Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onMUC LUC Trang Cng thc lng giac cn nm vng ------------------------------------------------------------------------2A Phng trinh lng giac c ban ------------------------------------------------------------------5Bai tp ap dung ------------------------------------------------------------------------------------------5Hng dn giai bai tp ap dung -----------------------------------------------------------------------8Bai tp ren luy n ----------------------------------------------------------------------------------------29B Phng trinh bc hai va bc cao i vi mt ham lng giac ---------------------------32Bai tp ap dung ------------------------------------------------------------------------------------------33Hng dn giai bai tp ap dung -----------------------------------------------------------------------35Bai tp ren luy n ----------------------------------------------------------------------------------------56C Phng trinh bc nht theo sin va cos ----------------------------------------------------------59Bai tp ap dung ------------------------------------------------------------------------------------------59Hng dn giai bai tp ap dung -----------------------------------------------------------------------62Bai tp ren luy n ----------------------------------------------------------------------------------------81D Phng trinh lng giac ng cp ---------------------------------------------------------------84Bai tp ap dung ------------------------------------------------------------------------------------------85Hng dn giai bai tp ap dung -----------------------------------------------------------------------87Bai tp ren luy n ----------------------------------------------------------------------------------------92E Phng trinh lng giac i xng ---------------------------------------------------------------93Bai tp ap dung ------------------------------------------------------------------------------------------94Bai tp ren luy n ----------------------------------------------------------------------------------------96F Phng trinh lng giac cha cn thc va tri tuyt i -----------------------------------97Bai tp ap dung ------------------------------------------------------------------------------------------97Bai tp ren luy n ----------------------------------------------------------------------------------------99G Phng trinh lng giac khng mu mc ----------------------------------------------------101Bai tp ap dung ------------------------------------------------------------------------------------------102Bai tp ren luy n ----------------------------------------------------------------------------------------104H Phng trinh lng giac cha tham s Hai phng trinh tng ng ----------106Bai tp ap dung ------------------------------------------------------------------------------------------106Bai tp ren luy n ----------------------------------------------------------------------------------------112I H phng trinh lng giac ------------------------------------------------------------------------116Bai tp ap dung ------------------------------------------------------------------------------------------117J H thc lng trong tam giac Nhn dang tam giac ---------------------------------------121Bai tp ap dung ------------------------------------------------------------------------------------------122Bai tp ren luy n ----------------------------------------------------------------------------------------125" Cn cu bu thng minh"Page 1Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)CAC CNG THC LNG GIAC NM VNGCCng thc c ban2 2sin x cos x 1 + = tanx.cot x 1 = sinxtanxcosx=cosxcot xsinx= os2211 tan xc x+ = 2211 cot xsin x+ =CCng thc cung nhn i Cng thc ha bc Cng thc cung nhn basin2x 2sinx.cosx =2 22 2cos x sin xcos2x2cos x 1 1 2sin x-=- =-os21 c2xsin x2-= osos21 c2xc x2+=3sin3x 3sinx 4sin x = -3cos3x 4cos x 3cosx = -CCng thc cng cung ( )sin a b sina.cosb cosa.sinb = ( )os c a b cosa.cosb sina.sinb = m( )tana tanbtan a b1 tana.tanb++ =-( )tana tanbtan a b1 tana.tanb-- =+ 1 tanxtan x4 1 tanx + + = - 1 tanxtan x4 1 tanx - - = + CCng thc bin i tng thanh ticha b a bcosa cosb 2cos .cos2 2+ -+ = a b a bcosa cosb 2sin .sin2 2+ -- =-a b a bsina sinb 2sin .cos2 2+ -+ = a b a bsina sinb 2cos .sin2 2+ -- =( )sin a btana tanbcosa.cosb++ =( )sin a btana tanbcosa.cosb-- =CCng thc bin i tich thanh tng( ) ( )cos a b cos a bcosa.cosb2+ + -=( ) ( )sin a b sin a bsina.cosb2+ + -=( ) ( )cos a b cos a bsina.sinb2- - +=Mt s cng thc thng dung khac sinx cosx 2sin x 2cos x4 4 + = + = - sinx cosx 2sin x 2cos x4 4 - = - = + 4 4 21 cos4xcos x sin x 1 s3 1in 2x2 4++ =- = 6 6 23 cos4xcos x sin x 1 s5 3in 2x4 8++ =- =Page 2" All the flower of tomorrow are in the seeks of today"Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onMt s lu y :^ iu kin co nghi m cua phng trinh sinxcosx=a=a la:1 1 - a .^ Khi giai phng trinh cocha cac ham stan hoccot , comu s hoc cn bc chn thi nht thit phai t iu kin phng trinh xac inh. Phng trinh chatanx, iu kin: ( )cosx 0 x k k2p + p . Phng trinh chacot x, iu kin:( )sinx 0 x k k p . Phng trinh cha ca tanx vacot x, iu kin: ( )x k. k2p .^ Khi ti m c nghim phai ki m tra (so) vi iu kin. Ta thng dung mt trong cac cach sau y ki m tra iu kin: Kim tra trc tip bng cach thay giatri cua x vao biu thc iu kin. Nu khi thvao, giatriy lam ng thc ung thi nhn nghim, nu sai thiloai nghim. Dung ng tron lng giac, nghia labiu din cac ngon cung cua iu kin va cung cua nghim. Nu cac ngon cung nay trung nhau thita loai nghim, nu khng trung thita nhn nghim.Cach biu din cung goc lng giac trn ng tron: " Nu cung hoc goc lng giac AM cos o lak2npa +00k.360hay an + vik ,n+ thi con i mM trn ng tron lng giac cach u nhau".Vi du 1:Nu s AM k23p= + p thico mt imM tai vitri 3p (ta chonk 0 = ).Vi du 2:Nus AM k6p= + p thi co2 imM tai vitri 6pva76p (ta chon k 0,k 1 = =).Vi du 3:Nu s 2AM k.4 3p p= +thico 3 i mM tai cac vitri11;4 12p pva1912p, ( )k 0;1;2 =.Vi du 4:Nu s k2AM k.4 2 4 4p p p p= + = +thico 4 i mM tai cac vitri 4p,34p, 54p;74p (ng vi cac vitri k 0,1,2,3 =).Vi du 5:Tng hp hai cungx k6p=- + p vax k3p= + pBiu din cungx k6p=- + p trn ng tron thi co2 im tai cac vi tri:6p-va56p" Cn cu bu thng minh"Page 3 giai c phng trinh lng giac cung nh cac ng dung cua no, cac ban hoc sinh cn nm vng tt ca nhng cng thc lng giac. o la hanh trang, la cng cu cn thit nht chinh phuc th gii mang tn: "Phng trinh lng giac"Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Biu din cungx k3p= + p trn ng tron thico 2 im tai cac vi tri: 3p va 43p.Tng hp hai cung gm 4 i m nh hinh veva cung tng hp la:x k3 2p p= +^ i vi phng trinh 221 1cos x cosx2 21 1sin x sinx2 2 = = = = ta khng nn giaitrc tip vikhi o co ti 4 nghi m, khi kt hp va so sanh vi iu kin rt phc tap, ta nn ha bc la ti u nht. Nghia la: 22221cos x2cos x 1 0 cos2x 021 cos2x 0 2sin x 1 0sin x2 = - = = = - = = .Tng ti vi phng trinh 22sin x 1 sinx 1cosx 1 cos x 1 = = = = ta khng nn giai nh th, mann bin i da vao cng thc 2 2sin x cos x 1 + =. Luc o: 2 22 2sin x 1 cos x 0 cosx 0sinx 0 cos x 1 sin x 0 = = = = = = ^ S dung thanh thao cu thn chu:'' Cos i Sin bu Phu cheo '' y co thxem la cu thn chu''n gian, dnh'' trong lng giac nhng no lai ong vai tro lamt trong nhng nhn t cn thit, hiu qua nht khi giai phng trinh lng giac. Cos i, nghia lacos cua hai goc i nhau thi bng nhau, tc la( )cos cos - a = a, con cac cung goc lng giac con lai thibng '' '' chinh no: ( ) ( ) ( )sin sin , tan tan , cot tan - a =- a - a =- a - a =- a Sin bu, nghia lasin cua hai goc bu nhau thi bng nhau, tc la ( )sin sin p- a = a, con cac cung goc lng giac con lai thibng '' '' chinh no: ( ) ( ) ( )cos cos , tan tan , cot tan p- a =- a p- a =- a p- a =- a Phu cheo, nghia la vi hai goc phu nhau (cotng bng 900) thisin goc na y bng cos goc kia vangc lai, tc la: sin cos , cos sin , tan cot , cot tan2 2 2 2 p p p p - a = a - a = a - a = a - a = a Ta hay th n vi vidu nho sau y th y c hiu qua cua '' cu thn chu '' nay: Giai phng trinh lng giac:sinu cosv =Ro rang, phn phng trinh lng giac c ban, ta chibit cach giai sao cho phng trinh sinu sinv = , v y con phng trinhsinu cosv =thi sao ?Cu tra li y chinh laphu cheo, bi: sinu cosv sinu sin v2 p = = - ( )u v k2 u v k2 , k2 2p p= - + p = ++ p . Qua vidu nay, chc hn nu trong bai gp nhng phng trinh dang nh 2sinx cos x3 p = - thicac ban hoc sinh se khng con ca m th y lung tung na. Page 4" All the flower of tomorrow are in the seeks of today"/35/64/3/6OPhng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on Mt s cung goc hay dung khac:

( )( )sin x k2 sinxcos x k2 cosx+ p =+ p =va( )( )( ) sin x k2 sinxkcos x k2 cosx+p+ p =-+p+ p =-.A PHNG TRINH LNG GIAC C BANCDang: u v k2sinu sinvu v k2 = + p= =p- + pc bit: sinx 0 x ksinx 1 x k22sinx 1 x k22= = p p= = + pp =- =- + pCDang: u v k2cosu cosvu v k2 = + p= =- + pc bit: cosx 0 x k2cosx 1 x k2cosx 1 x k2 p= = + p= = p=- =p+ p CDang:tanu tanv u v kk : u, v k2= = + pp + pc bit: tanx 0 x ktanx 1 x k4 = = pp= = + pCDang:cot u cot v u v kk : u, v k= = + p p c bit: cot x 0 x k2cot x 1 x k4 p= = + p p= = + pBAI TP AP DUNGBai 1. Giai phng trinh:( )cos3x 4cos2x 3cosx 4 0 , x 0;14 - + - = * " Bai 2. Giai phng trinh: ( ) ( ) ( )2cosx 1 2sinx cosx sin2x sinx - + = - *Bai 3. Giai phng trinh:( ) cos3x cos2x cosx 1 0 + - - = *Bai 4. Giai phng trinh: ( ) sinx cosx 1 sin2x cos2x 0 + ++ + = *Bai 5. Giai phng trinh:( ) ( ) sinx 1 cos2x sin2x 1 cosx + + =+ *Bai 6. Giai phng trinh: ( ) 1 1 74sin xsinx 43sin x2 p + = - * p - Bai 7. Giai phng trinh:( )

4 47sin x cos x cot x cot x8 3 6 p p + = + - * " Cn cu bu thng minh"Page 5Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Bai 8. Giai phng trinh: ( ) 4 44sin 2x cos 2xcos 4xtan x tan x4 4+= * p p - + Bai 9. Giai phng trinh:( ) 3 x 1 3xsin sin 110 2 2 10 2 p p - = + Bai 10. Giai phng trinh:( ) sin 3x sin2xsin x 14 4 p p - = + Bai 11.( )

