x¿ sin (a  b)  sin (a  b)  2sin(a  b) sin(a  b) cos2a  sin 2a K¿ cos a  cos b  1  cos(a  b) cos(a  b) a  b) X¿ tan a  tan b  cos(a2sin( b)  cos(a  b) 2

2

2

2

2

2

2

2

2

2

2

139


g¿

tan x tan(

ð ð  x) tan(  x)  tan 3x 3 3

22-k¿ RsaybBa¢ak´ÉklkçN¼PaB ½ sin( x  y) sin( y  z) sin(z  x)   0 cos x cos y cos y cos z cos z cos x

x¿ TajbgHajfa ½ sin3 (x  y) sin3 (y  z) sin3 (z  x) sin(x  y) sin(y  z) sin(z  x)    3 cos3 x cos3 y cos3 y cos3 z cos3 z cos3 x cos2 x cos2 y cos2 z cot 2 a  1 cot 2a  2 cot a

23-cUrRsaybBa¢ak´fa 24-cUrRsaybBa¢ak´fa sin 4x  4 sin x cos x  8 sin x cos x nig cos 4x  8 cos x  8 cos x  1 . 25-cUrbgHajfa 2(tan a  x)(1  x tan a) (1  x ) sin 2a  2 x cos 2a  . 1  tan a 26-RsaybBa¢ak´smPaBxageRkam ½ 2 tan x  cot x  k¿ sin 2 x x¿ cot x  tan x  2 cot 2x 3

4

2

2

2

140


K¿ tan(ð4  x)  tan(ð4  x)  2 tan 2x X¿ 1 sincosx x  tan 2x g¿ tan(ð4  x)  tan(ð4  x)  cos22x 27-cUrsRmYlkenßamxageRkam ½ 1  cos x  sin x k¿ 1  cos x  sin x cos x  sin x x¿ 11  cos x  sin x K¿ sin12xcos sin2x4xcos sin4x6x X¿ sinsin(x x siny)y u  cos v g¿ 1cos cos( u  v) 28-cUrbMElgCaplKuNénkenßam S  cos x  cos 2 x  cos 3x  cos 4 x . 29-eK[ a nig b CamMuRsYcEdl sin a  21 nig sin b  6 4 2 cUrKNnatémøGnuKmn_rgVg´énmMu a  b nig a  b 141


rYcTajrktémøén b Car¨adüg´. 30-eK[ a CamMuRsYcEdl cos a  2 2 2 . cUrKNna cos 2a rYcTajrktémøénmMu a Cara¨düg´ . 31-cUrbMElgplbUk S  sin a  sin b  sin c  sin(a  b  c) CaplKuNktþa . 21 ni g cos C  32-RtIekaN ABC mYyman cos B  20 29 29 cUrrkRbePTénRtIekaN

ABC

.

33-cUrkMnt´rktémøGtibrma nig Gb,brma ( ebIman ) énGnuKmn_xageRkam ½ k¿ y  3 sin x  4 cos x  7 x¿ y  5 sin x  12 cos x  17 K¿ y  sin x  cos x 4

4

142


2

2

X¿ y  3 sin x  4 sin x cos x  5 cos x  2 g¿ y  sin x  cos x c¿ y   sin x  sin1 x    cos x  cos1 x  q¿ y   sin x  sin1 x    cos x  cos1 x  C¿ y  sin x  cos x j¿ y  tan x  2 3 tan x  4 34-sRmYl A  2  2  2  ....  2  cos a Edl ð 0a . 2 6

6

2

2

2

2

2

2

2

3

2

3

3

8

3

8

2

n

35-eK[smIkardWeRkTIBIr (E) : x 2  (m2  m) x  m  2  0

eKsnµtfasmIkaren¼man¦sBIrtagerogKñaeday tan a nig tan b . ð a  b  k¿ cUrkMnt´témøéná¨r¨aEm¨Rt m edIm,I[ 3 x¿ eda¼RsaysmIkarxagelIcMeBa¼ m Edlán 143


rkeXIj . K¿edayeRbIlTæplxagelIcUrTajrktémøRákdén ð tan . 12 ð tan 36-k¿ cUrKNnatémøRákdén 8 . x¿ cUreda¼RsaysmIkar 100  10 ð 37-k¿ cUrKNnatémøRákdén tan 5 . x¿ cUreda¼RsaysmIkar

1 log 2 2 1(tan x )

0

5 ) sin2 2x  3 (5  2 5) cos4 x  0 sin x  (2  2 ð ð cos sin 10 10 4

38-k¿ cUrKNnatémø®ákdén nig x¿ cMeBa¼RKb´cMnYnBit x , y  IR cUrbgHajfa ½ ð x  ( x  y)  4( x  y ) sin 10 2

2

39-eK[sIVúténcMnYnBit

2

(Un )

2

2

kMnt´eday 144


ð     a U 2 cos a , 0  0 2   Un1  2  Un , n  IN 

edayeFVIvicartamkMenIncUrbgHajfa U  2 cos 2a . 40-eK[GnuKmn_ f(x, y)  (a cos x  b cos y)  (a sin x  b sin y) 2 2 2 cUrRsayfa f( x, y)  ( a  b ) . 41-k¿ cUrRsaybBa¢ak´TMnak´TMng tan x  cot x  2 cot 2x x¿ cUrKNnaplbUkxageRkam ½ n

n

2

1 a 1 a 1 a tan  2 tan 2  ....  n tan n 2 2 2 2 2 2 B n  tan b  2 tan 2b  2 2 tan 2 2 b  .....  2n tan 2n b A n  tan a 

2

42-k¿ cUrRsaybBa¢ak´fa tan2x .tan x  tan2x  2 tan x x¿ cUrKNnaplbUk S   21 tan 2 a tan 2 a 43-k¿ cUrRsayfa sin x  41 ( 3 sin x  sin 3x) x¿ cUrKNna n

k 1

n

2

k

k

k 0

3

S n  sin3 a 

1 1 1 sin3 3a  2 sin3 32 a  ....  n sin3 3n a 3 3 3 1 4 1   cos 2 x sin 2 2 x sin 2 x

44-k¿ cUrRsayfa

145

2


1 4 4n Sn    ....  2 2 cos a cos 2a cos 2 2 n a 1  cot x  cot 2 x sin 2 x 1 1 1 1 Sn     ....  sin a sin 2a sin 2 2 a sin 2 n a x cot 1 2  1 cos x cot x

x¿ cUrKNna 45-k¿ cUrRsayfa x¿ cUrKNna

46-k¿ cUrRsayfa x¿ cUrKNnaplKuN ½ Pn  (1 

1 1 1 1 )(1  )(1  ).....( 1  ) cos a cos 2a cos 2 2 a cos 2 n a

47-k¿ cUrRsayfa sin(nx)  cos(n  1) x cos(nx)    . cot x n n 1 n  cos x  cos x cos x 

x¿ cUrKNna Sn 

sin x sin 2 x sin 3x sin(nx)    ....  cos x cos2 x cos3 x cos n x

48-k¿ cUrRsayfa 1 1 tan(n  1)x  tan(nx)  cos(nx). cos(n  1)x sin x

x¿ KNna Sn 

1 1 1   ....  cos x cos 2 x cos 2 x cos 3x cos(nx) cos(n  1)x 146


49-k¿ cUrRsayfa tan(n  1)x  tan(nx)  tan x 1  tan(nx) tan(n  1)x 

x¿ KNnaplbUk Sn  tan x tan 2 x  tan 2 x tan3x  .....  tan(nx) tan(n  1)x

50-k¿ cUrRsaybBa¢ak´fa 2 cos x x¿ cUrKNnaplKuN ½

 1

1  2 cos 2 x 1  2 cos x

Pn  (2 cos a  1)(2 cos 2a  1)(2 cos 2 2 a  1).....(2 cos 2 n a  1)

51-KNnaplKuN x x x Pn  cos x . cos . cos 2 ......... cos n 2 2 2

52-KNnaplKuN x 2 x 2 x Pn  (1  tan x)(1  tan )(1  tan 2 ).........(1  tan n ) 2 2 2 2

2

53-KNnaplKuN ½ Pn  (cos a  cos b)(cos

a b a b a b  cos )(cos 2  cos 2 ).....(cos n  cos n ) 2 2 2 2 2 2

54-KNnaplKuN x x x Pn  (1  4 sin2 x)(1  4 sin2 )(1  4 sin2 2 ).....(1  4 sin2 n ) 3 3 3 147