38cos x cos3x 13 p + = Bai 12. Giai phng trinh:( ) 32sinx 2sinx 14 p + = Bai 13. Giai phng trinh:( )

3sinx 2sinx 14 p - = Bai 14. Giai phng trinh:( ) cosx cos2x cos3x cos4x 0 + + + = *Bai 15. Giai phng trinh: ( )

2 2 23sin x sin 2x sin 3x2+ + = * .Bai 16. Giai phng trinh:( )

2 2 2sin x sin 2x sin 3x 2 + + = *.Bai 17. Giai phng trinh:( )

2 2 2 2sin x sin 3x cos 2x cos 4x + = + *Bai 18. Giai phng trinh:( ) 2 2 2 2sin 3x cos 4x sin 5x cos 6x - = - *Bai 19. Giai phng trinh:( )sin 2 25x 9xcos3x sin7x 2 2cos4 2 2 p + = + - * Bai 20. Giai phng trinh: ( ) 2 2 2sin x cos 2x cos 3x = + *Bai 21. Giai phng trinh:( ) 22sin 2x sin7x 1 sinx + - = *Bai 22. Giai phng trinh: ( ) sinx sin2x sin3x 1 cosx cos2x + + =+ + *Bai 23. Giai phng trinh: ( )

3 3 3sin xcos3x cos xsin3x sin 4x + = *Bai 24. Giai phng trinh:( )2 3cos10x 2cos 4x 6cos3xcosx cosx 8cosxcos 3x + + = + *Bai 25. Giai phng trinh: ( ) 3 3 24sin x 3cos x 3sinx sin xcosx 0 + - - = *Bai 26. Giai phng trinh:( ) ( ) ( )

22sinx 1 3cos4x 2sinx 4 4cos x 3 + + - + = *Bai 27. Giai phng trinh:( ) ( ) 6 6 8 8sin x cos x 2 sin x cos x + = + *Bai 28. Giai phng trinh:( ) ( ) 8 8 10 105sin x cos x 2 sinx cos x cos2x4+ = + + *Bai 29. Giai phng trinh:( ) ( )

3 3 5 5sin x cos x 2 sin x cos x + = + *Bai 30. Giai phng trinh: ( )

4 2 2 43cos x 4cos xsin x sin x 0 - + = *Bai 31. Giai phng trinh:( ) 3 32 3 2cos3xcos x sin3xsin x8-- = *Page 6" All the flower of tomorrow are in the seeks of today"Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onBai 32. Giai phng trinh:( ) 1cosxcos2xcos4xcos8x16= *Bai 33. Giai phng trinh: ( ) 34sin3xcos2x 1 6sinx 8sin x =+ - *Bai 34. Giai phng trinh:( ) 1cosx cos2x cos3x cos4x cos5x2+ + + + =- *Bai 35. Giai phng trinh: ( ) sin2x 2cosx sinx 10tanx 3+ - -= *+Bai 36. Giai phng trinh:( ) 21 sin2x cos2x2sinxsin2x1 cot x+ += *+Bai 37. Giai phng trinh: ( ) ( ) tanx cot x 2 sin2x cos2x + = + *Bai 38. Giai phng trinh: ( ) 2tan x tanxtan3x 2 - = *Bai 39. Giai phng trinh:( )

2 2 211tan x cot x cot 2x3+ + = *Bai 40. Giai phng trinh: ( ) 2 2 2x xsin tan x cos 02 4 2 p - - = * Bai 41. Giai phng trinh: ( ) ( )

2sin2x cot x tan2x 4cos x + = *Bai 42. Giai phng trinh:( ) ( ) 2 2cot x tan x16 1 cos4xcos2x-= + *Bai 43. Giai phng trinh:( )

12tanx cot2x 2sin2x2sin2x+ = + *Bai 44. Giai phng trinh:( )( ) ( ) 3 sinx tanx2 1 cosx 0tanx sinx+- + = *-Bai 45. Giai phng trinh:( ) ( )( )( ) ( )2 22 21 cosx 1 cosx1tan xsinx 1 sinx tan x24 1 sinx- + +- = + + *-Bai 46. Giai phng trinh: ( ) cos3xtan5x sin7x = *Bai 47. Giai phng trinh:( ) 1 1sin2x sinx 2cot x2sinx sin2x+ - - = *Bai 48. Giai phng trinh:( ) ( ) 4 4sin x cos x 1tanx cot2xsin2x 2+= + *Bai 49. Giai phng trinh: ( ) 2 2 2 2tan x.cot 2x.cot3x tan x cot 2x cot 3x = - + *Bai 50. Giai phng trinh: ( )

xcot x sinx 1 tanxtan 42 + + = * " Cn cu bu thng minh"Page 7Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)HNG D N GIAI PHNG TRINH LNG GIAC C BAN OLi bi nh: T vic xut hin ba cung x,2x,3x, giup ta lin tng n vic a chung vcung mt cung. Nhng a v cung xhay cung2x? Cac ban co thtrali cu hoi o da vao quan nim sau:" Trong phng trinh lng giac tn tai ba cung x,2x,3x, ta nn a vcung trung gian2xnu trong biu thc co cha sin2x (hoc cos2x). Con khng cha sin2x (hoc cos2x), nn a vcung x".Bai giai tham khao( ) ( ) ( )3 2 3 24cos x 3cosx 4 2cos x 1 3cosx 4 0 4cos x 8cos x 0 * - - - + - = - =( )( )( )( )

2cosx 0 N4cos x cosx 2 0 x k, kcosx 2 L 2=p - = = + p =. 0,5 k 3,93 5 7Do x 0;14,k 0 k 14 x ; ; ;k2 22 2 2 - p p p p p + p .Bai giai tham khao( ) ( ) ( )2cosx 1 2sinx cosx 2sinxcosx sinx * - + = -( ) ( ) ( )2cosx 1 2sinx cosx sinx 2cosx 1 0 - + - - =( ) ( ) ( ) ( )2cosx 12sinx cosx sinx 0 2cosx 1 sinx cosx 0 - + - = - + = ( ) x k22cosx 1 0cosx cos3k;l3sinx cosx 0tanx 1x l4pp = + p - = = + = p =- =- + p .OLi bi nh: T vic xut hin cac cung3xva2x, chung ta nghi ngay n vic a chung vcung mt cung x bng cng thc nhn ba va cng thc nhn i cua ham cosBai giai tham khao( )3 2 3 24cos x 3cosx 2cos x 1 cosx 1 0 2cos x cos x 2cosx 1 0 * - + - - - = + - - =( ) ( ) ( ) ( ) 2 2cos x 2cosx 1 2cosx 1 0 2cosx 1 cos x 1 0 + - + = + - =Page 8" All the flower of tomorrow are in the seeks of today"Bai 1.Giai phng trinh:( )cos3x 4cos2x 3cosx 4 0 , x 0;14 - + - = * " Trich thi tuyn sinh ai hoc khi D nm 2002Bai2.Giai phng trinh: ( ) ( ) ( )2cosx 1 2sinx cosx sin2x sinx - + = - *Trich thi tuyn sinh ai hoc khi D nm 2004Bai3.Giai phng trinh:( ) cos3x cos2x cosx 1 0 + - - = *Trich thi tuyn sinh ai hoc khi D nm 2006Bai4.Giai phng trinh: ( ) sinx cosx 1 sin2x cos2x 0 + ++ + = *Trich thi tuyn sinh ai hoc khi B nm 2005Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) ( ) 2sinx 0 x k2cosx 1 sin x 0 k;l1 2cosx x l22 3 = = p - + = p =- = +p .Bai giai tham khao( ) ( )2sinx cosx 2sinxcosx 2cos x 0 * + + + =( ) ( )sinx cosx 2cosx sinx cosx 0 + + + =( ) ( )sinx cosx 1 2cosx 0 + + =( ) sinx cosx tanx 1x k4k;l1 22cosx cosx cosx l22 33p =- =- =- + p p p=- = = +p .OLi binh: T vic xut hin cua cung 2xva cung xmata nghi n vic chuy n cung2xvcung xbng cng thc nhn i cua ha m sin va cos, to xut hin nhn t chung hai v( ) ( )2sinx 1 2cos x 1 2sinxcosx 1 cosx * + - + =+( ) ( ) 22sinxcos x 2sinxcosx 1 cosx 2sinxcosx cosx 1 1 cosx 0 + =+ + - + =( ) ( ) ( ) 21x k2cosx3cosx 1 sin2x 1 0 k,l2sin2x 1x l4p = + p=- + - = p== + p .OLi binh: T vic xut hin hai cung 3x2p-va7x4p-giup ta suy nghi n vic a hai cung khac nhau na y vcung mt cung chung lax. la m c iu o, ta co thdung cng thc cng cung hoc dung cu thn chu"cos i sin bu phu cheo''. Ta thc hin hai ytng o qua hai cach giai sau yBai giai tham khaoCach giai 1.S dung cng thc cng cung: ( )sin a b sina.cosb cosa.sinb = " Cn cu bu thng minh"Page 9Bai6.Giai phng trinh: ( ) 1 1 74sin xsinx 43sin x2 p + = - * p - Trich thi tuyn sinh ai hoc khi A nm 2008Bai5.Giai phng trinh:( ) ( ) sinx 1 cos2x sin2x 1 cosx + + =+ *Trich thi tuyn sinh ai hoc khi D nm 2008Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)( )1 1 7 74 sin cosx sinxcossinx 3 3 4 4sinxcos sin cosx2 2 p p * + = - p p -( ) 1 1 24. sinx cosxsinx cosx 2 + = - + iu kin:sinxcosx 0 sin2x 0 .( ) sinx cosx2 2 sinx cosxsinxcosx+ =- +( ) ( )sinx cosx 2 2sinxcosx sinx cosx 0 + + + =( ) ( )sinx cosx 1 2sin2x 0 + + =( )

x k4tanx 1sinx cosx 0x l k,l,m28 1 2sin2x 0sin2x5 2x m8p =- + p=- + = p =- + p + = =- p= + p .Cach giai 2.S dung "cos i sin bu phu cheo''Ta co:( )3sin x sin 2 x cosx2 27 1sin x sin 2 x sin x sinx cosx4 4 42 p p - = - p- - = p p p - = p- + =- + =- + ( ) ( )1 1 14. sinx cosxsinx cosx2 * + = - + .Giai tng t nh cach giai 1.OLi bi nh: T tng hai cungx x3 6 2p p p+ + - =giup ta lin tng n cu ''phu cheo'' , tht v y: cot x cot x cot x cot x cot x tan x 13 6 3 2 3 3 3 p p p p p p p + - = + - + = + + = . Cng vic con lai cua chung ta ladung cng thc: 4 4 21sin x cos x 1 sin 2x2+ =- . Nu khng co nhn xet nay, mata tin hanh bin i tancoscotsin= , ri qui ng thibai toan trnn rt phc tap, cha tinh n vic i chiu nghim vi iu kin.Bai giai tham khaoK: sin x 01 3sin x sin x 0 cos 2x 0 cos 2x 03 6 2 6 6sin x 06 p + p p p p + - - - p - .Page 10" All the flower of tomorrow are in the seeks of today"Bai7.Giai phng trinh:( )

4 47sin x cos x cot x cot x8 3 6 p p + = + - * Trich thi tuyn sinh ai hoc Giao Thng VnTai Tp. HCM nm 1999Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) ( ) 2 21 7 1 1 k1 sin 2x sin 2x 1 cos4x x , k2 8 4 2 12 2p p* - = = - = = + .Bai giai tham khaoK:cos x 01 4cos x cos x 0 cos2x cos 0 cos2x 04 4 2 2cos x 04 p - p p p - + + p + .Ta co:tan x tan x tan x tan x tan x cot x 14 4 4 2 4 4 4 p p p p p p p - + = - - + = - - = .( ) ( )2 4 2 4 4 21 11 sin 4x cos 4x 1 1 cos 4x cos 4x 2cos x cos 4x 1 02 2* - = - - = - - =( )( )

22 222t 1 N2t t 1 01cos 4x 1 sin 4x 0 sin4x 0t Lt cos 4x 02t cos 4x 0= - - = = = = =- = = ( )( )( )

sin2x 0 Nkx , kcos2x 0 L 2=p = =.OLu y , ta co ththc hin bin i mu s bng cng thc cng theo tan nh sautan tanx tan tanx1 tanx 1 tanx4 4tan x .tan x . . 14 4 1 tanx 1 tanx1 tan tanx 1 tan tanx4 4p p- + p p - + - + = = = p p + - + -.OLi binh: Nhin vao phng trinh na y, ta nghidung cng thc cng cung theo sin, hoc xet tng cung cua chung, . nhng ng vi lam nh th, no se kho i n kt qua. Ta hay xem gia hai cung 3 x10 2p-va3x10 2p+co mi lin h gi hay khng ? Tht v y: " Cn cu bu thng minh"Page 11Bai8.Giai phng trinh: ( ) 4 44sin 2x cos 2xcos 4xtan x tan x4 4+= * p p - + Trich thi tuyn sinh ai hoc Xy Dng nm 1997Bai9.Giai phng trinh:( ) 3 x 1 3xsin sin 110 2 2 10 2 p p - = + Trich thi tuyn sinh ai hoc Thu y Li nm 2001Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)3x 3x 9 3x 3 xsin sin sin sin310 2 10 2 10 2 10 2 p p p p + = p- + = - = - . T o, ta set 3 xt10 2p= -vas dung cng thc nhn ba la ti u nht.Bai giai tham khaoTa co: 3x 3x 9 3x 3 xsin sin sin sin310 2 10 2 10 2 10 2 p p p p + = p- + = - = - .( ) ( )

3 x 1 3 x1 sin sin3 210 2 2 10 2 p p - = - .t 3 xt10 2p= - . Va ( ) ( ) ( )3 21 12 sint sin3t sint 3sint 4sin t sint 1 sin t 02 2 = = - - =( ) 3 x 3t kk x k2sint 010 2 5k,lcost 0 3 x 2t ll x l2210 2 2 5 p p= p - = p = - p = p = p p p= + p - = + p = - p .Bai giai tham khaoTa co: 3sin 3x sin 3x sin 3x sin 3x sin3 x4 4 4 4 4 p p p p p - =- - =- p- - =- + =- + tt x x t4 4p p= + =- . Luc o ( )1 sin3t sin 2t .sint2 p - = - 32 2 2sint 0 sint 04sin t 3sint cos2tsint 04sin t 3 1 2sin t 0 sin t 1 = = - + = - + - = = ( ) t kx ksint 04x m , k,l,mcost 04 2t lx l24p= p =- + p = p p =- + p = p= + p = + p .Bai giai tham khaoTa co: ( )cos3x cos 3x cos 3 x3 p =- p+ =- + .Page 12" All the flower of tomorrow are in the seeks of today"Bai11.Giai phng trinh:( )

38cosx cos3x 13 p + = Trich thi tuyn sinh ai hoc Quc Gia Ha Ni nm 1999Bai10.Giai phng trinh:( ) sin 3x sin2xsin x 14 4 p p - = + Trich thi tuyn sinh Hoc Vin Bu ChinhVin Thng nm 1999Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onPhng trinh:( ) ( )

31 8cos x cos 3 x 23 3 p p + =- + .tt x3p= + . Luc o:( )3 3 32 8cos t cos3t 8cos t 4cos t 3cost =- =- +( ) ( ) 3 212cos t 3cost 0 cos3t 4cos t 1 0 cos3t 2cos2t 1 0 - = - = + =( ) t kx k2cos3t 06t l x l k;l;m13cos2t22x mt m33p= + p p= + p =p = + p = p =-p p= + p=- + p .Bai giai tham khaoCach giai 1.tt x x t4 4p p= + =- . Luc o:( )3 31 sin t 2sin t sin t sint cost4 p = - = - ( ) ( ) ( )

3 3 2 2sin t sint cost sin t sin t cos t sint cost = - = + - ( ) 2 2cost sin t sintcost cos t 0 - + - =( )( )( )

cost 0 N1cost sin2t 1 0 t k x k , ksin2t 2 L 2 2 4 =p p - = = + p = + p = .OLi binh: Trong ( ), ti as dung kithut ghep cng thc 2 21 sin t cos t = +. Vy trong giai phng trinh lng giac, du hiu nh th nao bit ghep nh th? Cu trali rt n gian: " Khi bc cua sin va cos khng ng bc va hn kem nhau hai bc, ta nn ghep 2 21 sin t cos t = + phng trinh tr nn n gian hn ".Cach giai 2. ( ) ( )3 31 11 2 . 2sin x 2sinx 2 sinx cosx 2sinx42 2 p + = + = ( ) ( ) ( ) 3 2sinx cosx 4sinx sinx cosxsinx cosx 4sinx + = + + =( ) ( )sinx cosx 1 2sinxcosx 4sinx + + =

2 23sinx 2cos xsinx 2sin xcosx cosx 0 - + + + =( ) ( ) 2 2sinx 3 2cos x cosx 2sin x 1 0 - + + + =( ) ( ) 2 2sinx 2sin x 1 cosx 2sin x 1 0 - + + + =" Cn cu bu thng minh"Page 13Bai12.Giai phng trinh:( ) 32sinx 2sinx 14 p + = Trich thi tuyn sinh Phn Vin Bao Chi Truyn Thng nm 1998Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)( ) ( )( )

220 2sin x 1 0 VN2sin x 1 cosx sinx 0cosx sinx 0 = + > + - = - =( )tanx 1 x k, k4p = = + p .Cach giai 3. ( ) ( )3 31 11 2 . 2sin x 2sinx 2 sinx cosx 2sinx42 2 p + = + = ( ) ( ) 3sinx cosx 4sinx 2 + =Vi( )cosx 0 hay sinx 1 = = khng phai langhim cua phng trinh ( )2 nn chia hai v cua phng trinh ( )2 cho 3cos x, ta c: ( ) ( ) ( )322 tanx 1 4tanx. 1 tan x + = +Giai phng trinh theo tanx ta c nghim: ( )tanx 1 x k, k4p= = + p .Bai giai tham khaoCach giai 1.tt x x t4 4p p=- =+ . Luc o:( )( )3 31 sin t 2sin t 4 sin t sint cost = + = +( ) ( ) 3 2 2sin t sint cost sin t cos t = + +( ) 3 3 2 2 3sin t sin t sintcos t costsin t cos t cost sintcost 1 0 = + + + + =( )( ) cost 03t k x k k, ksin2t 2 L 2 4 4=p p p = + p = + p - + p =-.Cach giai 2 va cach giai 3 (tng t vidu 13). Ban oc tgiaiOLi bi nh: Bai toan co cac cung khac nhau theo m t ham bc nht lng giac cos (hoc sin hoc casin va cos) dang tng (hoc hiu). Ta nn ghep cac s hang na y thanh cp sao cho hiu (hoc tng) cac cung cua chung bng nhau, tc latrong trng hp nay y( )x 4x 5x + = va ( )2x 3x 5x + =. Tai sao phai ghep nh vy ? Lydo rt n gian, chung ta cn nhng "tha s chung" nhom ra ngoai, a bai toan vdang phng trinh tich s.Bai giai tham khao( ) ( ) ( )5x 3x 5x xcosx cos4x cos2x cos3x 0 2cos cos 2cos cos 02 2 2 2* + + + = + =Page 14" All the flower of tomorrow are in the seeks of today"Bai14.Giai phng trinh:( ) cosx cos2x cos3x cos4x 0 + + + = * Bai13.Giai phng trinh:( )

3sinx 2sinx 14 p - = Trich thi tuyn sinh ai hoc Quc Gia Tp.HCM nm 1998Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on

5x 3x x 5x x2cos cos cos 0 4cos cosxcos 02 2 2 2 2 + = = ( ) 5x k25x k xcos 02 2 5 52cosx 0 x l x l k;l;m2 2xx x 2mcos 0m22 2 p p p = + p = + = p p = = + p = + p p =p+ p== + p .OLi binh: Vi nhng phng trinh co nhng hang t bc hai theo sin va cos, ta thng dung cng thc ha bc bai toan trnn n gian hn.Bai giai tham khao( ) ( ) ( ) ( ) ( )1 1 1 31 cos2x 1 cos4x 1 cos6x cos2x cos6x cos4x 02 2 2 2* - + - + - = + + =( )2cos4xcos2x cos4x 0 cos4x 2cos2x 1 0 + = + =( )

kcos4x 04x k x2 8 4k,l22cos2x cos2x l2 x l33 3 p p p= = + p = + p p p= = +p = + p .Bai giai tham khao( ) ( ) ( ) ( )1 1 11 cos2x 1 cos4x 1 cos6x 22 2 2* - + - + - =( ) ( ) ( ) 1 1cos2x cos4x cos6x cos2x cos6x cos4x 1 02 2 - + + = + + + =( ) 22cos4xcos2x 2cos 2x 0 2cos2x cos4x cos2x 0 + = + =( )

x k2cosx 04cos2xcos3xcosx 0 cos2x 0 x l k,l,m4 2cos3x 0x m6 3p = + p = p p = = = + = p p= + ." Cn cu bu thng minh"Page 15Bai15.Giai phng trinh: ( )

2 2 23sin x sin 2x sin 3x2+ + = * .Trich thi tuyn sinh Cao ng S Pham Hng Yn khi A nm 2000Bai16.Giai phng trinh:( )

2 2 2sin x sin 2x sin 3x 2 + + = *.Trich thi tuyn sinh ai hoc S Pha m KiThutTp. HCM khi A nm 2001Bai17.Giai phng trinh:( )