55-KNnaplKuN x x x Pn  (3  4 sin2 x)(3  4 sin2 )(3  4 sin2 2 )......(3  4 sin2 n ) 3 3 3

56-KNnaplKuN x 2 x 2 x 3  tan 3  tan 2 3  tan n 3  tan2 x 3 3 3 Pn  . . ..... 1  3 tan2 x 1  3 tan2 x 1  3 tan2 x 2 x 1  3 tan n 3 3 3 2

57-k¿ cUrRsayfa

tan3 x 1  ( tan 3x  3 tan x) 1  3 tan2 x 8  1  3 k tan 3 a n  k  Sn    3 2 k  1 3 tan 3 a  k0   

x¿ cUrKNnaplbUk 58-k¿ cUrRsayfa

tan3 x 1  tan 2 x  tan x 2 1  tan x 2 n

x¿ cUrKNnaplbUk

Sn 

 k0

 1 3 k   k tan 2 a  2  2 k  1  tan 2 a     

59-KNnaplKuN n

n

Pn  (tana  cota)(tan2 a  cot2 a)(tan4 a  cot4 a)......(tan2 a  cot2 a)

60-bgHajfa

cos a cos 2a cos 4a.... cos 2

n 1

sin 2 n a a n 2 sin a

.

148


(a n )

61- eK[sIVúténcMnYnBit     

kMnt´eday

1 a1  2 3

an 1 

a n  3a n 1 , n  IN 4

k¿ cUrRsaybBa¢ak´fa 0  a  1 . x¿ eKtag a  cos è . cUrrkRbePTénsIVút K¿ KNna è nig a CaGnuKmn_én n . n

n

n

n

62-eKmansIVút

( n )

n

(b n )

kMnt´eday

3 b0  2

bn

nig b  1  1  4b , n  IN k¿ eKBinitüsIVút (è ) Edl 0  è  ð2 n 1

2

n

n

n

, n  IN

tan èn bn  2

ehIy . cUrkMnt´rkRbePTénsIVút (è ) ? x¿ KNna è nig b CaGnuKmn_én n . ð t  tan ( t ) 63-eKmansIVút n kMnt´eday 3 n

n

n

0

149


tn 1  2  4  tn2 , n  IN

nig k¿ cUrbgHajfa 0  t n  2 . x¿ tag tn  2 sin èn . cUrkMnt´rkRbePTénsIVút ( ) ? K¿ KNna èn nig tn CaGnuKmn_én n

64-eKmansIVút

1 u0  3

n

.

nig

un 1  un  1  un2 , n  IN

k¿ tag u  cot è Edl 0  è  ð2 , n  IN . cUrkMnt´rkRbePTénsIVút (èn ) x¿ KNna è nig u CaGnuKmn_én n . 65-eK[ (u ) CasIVútviC¢manEdl u0  2 2u nig u  1  u . KNna un . 66-eK[RtIekaN ABC mYYymanRCúg a , b , c . n

n

n

n

n

n

2 n 1

n

n

150


cUrRsaybBa¢ak´fa cosa A  cosb B  cosc C  21 ( 1a  b1  1c ) 67-eda¼RsaysmIkarxageRkam ½ k¿ cos(2x  ð3)  cos x x¿ sin 2x  sin( ð3  x) K¿ sin( ð4  x)  23 X¿ sin(2x  ð3)  sin(23ð  x) g¿ tan3x  tan(x  ð4) 68-eda¼RsaysmIkar ½ k¿ sin 3x  cos(x  ð6) x¿ sin(2x  ð4)  cos( ð6  x) K¿ tan(ð4  x)  cot( ð3  2x) X¿ tan(ð3  2x)  tan(3x  ð6) g¿ tan(3x  ð3 )  cot ð5 69-eda¼RsaysmIkarxageRkam ½ k¿ 2 sin x  3 sin x  1  0 2