2 2 2 2sin x sin 3x cos 2x cos 4x + = + *Trich thi tuyn sinh ai hoc Kinh t Quc Dn nm 1999Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Bai giai tham khao( ) ( ) ( ) ( ) ( )1 1 1 11 cos2x 1 cos6x 1 cos4x 1 cos8x2 2 2 2* - + - = + + +( )cos2x cos6x cos4x cos8x 2cos4xcos2x 2cos6xcos2x - + = + - =( )2cos2x cos6x cos4x 0 4cos2xcos5xcosx 0 + = =( )

cosx 0mcos2x 0 x k x l x ; k,l,m2 4 2 10 5cos5x 0= p p p p p = = + p = + = + =.Bai giai tham khao( ) ( ) ( ) ( ) ( )1 1 1 11 cos6x 1 cos8x 1 cos10x 1 cos12x2 2 2 2* - - + = - - + cos6x cos8x cos10x cos12x 2cos7xcosx 2cos11xcosx + = + =( ) ( ) x k2cosx 0lcosx cos7x cos11x 0 x k,l,mcos7x cos11x2mx9p = + p=p - = =- = p= .Bai giai tham khao( )x x cos3x sin7x 1 cos 5x 1 cos9 cos3x sin7x sin5x cos92 p * + =- + - - + = - cos3x cos9x sin7x sin5x 0 2cos6xcos3x 2cos6xsinx 0 + + - = + =( ) ( ) x k12 6cos6x 0cos6x cos3x sinx 0 x l k,l,mcos3x cos x 42mx8 2p p = +=p + = = + p p = + p p =- + .Bai giai tham khaoPage 16" All the flower of tomorrow are in the seeks of today"Bai18.Giai phng trinh:( ) 2 2 2 2sin 3x cos 4x sin 5x cos 6x - = - *Trich thi tuyn sinh ai hoc khi B nm 2002Bai19.Giai phng trinh:( )sin 2 25x 9xcos3x sin7x 2 2cos4 2 2 p + = + - * Trich thi tuyn sinh ai hoc Th Duc Th Thao nm 2001Bai20.Giai phng trinh: ( ) 2 2 2sin x cos 2x cos 3x = + *Trich thi tuyn sinh ai hoc Quc Gia Ha Ni nm 1998Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) ( ) ( )1 cos2x 1 cos4x 1 cos6xcos2x cos4x 1 cos6x 02 2 2- + +* = + + + + =( ) 22cos3xcosx 2cos 3x 0 2cos3x cosx cos3x 0 + = + =( ) kxk6 3cosx 0xl6 34cos3xcos2xcosx 0 cos2x 0 x k,l,ml4 2xcos3x 04 2x m2p p = + p p= = +p p = = = + p p = += p= + p .OLi binh: T vic xut hin cac cung ( ) ( ) ( )x , 2x , 7x vanhn xet 7x x4x2+=, ta co thinh hng nhom ( )sin7x sinx -, ( )22sin 2x 1 -lai vi nhau, sau khi dung cng thc tng thanh tich va ha bc nhm xut hin nhn t chung va cui cung a ta c phng trinh tich s n gian hn.Bai giai tham khao( ) ( ) ( )2sin7x sinx 1 2sin 2x 0 2cos4xsin3x cos4x 0 * - - - = - =( ) ( ) k2cos4x 0x18 3cos4x 2sin3x 1 0 k,l15 l2sin3xx218 3p p= = + - = p p= = + .Bai giai tham khao( ) ( ) ( )sinx sin3x sin2x 1 cos2x cosx * + + = + +( ) ( ) 22sin2xcosx sin2x 2cos x cosx sin2x 2cosx 1 cosx 2cosx 1 0 + = + + - + =( ) ( ) ( ) ( )2cosx 1 sin2x cosx 0 2cosx 1 2sinxcosx cosx 0 + - = + - =( ) ( ) ( )

x k2cosx 0x l216cosx 2sinx 1 2cosx 1 0 sinx k,l,m,n52x m216cosx2 2x n23p = + p =p = +p - + = = p = + p =- p = + p." Cn cu bu thng minh"Page 17Bai22.Giai phng trinh: ( ) sinx sin2x sin3x 1 cosx cos2x + + =+ + *Bai23.Giai phng trinh: ( )

3 3 3sin xcos3x cos xsin3x sin 4x + = *Trich thi Tuyn sinh ai hoc Ngoai Thng nm 1999Bai21.Giai phng trinh:( ) 22sin 2x sin7x 1 sinx + - = *Trich thi tuyn sinh ai hoc nmkhi A nm 2007Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Bai giai tham khao( ) ( ) ( )3 3 3 3 3sin x 4cos x 3cosx cos x 3sinx 4sin x sin 4x * - + - =

3 3 3 3 3 3 34sin xcos x 3sin x3cosx 3cos xsinx 4cos xsin x sin4x - + - =( ) 2 2 33sinxcosx cos x sin x sin 4x - =

3 33 3sin2xcos2x sin 4x sin4x sin 4x2 4 = =( ) 3k3sin4x 4sin 4x 0 sin12x 0 12x k x , k12p - = = = p = .Bai giai tham khao( ) ( ) ( )3cos10x 1 cos8x cosx 2cosx 4cos 3x 3cos3x * + + = + -( )cos10x cos8x 1 cosx 2cosxcos9x + += + 2cos9xcosx 1 cosx 2cosxcos9x += +( )cosx 1 x k2 , k = = p .Bai giai tham khao( ) ( ) ( )2 2 2sinx 4sin x 3 cosx sin x 3cos x 0 * - - - =( ) ( ) 2 2 2sinx 4sin x 3 cosx sin x 3 1 sin x 0 - - - - = ( ) ( ) 24sin x 3 sinx cosx 0 - - =( ) ( )2 1 cos2x 3 sinx cosx 0 - - - = ( ) 21 2 2x kcos2x cos 2x k23k;l2 3 3sinx cosx tanx 1x l4p p p = + p =- = = + p p= == + p .Bai giai tham khao( ) ( ) ( ) ( )22sinx 1 3cos4x 2sinx 4 4 1 sin x 3 0 * + + - + - - =( ) ( ) ( ) ( )2sinx 1 3cos4x 2sinx 4 1 2sinx 1 2sinx 0 + + - + - + =( ) ( )2sinx 1 3cos4x 2sinx 4 1 2sinx 0 + + - + - =Page 18" All the flower of tomorrow are in the seeks of today"Bai24.Giai phng trinh: ( )2 3cos10x 2cos 4x 6cos3xcosx cosx 8cosxcos 3x + + = + *Bai25.Giai phng trinh: ( ) 3 3 24sin x 3cos x 3sinx sin xcosx 0 + - - = *Bai26.Giai phng trinh:( ) ( ) ( )

22sinx 1 3cos4x 2sinx 4 4cos x 3 + + - + = *Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) ( )3 cos4x 1 2sinx 1 0 - + =( )

kx4x k22cos4x 1x l2 x l2 k;l;m16 6sin2x27 7x m2 x m26 6p == p=p p =- +p =- +p =- p p = + p = + p .Bai giai tham khao( ) ( ) ( )6 8 6 8 6 2 6 2sin x 2sin x cos x 2cos x 0 sin x 1 2sin x cos x 2cos x 1 0 * - + - = - - - =( ) 6 6 6 6sin xcos2x cos xcos2x 0 cos2x sin x cos x 0 - = - =( ) 6 6kxcos2x 0 cos2x 0k4 2x , ktanx 1 sin x cos x 4 2x k4p p = + = =p p = + = p = = + p.Bai giai tham khao( ) ( ) ( )10 8 8 1052cos x cos x sin x 2sinx cos2x 04* - - - + =( ) ( ) 8 2 8 25cos x 2cos x 1 sin x 1 2cos x cos2x 04 - - - + =

8 8 8 85 5cos x.cos2x sin xcos2x cos2x 0 cos2x cos x sin x 04 4 - + = - + = ( )( )

8 88 8cos2x 02x kk2x , k554 2cos x sin x 0sin x cos x 1 VN44p= = + pp p = + - + = = + > .Bai giai tham khaoO Cach giai 1( ) ( ) ( )3 5 5 3sin x 2sin x 2cos x cos x 0 * - - + =( ) ( ) 3 2 3 2 3 3sin x 1 2sin x cos x 2cos x 1 0 sin xcos2x cos xcos2x 0 - - - = - =" Cn cu bu thng minh"Page 19Bai27.Giai phng trinh:( ) ( ) 6 6 8 8sin x cos x 2 sin x cos x + = + *Trich thi tuyn sinh ai hoc Quc Qua Ha Ni Khi B nm 1999Bai28.Giai phng trinh:( ) ( ) 8 8 10 105sin x cos x 2 sinx cos x cos2x4+ = + + *Trich thi tuyn sinh ai hoc Ngoai Thng Tp.HCMkhi D 2000Bai29.Giai phng trinh:( ) ( )

3 3 5 5sin x cos x 2 sin x cos x + = + *Trich thi tuyn sinh ai hoc Qucgia HaNi khi D 1998Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)( ) ( ) 3 33cos2x 0mcos2x sin x cos x 0 x , mtan x 1 4 2=p p - = = + =.O Cach giai 2( ) ( ) ( )3 3 2 2 5 5sin x cos xsin x cos x 2sin x 2cos x 0 * + + - - =( ) ( ) 3 2 5 5 3 2sin xcos x sin x cos x cos sin x 0 - - - =( ) ( ) ( ) 3 2 2 3 2 2 3 3sin x cos x sin x cos x cos x sin x 0 cos2x sin x cos x 0 - - - = - =( ) 3 3 3cos2x 0 cos2x 0mx , msin x cos x 0 tan x 1 4 2 = =p p = + - = = .Bai giai tham khao( )2 221 cos2x 1 cos2x3 sin 2x 02 2 + - * - + = ( ) ( ) ( ) 2 2 23 1 2cos2x cos 2x 4 1 cos 2x 1 2cos2x cos 2x 0 + + - - + - + =( ) 28cos 2x 4cos2x 0 4cos2x 2cos2x 1 0 + = + =( )

kcos2x 0x k4 2 4k,m1cos2xx m23p p p= = + + p p=- = + p .O Cach khacDo cosx 0hay sinx 1 = = khng langhim cua phng trinh ( )*Chia hai v cua ( )* cho 4cos x, ta c: ( )22 42t 4t 3 03 4tan x tan x 0t tan x 0- +=* - + = = ( )222t 1x ktanx 1tan x 14t 3k,mtan x 3 tanx 3x mt tan x3 p= = + p= = = p = = = + p = .Bai giai tham khaoOLi bi nh: Ta nhn th y trong phng trinh cocha cos3x lnsin3x, nu ta s dung cng thc nhn ba khai trin cung i n kt qua cui cung, nhng no tng i phc tap. Chinh vi th, y ta kheo leo phn tich ap dung cng thc tich thanh tng co xut hin s 12 nhm ti gian c vi s 2 3 28-phc tap bn v phai cua phng trinh.Page 20" All the flower of tomorrow are in the seeks of today"Bai30.Giai phng trinh: ( )

4 2 2 43cos x 4cos xsin x sin x 0 - + = *Trich thi tuyn sinh ai hoc Quc Qua Tp.HCM 1998 1999 t 1Bai31.Giai phng trinh:( ) 3 32 3 2cos3xcos x sin3xsin x8-- = *Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) ( ) ( )2 22 3 2cos3xcosx cos x sin3xsinx sin x8-* - =( ) ( ) 2 21 1 2 3 2cos4x cos2x cos x cos2x cos4x sin x2 2 8- + - - =

2 2 2 22 3 2cos4xcos x cos2xcos x cos2xsin x cos4xsin x4- + - + =( ) ( ) 2 2 2 22 3 2cos4x cos x sin x cos2x cos x sin x4- + + - =( ) 22 3 2 1 2 3 2cos4x cos 2x cos4x 1 cos4x4 2 4- - + = + + =( ) ( ) 2 k4cos2x 2 1 cos4x 2 3 2 cos4x x , k2 16 2p p + + =- =- = + .Bai giai tham khaoOLi binh: Trong bai toan xut hin bn cung x,2x,4x,8x khac nhau, giup ta lin tng n vic a chung vcung mt cung. lam vic na y ta se suy nghi n vic dung cng thc 2 2cos2x 2cos x 1 1 2sin x = - =-, nhng no thi khng khaquan cho m y, bi thphng trinh se trthanh phng trinh bc cao, vic giai se gy kho khn. Nhng yrng, cac cung na y ln lt gp i nhau, ta cht nhn cng thc nhn i cuasin, bng cach nhn thm hai v cua ( )* chosinx. am trong phep nhn, ta nn kim tra xemsinx 0 =cophai langhim hay khng trc khi nhn.Nhn thy:( )sinx 0 x k hay cosx 1 cos2x cos4x cos8x 1 = = p = = = = nn ( )1116* =(v nghim) nnsinx 0 x k = = p khng la nghi m cua ( )*Nhn ca 2 v cua phng trinh ( )* cho 16sinx 0 , ta c: ( )16sinxcosxcos2xcos4xcos8x sinx 8sin2xcos2xcos4xcos8x sinxsinx 0 sinx 0 = = *

4sin4xcos4xcos8x sinx 2sin8xcos8x sinx sin16x sinxsinx 0 sinx 0 sinx 0 = = =

k2xk2x 1515lxlx 17 1717 17x mp = p = p p = + p p = + p vi ( ) 17p 1k 15n; l ; k,l,m,n,p2- ." Cn cu bu thng minh"Page 21Bai32.Giai phng trinh:( ) 1cosxcos2xcos4xcos8x16= *Trich thi tuyn sinh ai hoc Kinh t Quc Dn nm 1998Bai33.Giai phng trinh: ( ) 34sin3xcos2x 1 6sinx 8sin x =+ - *Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Bai giai tham khao( ) ( )34sin3xcos2x 1 2 3sinx 4sin x 4sin3xcos2x 1 2sin3x * =+ - =+( ) ( ) ( ) 22sin3x 2cos2x 1 1 2sin3x 4cos x 3 1 - = - = oDo ( )cosx 0 x k , k2p= = + p khng langhim phng trinh ( )o, nn nhn hai v ( )o cho cosx 0 , ta c:( ) ( )32sin3x 4cos x 3cosx cosx 2sin3xcos3x cosx - = = o( )

l2x14 7sin6x cosx cosx cos 6x l,km22x10 5p p = + p = = - p p = +.Bai giai tham khaoKhi( )x k2 , k = p thi( ) ( )152* =- *khng co nghim( )x k2 , k = p .Khi ( )xx k2 , k sin 02 p . Nhn hai v cua ( )* cho x2sin 02 , ta c:( )x x x x x x2sin cosx 2sin cos2x 2sin cos3x 2sin cos4x 2sin cos5x sin2 2 2 2 2 2* + + + + =-

3x x 5x 3x 7x 5x 9x 7x 11x 9x xsin sin sin sin sin sin sin sin sin sin sin2 2 2 2 2 2 2 2 2 2 2 - + - + - + - + - = - .( ) 11x 11x 2msin 0 m x , m 11, m2 2 11p = = p = .Bai giai tham khaoOLi bi nh: Khi giai phng trinh lng giac co cha tan hoc cot, co n mu hay cn bc chn, ta phai c iu kin phng trinh xac inh. c bit i vi nhng bai toan cocha tan (hoc cot), ta ha y thay th chung bng sin cos,cos sin nhm muc ich " n gian hoa " va chicon lai hai gia trilng giac la sin va cos mathi.Ta se dung cac cach sau y kim tra xem co nhn nghi m hay khng Thay cac giatri x ti m c vao iu kin xem co thoa khng. Nu thoa thighi nhn nghim y, nu khng thoa thi loai. Hoc biu din cac ngon cung iu kin vangon cung nghim trn cung mt ng tron lng giac. Ta se loai bo ngon cung cua nghim khi co trung vi ngon cung cua iu kin. Hoc so vi iu kin trong qua trinh giai phng trinh.Page 22" All the flower of tomorrow are in the seeks of today"Bai34.Giai phng trinh:( ) 1cosx cos2x cos3x cos4x cos5x2+ + + + =- *Bai35.Giai phng trinh: ( ) sin2x 2cosx sinx 10tanx 3+ - -= *+Trich thi tuyn sinh ai hoc khi D nm 2011Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn oniu kin:tanx 3cosx 0 -( ) ( ) ( )sin2x 2cosx sinx 1 0 2cosx sinx 1 sinx 1 0 * + - - = + - + =( ) ( ) ( ) 1x k2cosx3sinx 1 2cosx 1 0 k,l2sinx 1x l22p = + p= + - = p=-=- +p .So vi iu kin, ho nghi m cua phng trinh la ( )x k2 , k3p= + p .Bai giai tham khaoiu kin:sinx 0 ( )2 2sin x(1 sin2x cos2x) 2 2sin xcosx 1 sin2x cos2x 2 2cosx * + + = + + =( ) 22cos x 2cosxsinx 2 2cosx 0 2cosx cosx sinx 2 0 + - = + - =( ) cosx 0x kcosx 02k,lcos x 1 cosx sinx 2x l244 p== + p = p p - = + = = +p .So vi iu kin, ho nghi m phng trinh la( )x k x l2 , k,l2 4p p= + p = +p .Bai giai tham khaoiu kin:( )sinx 02sinxcosx 0 sin2x 0 2x k x k , kcosx 02 p p .( ) ( ) ( )2 2sinx cosx sin x cos x2 sin2x cos2x 2 sin2x cos2xcosx sinx sinxcosx+* + = + = +( ) ( ) 1 12 sin2x cos2x sin2x cos2xsinxcosx sin2x = + = +( ) 2sin2x sin2x cos2x 1 sin 2x sin2xcos2x 1 0 + = + - =( ) 2sin2xcos2x cos 2x 0 cos2x sin2x cos2x 0 - = - =( )

cos2x 0x k42cos2xsin 2x 0 k,lsin 2x 0 4x l48 2 p= = + p p - = p p p - = = + ." Cn cu bu thng minh"Page 23Bai36.Giai phng trinh:( ) 21 sin2x cos2x2sinxsin2x1 cot x+ += *+Trich thi tuyn sinh ai hoc khi A nm 2011Bai37.Giai phng trinh: ( ) ( ) tanx cot x 2 sin2x cos2x + = + *Trich thi tuyn sinh ai hoc Giao Thng Vn Tai Tp.HCM nm 1998Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Kt hp vi iu kin, phng trinh co 2 ho nghi m: ( )x k x l , k,l4 8 2p p p= + p = + .Bai giai tham khaoiu kin:( )3cosx 0kcos3x 0 x , kcos3x 4cos x 3cosx 0 6 3 p p + = - .( ) ( )sinx sinx sin3xtanx tanx tan3x 2 2cosx cosx cos3x * + = + = ( ) 2sinx sinxcos3x cosxsin3x 2cos xcos3x + =( ) 2sinxsin 2x 2cos xcos3x - =

2 22sin xcosx 2cos xcos3x - =( ) 2sin x cosxcos3xdo cosx 0 - = ( ) ( ) 1 11 cos2x cos4x cos2x2 2 - - = +( ) lcos4x 1 x , l4 2p p =- = + So nghim vi iu kin: Cach 1 :Khi lx4 2p p= +thi3 l3 2cos3x cos 04 2 2 p p = + = (nhn). Cach 2 :Biu din cac ngon cung iu kin va ngon cung nghi m, ta th y khng co ngon cung nao trung nhau. Do o: lx4 2p p= +la nghi m cua phng trinh. (Cach 2 nay m t nhiu thi gian). Cach 3 :Nu 3 l33x k4 2 2p p p= + = + pthi

3 6l 2 4k 2k 3l 0,5 + =+ - = (v li vik,l ).V y ho nghi m cua phng trinh la: ( )lx , l4 2p p= + .Bai giai tham khaoiu kin: cosx 0sinx 0 sin2x 0sin2x 0 .Page 24" All the flower of tomorrow are in the seeks of today"Bai38.Giai phng trinh: ( ) 2tan x tanxtan3x 2 + = *Trich thi tuyn sinh ai hoc Quc Gia HaNi nm 1996/4/6/23/45/67/65/43/27/411/6Bai39.Giai phng trinh:( )

2 2 211tan x cot x cot 2x3+ + = *Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( )2 2 2 2 2 2 21 1 1 11 1 1 1 201 1 13 3 cos x sin x sin 2x cos x sin x 4sin xcos x * - + - + - = + + = ( ) 2 222 2 24sin x 4cos x 1 20 5 20 3 1 3sin 2x 1 cos4x3 3 4 2 4 4sin xcos x sin 2x+ + = = = - =( ) 1 2 kcos4x cos x , k2 3 6 2 p p p =- = = + .Thay vao nghi m vao iu kin, thoa. V y ho nghi m cua phng trinh la( ) kx , k6 2p p= + .Bai giai tham khaoiu kin:cosx 0 sinx 1 .( ) ( )( ) ( )( )222 21 sinx 1 cos x1 sin x 11 cos x 1 cosx 0 1 cosx 02 2 2 cos x 1 sin x - - p * - - - + = - + = - ( ) ( ) 21 cos x 1 cosx1 cosx 0 1 cosx 1 01 sinx 1 sinx - - - + = + - = + + ( ) ( )( )( )( )

x k2cosx 1 N1 cosx cosx sinx 0 k;ltanx 1 Nx l24 =p+ p=- + - - = p =-=- +p .Bai giai tham khaoiu kin: 2cosx 1sinx 0 sinx 02cos2x 0 2cos x 1 0cosx2 - Ta co: cosx sin2x cos2xcosx sin2xsinx cosxcot x tan2xsinx cos2x sinxcos2x sinxcos2x++ = + = = .Luc o:( )22 2 2sin2xcosx 2cos x 24cos x 4cos x 0 cos x 4 0sinxcos2x cos2x cos2x * = - = - = ( )( )( ) cosx 0 cosx 0 Nx k2k;l1 12 cos2x Nx lcos2x 26p = == + p p= = = + p ." Cn cu bu thng minh"Page 25Bai40.Giai phng trinh: ( ) 2 2 2x xsin tan x cos 02 4 2 p - - = * Trich thi tuyn sinh ai hoc khi D nm 2003Bai41.Giai phng trinh: ( ) ( )

2sin2x cot x tan2x 4cos x + = *Trich thi Tuyn Sinh ai hoc Mo ia cht nm 2000Bai42.Giai phng trinh:( ) ( ) 2 2cot x tan x16 1 cos4xcos2x-= + *Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Bai giai tham khaoiu kin:sin2x 0sin4x 0cos2x 0 .Ta co:2 2 4 42 22 2 2 2 2cos x sin x cos x sin x 4cos2xcot x tan xsin x cos x sin xcos x sin 2x-- = - = = .( ) ( ) ( ) ( ) ( )22416 1 cos4x 1 4 1 cos4x sin 2x 1 2 1 cos4x 1 cos4xsin 2x* = + = + = + -( ) 2 2 211 2 1 cos 4x 1 2sin 4x sin 4x2 = - = = (Nhn dosin4x 0 )( ) ( ) 1 1 k1 cos8x cos8x 0 x , k2 2 16 8p p - = = = + .Bai giai tham khaoiu kin:sin2x 0 sin2x 0cosx 0 cos2x 1 ( )2 22sinx cos2x 12sin2x 4sin x cos2x 2sin 2x 1cosx sin2x sin2x* + = + + = +( ) ( ) 2 2 2 2 2 24sin x 1 2sin x 8cos xsin x 1 2sin x 1 4cos x 0 + - = + - =( )( )( )( )