151


2

x¿ 4 sin x  2( 2  1) sin x  2  0 K¿ 2 cos x  (1  3) cos x  23  0 X¿ tan x  (1  3) tan x  3  0 g¿ 3 tan x  4 tan x  3  0 70-eda¼RsaysmIkar ½ k¿ 4 sin x  2( 3  1) sin x  3  0 x¿ 3 sin x  (1  3) sin x cos x  cos x  0 K¿ tan x  ( 3  1) tan x  3  0 X¿ 2 log cos x  3 log cos x  1  0 g¿ log sin x  3 log sin x  2  0 71-eda¼RsaysmIkarxageRkam ½ k¿ ( 3  1) sin x  2 3 sin x cos x  ( 3  1) cos x¿ 3 cos x  sin x  2 K¿ sin x  cos x  2 X¿ cos x  3 sin x  1 2

2

2

2

2

2

2

2 2

2

2

2

2

2

2

x0

152


72-eda¼RsaynwgBiPakßasmIkar m cos x  sin x  2m . 73-eKmansmIkar (2 cos á  1)x  2x  cos á  0 Edl 0  á  ð . kMnt´témø á edIm,I[smIkarman¦sBIrepÞógpÞat´ 1 1   4 sin á  0 . x' x' ' 74-eK[smIkar ½ 2

x 2  2 x sin   ( 2  3 )(1  4 cos 2 )  0 , 0    2

k¿ cUrRsayfasmIkaren¼man¦sCanic©RKb´  . x¿ cUrrkTMnak´TMngrvag¦s x' nig x'' minGaRs&ynwg 

75-eda¼RsaysmIkar ½ k¿ sin x  3 sin(72ð  x)  tan x  3 x¿ 2 cot x cos x  4 cos x  1  sin1 x K¿ 2 cos 2x  2 cos x sin x  cos x X¿ sin (x  ð4)  2 sin x 2

2

2

2

2

3

153


g¿ tan2x  tan x tan(ð4  x) tan(x  ð4) 76-eda¼RsaysmIkar ½ k¿ cos x cos 3x  sin x sin 3x  42 x¿ cos x(2 sin x13sin22) x 2 cos x  1  1 K¿ 4 sin(x  ð4 ) cos(x  12ð )  3  1 X¿ (1  3) tan x tan(ð3  x) tan(ð3  x)  3  1  cos1 3x 5ð 77-k¿ cUrKNnatémø®ákdén tan12ð nig tan 12 x¿ eda¼RsaysmIkar tan x  5 tan x  5 tan x  1  0 78-k¿ KNnatémø®ákdén tan ð8 x¿ eda¼RsaysmIkar 1  log tan x  0 . 79-eda¼RsayRbB&næsmIkar 3

3

2

2

3

2

2

1 2

7  3 2   sin x 3 sin x sin y  8   3 sin 2 x sin y  sin 3 y  5 2  4

80-eda¼RsaysmIkar 81-eda¼RsaysmIkar

logsin x cos x sin x . logsin x.cos x cos x x2  x 2 cos  2x  2x 6 2

.

154

1 4


82-eda¼RsaysmIkar ½ a)

ð sin3 ( x  )  2 sin x 4 cos x

b)

 7  4 3

cos x

 7  4 3

4

( RbLgGaharUbkrN_eTArusßI éf¶ 05 emsa qñaM 2000 ) 83-eK[smIkar m sin x  (m  1) cos x  cosm x k¿ kMnt´ m edIm,I[smIkaren¼man¦s . x¿ eKtag x , x Ca¦sBIrénsmIkarxagelI ehIyepÞógpÞat´ x  x  ð2  kð cUrKNna cos 2(x  x ) . ( k  Z ) 84-RsaybBa¢ak´faebI 1

2

1

2

1

tan x 

1 y  1 y

2

, 1 y  1 , y  0

1 y  1 y

ena¼eKán y  sin 2x . 85-eK[RtIekaN ABC mYYyman cos A  cos B  cos C  1 cUrkMnt´RbePTénRtIekaN ABC ? 86-eKmansmPaB sina x  cosb x  a 1 b . 2

4

2

2

4

155


sin10 x cos10 x 1   a4 b4 (a  b) 4

cUrRsayfa 87-eK[ cos a  31 , cos b  38 nig cos c  75 . cUrRsayfa tan 2a  tan 2b  tan 2c  1 ? 88-eKtag a , b , c CaRCúgrbs´RtIekaN ABC Edl tan A2 . tan B2  31 . k¿ cUrRsayfa c  a 2 b . x¿ cUrbgHajfa a  b  6abc  8c . 89-eK«bmafasmIkar ax  bx  c  0 man¦BIrtag eday tan  nig tan  . cUrKNna 2