2sinx 0 L2sin x 1 2 1 cos2x 0 x k , k13cos2x N2=p - + = = + p =-.Bai giai tham khaoK: ( )sinx 0sinx 1 cosxsinxtanx sinx 0 sinx 0 0 cosx 0 sin2x 0cosx cosxcosx 1- - - ( )( )( )( )( )( )3 sinx tanx 3 cosx 1cot x. 2 1 cosx 0 2 1 cosx 0tanx sinx cot x1 cosx+ +* - + = - + =--( ) cosx 11 cosx 031 cosx 2 011 2cosx 01 cosxcosx2=- + = + - = + =- =- Page 26" All the flower of tomorrow are in the seeks of today"Bai43.Giai phng trinh:( )

12tanx cot2x 2sin2xsin2x+ = + *Trich thi tuyn sinh ai hoc Quc Gia Tp.HCM nm 1998 1999Bai44.Giai phng trinh:( )( ) ( ) 3 sinx tanx2 1 cosx 0tanx sinx+- + = *-Trich thi tuyn sinh ai hoc Tai Chinh K Toan nm 2000(Loai dosinx 0 )(Nhn)Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) 2x k2 , k3p = + p .Bai giai tham khaoiu kin: sinx 1 cosx 0 .( )( )( )( )23 22 22 1 cos xsin x 1 sin x1 sinx2 1 sin x 1 sin x 4 1 sinx+* - = + +- - -( ) ( ) ( ) ( ) 2 3 2 21 cos x 1 sinx 2sin x 1 sinx 1 sin x 2sin x + + - = + - +( ) ( ) ( ) ( ) 2 2 21 cos x 1 sinx 1 sinx cos x 2sin x 1 sinx + + = + + +( )

2 2 21 sinx 0 sinx 1 Lcos2x 01 cos x cos x 2sin x 1 1 2cos2x + = =- = + = + =-

k2x k x2 4 2p p p = + p = +(nhn docosx 0 )Bai giai tham khaoiu kin: cos5x 0 .( )sin5xcos3x sin7x sin5xcos3x sin7xcos5xcos5x* = =( ) ( ) 1 1sin8x sin2x sin12x sin2x sin8x sin12x2 2 + = + =( ) kx12x 8x k22k;l12x 8x l2 lx20 10p = = + p =p- +p p p = +.So vi iu kin: Vi kx2p=thi 5k kcos5x cos cos2 2p p= =loa i nu k le.Vi lx20 10p p= +thilcos5x cos 04 2 p p = + (nhn).Bai giai tham khao" Cn cu bu thng minh"Page 27Bai45.Giai phng trinh:( ) ( )( )( ) ( )2 22 21 cosx 1 cosx1tan xsinx 1 sinx tan x24 1 sinx- + +- = + + *-Bai46.Giai phng trinh: ( ) cos3xtan5x sin7x = *Trich thi Tuyn sinh Cao ng Kinh t Cng Nghip Tp. HCM nm 2007Bai47.Giai phng trinh:( ) 1 1sin2x sinx 2cot x2sinx sin2x+ - - = *Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)iu kin: sinx 0 cosx 0 .( )2sin 2x sin2xsinx cosx 1 2cos2x * + - - =

2 2 2 24cos xsin x 2cosxsin x cosx 1 4cos x 0 + - +- =( ) ( ) 2 2 2 24cos x 1 cos x 2cosx 1 cos x cosx 1 4cos x 0 - + - - + - =( ) ( ) 2 3 3 24cos x 2cos x cosx 1 0 cosx 1 4cos x 2cos x 2cosx 1 0 + + - = + - + - =( ) ( ) ( ) ( ) 2cosx 1 x k2cosx 1 2cosx 1 2cos x 1 0 k,l1cosx x l22 3 =- =p+ p + - + = p = = +p .Bai giai tham khaoiu kin: sin2x 0 .Ta co: ( )( )24 4 2 2 2 2 21sin x cos x sin x cos x 2sin xcos x 1 sin 2x2cos 2x xsinx cos2x sin2xsinx cosxcos2x 1tanx cot2xcosx sin2x cosxsin2x cosxsin2x sin2x+ = + - =-- ++ = + = = = ( )22 211 sin 2x1 1 121 sin 2x sin 2x 1sin2x 2sin2x 2 2-* = - = = (Nhn dosin2x 0 )( ) 2cos 2x 0 2x k x k , k2 4 2p p p = = + p = + .Bai giai tham khaoiu kin:cosx 0sin2x 0sin2x 0sin3x 0sin3x 0 ( ) ( )2 2 2 2cot 3x tan xcot 2x 1 tan x cot 2x * - = -

1 cos2x 1 cos4x 1 cos2x 1 cos4xcot3x . 11 cos2x 1 cos4x 1 cos2x 1 cos4x - + - + - = - + - + - ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

cot3x 1 cos2x 1 cos4x 1 cos2x 1 cos4x1 cos2x 1 cos4x 1 cos2x 1 cos4x - + - + - = - - - + +( ) ( )cot3x 2cos4x 2cos2x 2 cos4x cos2x - =- +( ) cos3x4sin3xsinx 4cos3xcosxsin3x - =-Page 28" All the flower of tomorrow are in the seeks of today"Bai48.Giai phng trinh:( ) ( ) 4 4sin x cos x 1tanx cot2xsin2x 2+= + *Trich thi tuyn sinh ai hoc Bach Khoa nm 2000Bai49.Giai phng trinh: ( ) 2 2 2 2tan x.cot 2x.cot3x tan x cot 2x cot3x = - + *Trich thi tuyn sinh ai hoc Dc Ha Ni nm 2001Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( ) kxcos3x 06 3cos3xsinx cos3xcosx k,ltanx 1x l4p p = + = = = p = + p.Cach khac:( ) ( )2 2 2 2cot 3x tan xcot 2x 1 tan x cot 2x * - = -

( ) ( )( ) ( )2 2 2 22 2 2 21 tan2xtanx 1 tan2xtanxtan x cot 2x tan 2xtan x 1cot3xtan xcot 2x 1 tan x tan 2x tan2x tanxtan2x tanx+ -- - = = =- - - +

cos3x 0cot3x cot xcot3xsinx cosx= = =(Giai tng t nh trn)Bai giai tham khaoiu kin: ( )sinx 0sin2x 0x kcos 0 x , kx2 2cos 02cosx 0 p .( )x x xsin cosxcos sinxsincosx sinx cosx2 2 2sinx 1 . 4 sinx 4sinx cosx x sinx xcos cosxcos2 2 + * + + = + =

xcos x2cosx cosx sinxsinx. 4 4 1 4sinxcosxsinx x sinx cosxcosxcos2 - + = + = =( )

2x k2 x k16 12sin2x k,l5 522x l2 x l6 12 p p = + p = + p = p p = +p = + p .So vi iu kin, ho nghim cua phng trinh la( ) 5x k x l ; k,l12 12p p= + p = + p .BAI TP REN LUYNCu 1.Giai phng trinh:2sinxcosx 2cosx 3 3sinx - + =.Cu 2.Giai phng trinh: 2tanxcosx 1 2cosx tanx += + ." Cn cu bu thng minh"Page 29Bai50.Giai phng trinh: ( )

xcot x sinx 1 tanxtan 42 + + = * Trich thi tuyn sinh ai hoc khi B nm 2006Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Cu 3.Giai phng trinh:3 32sin xcosx cos xsinx8- =.Cu 4.Giai phng trinh:2 2 2cos x cos 2x cos 3x 1 + + =.Cu 5.Giai phng trinh:2 217sin 2x cos 8x sin 10x2 p - = + .Cu 6.Giai phng trinh:4 6cos x sin x cos2x + =.Cu 7.Giai phng trinh:1 cos4x sin4x02sin2x 1 cos4x-- =+.Cu 8.Giai phng trinh:22 1sinxcosx cos x2++ =.Cu 9.Giai phng trinh:( )2 x2 3 cosx 2sin2 412cosx 1 p - - - =-.Cu 10. Giai phng trinh: sin4x 3sin2x tanx + = .Cu 11. Giai phng trinh:2 3cos10x 2cos 4x 6cos3xcosx cosx 8cosxcos 3x + + = +.Cu 12. Giai phng trinh: ( )2 2 22cos x 2cos 2x 2cos 3x 3 cos4x 2sin2x 1 + + - = +.Cu 13. Giai phng trinh: 5x 7sin 2x 3cos x 1 2sinx , ;32 2 3 p p + - - =+ " p .Cu 14. Giai phng trinh: ( ) 2 2sin 4x cos 6x sin 10,5 10x , 0;2p - = p+ " .Cu 15. Giai phng trinh: tan2x tan3x tan5x tan2xtan3xtan5x - - = .Cu 16. Giai phng trinh:sinx sin2x sin3x3cosx cos2x cos3x+ +=+ +.Cu 17. Giai phng trinh:21 cosxtan x1 sinx+=-.Cu 18. Giai phng trinh:24cos x cos x3= .Cu 19. Giai phng trinh:1 12 2sin x4 sinx cosx p + = + .Cu 20. Giai phng trinh:22tanx cot2x 3sin2x+ = + .Cu 21. Giai phng trinh:23tan3x cot2x 2tanxsin4x+ = + .Cu 22. Giai phng trinh:2 2 2sin x sin 2x sin 3x 2 + + =.Page 30" All the flower of tomorrow are in the seeks of today"Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onCu 23. Giai phng trinh:( )25 4x 3sin2 x 8sin x 0 - p + p = .Cu 24. Giai phng trinh:sin2x2cosx 01 sin2x + =+.Cu 25. Giai phng trinh:sinxcot5x1cos9x= .Cu 26. Giai phng trinh:23tan6x 2tan2x cot4xsin8x- = - .Cu 27. Giai phng trinh:21 cosxtan x1 sinx+=-.Cu 28. Giai phng trinh:3 32cos xcos3x sin xsin3x4+ =.Cu 29. Giai phng trinh:4 4x x 5sin cos3 3 8 + = .Cu 30. Giai phng trinh:( )22sin3x 1 4sin x 1 - = .Cu 31. Giai phng trinh:3 3 2cos x 4sin x 3cosxsin x sinx 0 - - + =.Cu 32. Giai phng trinh:4 4x xsin cos 1 2sinx2 2+ =- .Cu 33. Giai phng trinh:sin 3x sin2xsin x4 4 p p - = + .Cu 34. Giai phng trinh:( )2442 sin x sin3xtan x 1cos x-+=.Cu 35. Giai phng trinh:2xtanx cosx cos x sinx 1 tan tanx2 + - = + .Cu 36. Giai phng trinh: 2 2x 7sinxcos4x 2sin 2x 4sin x , x 1 34 2 2 p - = - - " - < .Cu 37. Giai phng trinh: sinx sin2x sin3x 1 cosx cos2x + + =+ + .Cu 38. Giai phng trinh:2 2 2 2cos x cos 2x cos 3x cos 4x 2 + + + =.Cu 39. Giai phng trinh:( )( )2cos x cosx 12 1 sinxsinx cosx-= ++.Cu 40. Giai phng trinh: sinx sin2x sin3x sin4x sin5x sin6x 0 + + + + + = .Cu 41. Giai phng trinh: cosx cos3x 2cos5x 0 + + = .Cu 42. Giai phng trinh: 9sinx 6cosx 3sin2x cos2x 8 + - + = .Cu 43. Giai phng trinh:( )cosx sinx cosxsinx cosxcos2x - =.Cu 44. Giai phng trinh:( ) ( )22sinx 1 3cos4x 2sinx 4 4cos x 3 + + - + =.Cu 45. Giai phng trinh:3 32sin x sinx 2cos x cosx cos2x - = - +." Cn cu bu thng minh"Page 31Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Cu 46. Giai phng trinh:2 3 4 2 3 4sin x sin x sin x sin x cosx cos x cos x cos x + + + = + + +.Cu 47. Giai phng trinh: ( ) ( )2sin x tanx 1 3sinx cosx sinx 3 + = - +.Cu 48. Giai phng trinh:2tan2x cot x 8cos x + =.Cu 49. Giai phng trinh: ( ) ( )3 cot x cosx 5 tanx sinx 2 - - - =.Cu 50. Giai phng trinh:1 12 2sin x4 sinx cosx p + = + .B PHNG TRINH B C HAI VA BC CAO I VI MT HAMLNG GIACDangt n phu iu kin2asin x bsinx c 0 + +=t sinx = 1 t 1 - 2acos x bcosx c 0 + +=t cosx =1 t 1 - 2atan x btanx c 0 + +=t tanx =x k, (k )2p + p 2acot x bcot x c 0 + +=t cot x =( )x k, k p Nu t2t sin x = hoct sinx =th iu kin l0 t 1 (tng t cho cos)Mt s hng ng thc lng giac va m i lin h( )22 21 sin2x sin x cos x 2sinxcosx sinx cosx + = + + = +( )22 21 sin2x sin x cos x 2sinxcosx sinx cosx - = + - = -sin2xsinxcosx2= ( ) ( )3 3sin x cos x sinx cosx 1 sinxcosx + = + - ( ) ( )x3 3sin x cos x sinx cosx 1 sinxcos - = - +2 2sinx cosx sin x cos x 2tanx cot xcosx sinx sinxcosx sin2x++ = + = =2 2cosx sinx cos x sin x 2cos2xcot x tanx 2cot xsinx cosx sinxcosx sin2x-- = - = = =4 4 2 21 1 1 3 1cos4xsin x cos x 1 sin 2x cos 2x2 2 2 4++ =- = + =( ) ( )4 4 2 2 2 2cos x sin x sin x cos xcos x sin x cos2x - = + - =6 6 4 4 2 2 23 5 3cos4xsin x cos x sin x cos x sin xcos x 1 sin 2x4 8++ = + - =- =Page 32" All the flower of tomorrow are in the seeks of today"Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( )6 6 4 4 2 2cos x sin x cos2x sin x cos x sin xcos x - = + +x 11 tanxtan2 cosx+ =sinx cosx 2sin x 2cos x4 4 p p = = m( ) ( )2 2cosx cos x 1 sin x 1 sinx1 sinx cosxcosx 1 sinx cosx 1 sinx- += = =-- - (mi lin hgia sinx va cosx)BAI TP AP DUNGBai 51. Giai phng trinh: ( )

2cos4x 12sin x 1 0 + - = *Bai 52. Giai phng trinh: ( )

4 4cos x sin x cos4x 0 - + = *Bai 53. Giai phng trinh: ( ) ( )

cos3x sin3x5 sinx 3 cos2x , x 0;21 2sin2x + + =+ * " p + Bai 54. Giai phng trinh:( ) sin3x sin5x3 5= *Bai 55. Giai phng trinh:( )

sin5x15sinx = *Bai 56. Giai phng trinh: ( ) 2 2cos 3xcos2x cos x 0 1 - =Bai 57. Giai phng trinh: ( ) 4 43cos x sin x cos x sin 3x 04 4 2 p p + + - - - = * Bai 58. Giai phng trinh: ( ) 5 7sin 2x 3cos x 1 2sinx; x ;22 2 2 p p p + - - =+ " p * Bai 59. Giai phng trinh: ( ) ( ) 25sinx 2 3 1 sinx tan x - = - *Bai 60. Giai phng trinh:( )( ) 2sin2x 3cos2x 5 cos 2x6 p + - = - * Bai 61. Giai phng trinh:( )( ) 6 62 cos x sin x sinxcosx02 2sinx+ -= *-Bai 62. Giai phng trinh:( )( ) 1 sinx cos2x sin x41cosx1 tanx2 p + + + = *+Bai 63. Giai phng trinh:( ) 1 12sin3x 2cos3xsinx cosx- = + *Bai 64. Giai phng trinh:( )( ) 2cosx 2sinx 3 2 2cos x 111 sin2x+ - -= *+Bai 65. Giai phng trinh:( ) x 3x x 3x 1cosxcos cos sinxsin sin2 2 2 2 2- = *" Cn cu bu thng minh"Page 33Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)Bai 66. Giai phng trinh:( )

4 4sin x cos x 1 1cot2x5sin2x 2 8sin2x+= - *Bai 67. Giai phng trinh:( )( )

2 23cot x 2 2sin x 2 3 2 cosx + = + *Bai 68. Giai phng trinh: ( )

6 23cos4x 8cos x 2cos x 3 0 - + += *Bai 69. Giai phng trinh:( ) 2cos4xcot x tanxsin2x= + *Bai 70. Giai phng trinh:( )

2cot x tanx 4sin2xsin2x- + = *Bai 71. Giai phng trinh: ( )

2 22sin x tan x 2 + = *Bai 72. Giai phng trinh:( )

8 81sin x cos x8+ = *Bai 73. Giai phng trinh:( )

8 8 217sin x cos x cos 2x16+ = *Bai 74. Giai phng trinh:( ) 35x xsin 5cos xsin2 2= *Bai 75. Giai phng trinh: ( ) ( ) 2sin2x cot x tan2x 4cos x + = *Bai 76. Giai phng trinh:( ) 26x 8x2cos 1 3cos5 5+= *Bai 77. Giai phng trinh: ( )

3tan x tanx 14 p - = - * Bai 78. Giai phng trinh:( ) 4 44sin 2x cos 2xcos 4xtan x tan x4 4+= * p p - + Bai 79.Giai phng trinh:( ) ( )

4 21 248 1 cot2xcot x 0cos x sin x- - + = *Bai 80.Giai phng trinh:( ) ( )

8 8 10 105sin x cos x 2 sinx cos x cos2x4+ = + + *Bai 81.Giai phng trinh:( )

2cos2x 1cot x 1 sin x sin2x1 tanx 2- = + - *+Bai 82.Giai phng trinh: ( )sin2x 2tanx 3 + = *Bai 83.Giai phng trinh:( ) ( ) ( ) 1 tanx 1 sin2x 1 tanx - + =+ *Bai 84.Giai phng trinh:( ) ( ) 2cos2x cosx 2tan x 1 2 + - = *Bai 85.Giai phng trinh: ( )( )3sin2x cosx 3 2 3cos x 3 3cos2x 8 3cosx sinx 3 3 + - - + - =Bai 86.Giai phng trinh:221 14 sin x 4 sinx 7 0sinx sin x + + + - = ( ) *Bai 87.Giai phng trinh: ( )

2tan x tanxtan3x 2 - = *Bai 88.Giai phng trinh:( )

1 1sin2x sinx 2cot2x2sinx sin2x+ - - = *Page 34" All the flower of tomorrow are in the seeks of today"Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onBai 89.Giai phng trinh:( )

12cos2x 8cosx 7cosx- += *Bai 90.Giai phng trinh: ( ) ( ) ( )4 sin3x cos2x 5 sinx 1 - = - *Bai 91.Giai phng trinh:221 1sinx sin xsinx sin x- = - ( ) *Bai 92.Giai phng trinh:221 1cos x 2 cosx 1cosx cos x + - + = ( ) *Bai 93.Giai phng trinh:12tanx cot x 2sin2xsin2x+ = + ( ) *Bai 94.Giai phng trinh: ( )2 2tan x cot x 2 1 tanx cot x 0 + + + + =( ) *Bai 95.Giai phng trinh:( ) 4 411sin x cos x sin2x8+ = - *Bai 96.Giai phng trinh:5 5 24sin xcosx 4cos xsinx sin 4x - =( ) *Bai 97.Giai phng trinh: ( ) ( )cos 2x cos 2x 4sinx 2 2 1 sinx4 4 p p + + - + =+ - * Bai 98.Giai phng trinh:2 2 2 23cos x cos 2x cos 3x cos 4x2+ + + = ( ) *Bai 99.Giai phng trinh:6 62 2sin x cos x 1tan2x4 cos x sin x+=-( ) *Bai 100.Giai phng trinh:6 6sin x cos x sin2x + =( ) *HNG D N GIAI PHNG TRINH B C HAI VAB C CAO I VI MT HAM LNG GIACOLi binh: Trong bai toan toan co cha hai cung xva4xnn ta a vcung mt cung la2xbng cng thc nhn i cua 2cos4x 2cos 2x 1 = - vacng thc habc 21 cos2xsin x2-= . To, a ta vphng trinh bc hai theocos2x.Bai giai tham khao( ) ( )2 22cos 2x 1 6 1 cos2x 1 0 cos 2x 3cos2x 2 0 * - + - - = - +=( )

2t 1 t 2 t 3t 2 0cos2x 1 x k, kt cos2x,t 1 t cos2x,t 1 = = - += = = p = = ." Cn cu bu thng minh"Page 35Bai51.Giai phng trinh: ( )

2cos4x 12sin x 1 0 + - = *(Trich thi tuyn sinh Cao ng khi A, B, D nm 2011)Bai52.Giai phng trinh: ( )

4 4cos x sin x cos4x 0 - + = *(Trich thi tuyn sinh Cao ng Xy dng s 2 nm 2007)Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)OLi bi nh: Trong vi du na y, cung tn tai hai cung khac nhau xva 4x nn ta a vcung mt cung la 2x, nhng ln na y cn phai kt hp gia hng ng thc vacng thc nhn i: ( ) ( ) ( ) ( )2 22 2 2 2 2 2cos x sin x cos x sin xsin x cos x cos2x - = - + = . Concos4xta se ap dung cng thc nhn i nh trn c phng trinh bc hai theocos2x.Bai giai tham khao( ) ( ) ( )2 2 2 2 2cos x sin xcos x sin x 2cos 2x 1 0 * - + + - =( ) 2cos2x 1x k22cos 2x cos2x 1 0 k,l1cos2xx l26p=- = + p + - = p= = + p .OLi binh: Trong bai toan na y, cocha ng thi ba cung x,2x,3xvata khng tha cung xcua sinxv cung2xc (khng co cng thc lng giac nao), do o chi con cach duy nht la a ba cung na y vcung cungx. Nhn thy rng, trong v trai phng trinh co chacos3x sin3x + , ta nn phn tich hai thanh phn na y trc tranh lp lai va dai dong khi giai phng trinh. Concos2xtt nhin a v cung xbng cng thc nhn i: 2 2 2 2cos2x cos x sin x 2cos x 1 1 2sin x = - = - =-, nhng trong ba cng thc o, ta se ap dung cng thc nao ? Cu tra li la"da vao s bin i cua vtrai chon cng thc phu hp".Bai giai tham khaoiu kin:11 2sin2x 0 sin2x2+ - .Ta co:( ) ( )3 3cos3x sin3x 4cos x 3cosx 3sinx 4sin x + = - + -( ) ( ) 3 33 cosx sinx 4 cos x sin x =- - + -( ) ( ) 3 33 cosx sinx 4 cos x sin x =- - + -( ) ( ) 2 2cosx sinx 3 4 cos x cosxsinx sin x = - - + + + ( ) ( )cosx sinx 1 2sin2x = - +.( )( ) ( )cosx sinx 1 2sin2x5 sinx 3 cos2x1 2sin2x - +* + =+ + ( ) 25 sinx cosx sinx 2cos x 1 + - = -( )( ) 21cosx2cos x 5cosx 1 0 x k2 , k23cosx 2 L= p - - = = + p =.Page 36" All the flower of tomorrow are in the seeks of today"Bai53.Giai phng trinh: ( ) ( )