2

3

3

2

3

2

M  a sin 2 (  )  b sin(  ) cos(  )  c 2 cos(  )

CaGnuKmn_énelxemKuN a ,b , c . 90-cUrbgHajfacMnYn cos 7ð Ca¦ssmIkar 8x  4 x  4 x  1  0 . 91-cUrbgHajfacMnYn cos ð9 , cos 79ð , cos 139ð 3

2

156


3

Ca¦ssmIkar 8x  6x  1  0 . 92-cUrbgHajfa sin 7ð  sin 27ð  sin 37ð   27 . 93-cUrbgHajfa tan3 tan51 tantan57 tan63 tan69  tan9 94-KNnatémøénplKuN P  tan10 tan50 tan70 . 95-eK[smIkardWeRkTIBIr ½ o

o

o

o

0

o

0

o

0

(E) : x 2  (m  1) x  2m  3  0

k¿ kMnt´ m edIm,I[smIkaren¼man¦sBIrepßgKña . x¿ «bmafa tan a nig tanb Ca¦srbs´smIkar (E) kMnt´ m edIm,I[ sin(a  b)  cos(a  b) . 96-cUrKNnatémøén A  sin 7ð  sin 27ð  sin 47ð . 97-cUrbgHajfa tan20 tan 40 tan80  3 . 98-eK[smIkardWeRkTIbI ½ (E) : 3x  (m  3) x  (3  4 3) x  3  0 Edl m Caá¨ra¨Em¨Rt eK«bmafasmIkarman¦sbItagerogKñaeday 2

o

3

2

o

2

o

2

157


. k¿ kMnt´témø m edIm,I[ á  â  ã  34ð . x¿ cUrkMnt´ á , â , ã Edl 0  á  â  ã  ð2 cMeBa¼témø m EdlánrkeXIj xagelIen¼ . 99-eK[smIkar (E) : 3x  mx  m 3  9  0 Edl m Caá¨ra¨Em¨Rt . eK«bmafasmIkarman¦sbItagerogKñaeday tan  nig tan  . sin(  ) 2 3  k¿ kMnt´témø m edIm,I[ cos( .   ) 3 x¿ cUrkMnt´  ,  Edl 0  á  â  ð2 cMeBa¼témø EdlánrkeXIjxagelIen¼ . 100-KNnaplbUk ½ tan á , tan â , tan ã

2

m

1 x 1 x 1 x tan2  2 tan2 2  .........  n tan2 n 4 2 4 2 4 2 3ð 0 á,â,ã ð áâã  4 158

S n  tan2 x 

101-eK[mMubI

Edl

.


cUrRsaybBa¢ak´fa ½ k¿ sin á  sin â  sin ã  3 22 x¿ cos á  cos â  cos ã  3 22 102-eK[ n cMnYnBitviC¢man a , a , a , ......, a . cMeBa¼RKb´ x , x , x ,....., x  IR RsaybBa¢ak´fa ½ 1

2

3

n

n

1

2

3

n

2

n  n      x a sin x a cos  k  k   k  k  k 1   k 1 

103-eda¼RsaysmIkar CacMnYnKt´FmµCati .

2

 (a1  a 2  a 3  ....  a n )2

cosn x  sinn x  1

Edl

( 3rd IMO 1961 )

104-cUrkMnt´RKb´cemøIyBitsmIkar cos x  cos 2 x  cos 3x  1 . 2

2

2

( 4th IMO 1962 )

105-cUrbgHajfa

cos

ð 2ð 3ð 1  cos  cos  7 7 7 2

( 5th IMO 1963 )

159

n


106-cUrkMnt´RKb´ x éncenøa¼ 0 , 2 EdlepÞógpÞat´ 2 cos x  | 1  sin 2 x  1  sin 2 x | 

2

.

( 7th IMO 1965 )

107-bgHajjfa 1 1 1 n  .......... .     cot x cot 2 x n sin 2 x sin 4 x sin 2 x

cMeBa¼RKb´cMnYnKt´FmµCati n nigRKb´cMnYnBit x Edl sin 2 x  0 . ( 8th IMO 1966 ) n



160


Van Vannak DC