cos3x sin3x5 sinx 3 cos2x , x 0;21 2sin2x + + =+ * " p + (Trich thi tuyn sinh ai hoc khi A nm 2002)Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onKt hp vi iu kin, ta c ho nghim la( )x k2 , k3p= + p do 3 1sin2x2 2= -. ( ) x3Do x 0;2 nn5x3p = pp =OLi binh: T vic xut hin hai cung 3x, 5x x 4x = +, ta coth a chung v cung mt cung xtheo cng thc nhn ba va cng cung xut hin nhn t chung (cach giai 1). Hn na, trong bai xut hin s3 va5, ta cung co thtach5 2 3 =+va nhom chung mt cach kheo leo lai vi nhau, ap dung cng thc tng thanh tich (cach giai 2).Bai giai tham khaoCach giai 1( ) ( )5sin3x 3sin x 4x * = +( ) ( )35 3sinx 4sin x 3 sinxcos4x cosxsin4x - = +( ) ( ) 25sinx 3 4sin x 3 sinxcos4x 2cosxsin2xcos2x - = +( ) ( ) 2 25sinx 3 4sin x 3sinx cos4x 4cos xcos2x - = +( ) ( ) 2 2sinx 5 3 4sin x 3 cos4x 4cos xcos2x 0 - - + = ( ) 2sinx 12cos 2x 4cos2x 8 0 - + + =( )2sinx 3cos 2x cos2x 2 0 + - =

sinx 0 x kx k2cos2x 1 x l , k,l,m ; cos3x m222cos2x x m23 2 = =p = p = = p a = a = + pa = = + p .Cach giai 2( )5sin3x 3sin5x * =( )2sin3x 3 sin5x sin3x = -( ) 22sinx 3 4sin x 6cos4xsinx - =( )22sinx 3 4sin x 3cos4x 0 - - =( ) ( ) 2sinx 3 2 1 cos2x 3 2cos 2x 1 0 - - - - = ( )2sinx 3cos 2x cos2x 2 0 + - = .Giai tng t nh trn.Bai giai tham khaoiu kin: sinx 0 Cach giai 1" Cn cu bu thng minh"Page 37Bai54.Giai phng trinh:( ) sin3x sin5x3 5= *(Trich thi tuyn sinh ai hoc Thu y Li nm 2000)Bai55.Giai phng trinh:( )

sin5x15sinx = *(Trich thi tuyn sinh ai hoc Mo ia Cht nm 1997)Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)( )sin5x sinx 4sinx 2cos3xsin2x 4sinx 4cos3xsinx cosx 4sinx 0 * - = = - =( ) ( )4sinx cos3xcosx 1 0 2sinx cos4x cos2x 1 0 - = + - =( )( )

22sinx 0 Lsinx 2cos 2x cos2x 3 02cos 2x cos2x 3 0= + - = + - =( )( ) 2cos2x 11 cos2x 0 2sin x 0 sinx 0 Lcos2x 1,5 L= - = = ==-Vy phng trinh v nghim.Cach giai 2( ) ( )sin 2x 3x 5sinx sin2xcos3x sin3xcos2x 5sinx * + = + =( ) ( ) ( ) ( ) 23 3 2 2 2 22sinxcosx 4cos x 3cosx 3sinx 4sin x cos x sin x 5si nx sin x cos x - + - - = +( ) 5 2 3 3 2 212sin x 20cos xsin x 0 sin x 12sin x 20cos x 0 + = + =( )( )( )

33 22sin x 0 Lsin x 12 8cos x 00 12 8cos x 0 L= + = < + =Vy phng trinh v nghim.WWW.VNMATH.COMBai giai tham khao( ) ( )

1 cos6x 1 cos2x1 .cos2x 0 cos6xcos2x 1 0 22 2+ + - = - =Cach giai 1( ) ( )3 4 22 4cos 2x 3cos2x cos2x 1 0 4cos 2x 3cos 2x 1 0 - - = - - =( )( )

22cos 2x 1ksin2x 0 2x k x , k12cos 2x L4=p = = p = =-.Cach giai 2( ) ( )212 cos8x cos4x 1 0 cos8x cos4x 2 0 2cos 4x cos4x 3 02 + - = + - = + - =( )( )

cos4x 1k4x k2 x , k32cos4x L2=p = p = =- .Cach giai 3.Phng trinh lng giac khng mu mc( ) ( ) cos6x cos2x 1k2 x , kcos6x cos2x 12= =p = = =-.Page 38" All the flower of tomorrow are in the seeks of today"Bai56.Giai phng trinh: ( ) 2 2cos 3xcos2x cos x 0 1 - =(Trich thi tuyn sinh ai hoc khi A nm 2005)Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onCach giai 4.Phng trinh lng giac khng mu mc( ) ( ) k2 cos8x cos4x 2 0 cos8x cos4x 1 x , k2p + - = = = = .Bai giai tham khao( )21 1 31 sin 2x sin 4x sin2x 02 2 2 2 p * - + - + - = ( ) 21 1 1sin 2x cos4x sin2x 02 2 2 - - - + - =( ) 2 21 1 1 1sin 2x 1 2sin 2x sin2x 02 2 2 2 - - - + - =( )( )

2sin2x 1sin 2x sin2x 2 0 x k , ksin2x 2 L 4=p + - = = + p =-.OLi binh:T vic xut hin cac s 5 7,2 2p p la nhng bi s cua 2p, lam ta lin tng n cu "cos i sin bu phu cheo", tht v y, cac ban ycach giai sau:Bai giai tham khao( ) ( ) ( )sin 2x 2 3cos x 4 1 2sinx2 2 p p * - - - p - - - + p=+ ( ) ( )cos 2x 2 3sin x 4 1 2sinx cos2x 3sinx 1 2sinx - + p - - + p =+ + =+ ( ) 2x ksinx 0x l22sin x sinx 0 k;l;m16sinx52x m26 = p= p = +p - + = = p = + p .Vix k1k 2 k 2k 1 x2 2k , x ;2k k2 = p p < p < p +" Cn cu bu thng minh"Page 71Bai120.Giai phng trinh:( ) 3 34sin xcos3x 4cos xsin3x 3 3cos4x 3 + + = *Trich thi tuyn sinh Hoc Vin Bu Chinh Vin Thng nm 2001Bai121.Giai phng trinh:( ) ( ) 2 2 sinx cosx cosx 3 cos2x + =+ *Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao) Bi gii tham kho( )22sinxcosx 2cos x sinx cosx 1 * + - - =

2sin2x 2cos x 1 sinx cosx + - = + sin2x cos2x sinx cosx + = + sin 2x sin x4 4 p p + = + ( ) x k22x x k24 4k,2x2x x 26 34 4p p = p + = + + p p p p p= + + =p- - +p lll.Bi gii tham kho( ) ( ) ( )2 22 cos x sin x 6 cosx sinx 0 * - - - =( ) ( )( )( )

sinx cosx 0 1sinx cosx 2sinx 2cosx 6 02sinx 2cosx 6 2- = - + - = + =( ) ( )( ) ( )1 2cos x 0 x k x k , k4 4 2 45x 2 x 234 6 122 cos x cos k,4 2 6x m2 x m24 6 12 p p p p + = + = + p = + p p p p - = +p = +p p p - = = p p p - =- + p = + p l llBi gii tham kho( )1 3cosx sinx cos3x2 2* + =-( ) ( ) x 3x k2k3cos x cos 3x x , k,3 3 2x 3x 23p - =p- + p p p p - = p- = + p - =- p+ +pll.Bi gii tham khoPage 72" All the flower of tomorrow are in the seeks of today"Bai122.Giai phng trinh:( ) ( ) ( ) 2cosx 1 sinx cosx 1 - + = *Bai123.Giai phng trinh:( ) ( ) 2cos2x 6 cosx sinx = - *Bai124.Giai phng trinh:( ) 2cos3x 3sinx cosx 0 + + = *Bai125.Giai phng trinh:sinx 3cosx sinx 3cosx 2 + + + =( ) *Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn onTa co:1 3sinx 3cosx 2 sinx cosx 2sin x2 2 3 p + = + = + t t sinx 3cosx 2sin x 0 t 23 p = + = + ( )( )( ) 2t 1 Nt t 2 0t 2 L=* + - = =-( ) x k2162sin x 1 sin x k,3 3 2x 22p =- + p p p + = + = p = +pll.Bi gii tham khoTa co:1 3cosx 3sinx 2 cosx sinx 2cos x2 2 3 p + = + = - t t cosx 3sinx = +. Ta co:( ) ( )Bunhiacopxki22 2 2cosx 3sinx 1 3 sin x cos x 2 + + + = Do o:2 t 2t 1 - ( )( )( )

2t 1 L3t t t 2 0t 2 N t 1=* = + - = =- +( ) 2x k2x k23 32cos x 1 k,33x 2x 23 3p pp - = + p p = + p - = p p = p- =- +p lll.Bi gii tham kho( ) ( ) ( )2 2sinx cosx cos x sin x 0 * + - - =( ) ( ) ( )sinx cosx cosx sinxcosx sinx 0 + - - + =( ) ( )sinx cosx 1 cosx sinx 0 + - + =" Cn cu bu thng minh"Page 73Bai126.Giai phng trinh: ( )

2cosx 3sinxcosx 3sinx 1+ = *+ -Bai127.Giai phng trinh: sinx cosx cos2x + = ( ) *Ths. L Vn on Phng trinh lng giac va ng dung (Nng cao)

2cos x 0 cos x 0sinx cosx 04 4sinx cosx 132cos x 1 cos x cos4 4 4 p p - = - = + = - =-p p p + =- + = ( ) 3x k4x k4 2x 2 k, ,m3 32x 2 x m2x m24 4 4 4p = + pp p- = + pp = +p p p p p+ = +p + =- + p =- p+ pl ll.Bi gii tham kho( )33sinx 4sin x 3cos3x 1 * - - =-

3 1 1sin3x 3cos3x 1 cos3x sin3x2 2 2 - =- - =( ) k23x k2 x6 3 18 3cos 3x cos k,26 33x 2 x6 3 6 3 p p p p + = + p = + p p + = p p p p + =- +p =- + lllBi gii tham kho( ) ( )cos7xcos5x sin7xsin5x 3sin2x 1 * + - =( ) 1 3 1cos 7x 5x 3sin2x 1 cos2x 3sin2x 1 cos2x sin2x2 2 2 - - = - = - =( ) x kcos 2x cos k,3 3x3 = p p p + = p =- + pllBi gii tham kho( )28sinxcosx 6cos x 3 12sinx 3 0 * - +- +=

24sinxcosx 3cos x 6sinx 3 0 - - +=( ) 24sinxcosx 6sinx 3 cos x 1 0 - - - =

24sinxcosx 6sinx 3sin x 0 - + =Page 74" All the flower of tomorrow are in the seeks of today"Bai128.Giai phng trinh:( )

34sin x 1 3sinx 3cos3x - = - *Bai129.Giai phng trinh:( ) cos7xcos5x 3sin2x 1 sin7xsin5x - =- *Bai130.Giai phng trinh: ( )4sin2x 3cos2x 3 4sinx 1 - = -( ) *Phng trinh lng giac va ng dung (Nng cao)www.VNMATH.comThs. L Vn on( )sinx 4cosx 3sinx 6 0 + - =( )( ) 2 2 2osinx 0x k, k4cosx 3sinx 6 VN do: 4 3 6= = p + = +