Bồi dưỡng kiến thức học sinh giỏi lượng giác: Phần 2

Dau bang xay cac ba't dang thtfc tren va chi ABC la tam giac deu GAt^V3 (1) 2 A =B=C A B C B C tan — > 0, tan — > 0, tan — > tan — + tan — + tan — >3 2 2 2 A , B C ^ A B B C C A tan — + tan" — + lan" — + tan — tan — + tan — tan — + tan — tan — >3 2 2 2 2 2/ 2.3.4 ChiJng minh tu-dng tif 1/ ta co: 3^3 cos B ^ A B C ^ sm — + sm — + sm—< - - 2 C ^ 3^3 hcos—l-cos—< 2 ~ ^ I ^3 cotA + cotB + cotC > V3 o ABC la tam giac deu A A = B = C « ABC la tam giac deu A B C ^ tan —+ tan —+ tan —> V3 , 2 (2) ^ C A-B sm cos = (J 2 • A-B sm-—-— = sinA + sinB + sinC < < ^ _ Bai Trong moi tam giac ABC ta c6 cac he thiJc sau: Vi (2) hien nhicn diing, vay (1) dung => dpcm Bai Trong moi tam giac ABC, 3v^ 2/3 Chu-ng minh tu'dng tu" 1/ A-Bl^ A - B + sin" X) cos J - Da'u bang xay o A =B hi > 2 C+" 7t ^ A+B +C+ L_ phan 1/ vc'Ji chu y (2) V i the chang han de chiJng minh ba'l d^ng ^ colC - cot(B + C) > => cotB + cotC - col(B + C) > 0, vay (3) dung Sau (1) "tot h d n " neu nhu" Ve > 0, ton lai lam giac A||B„C(| cho hay Ta c6: colB > va 90" > B + C > C =^ col{B + C) < cotC V i Ic colA + cotB + colC > S vlft ^ Bat dang thiJc f(A, B , C) > P doi v d i m o i lam giac A B C g o i 1^ khong the ISm Trifdc hct ta chi?ng minh m o i tam giac i h i cot A + cotB + cotC > P VVn Khang jsjh^it xet: Ta hay hieu the nao la bat dang thufc khong the l a m " t o t hrin" >() + tan — - t a n — ivrv Icos A c o s B c o s C i = |COSA|.|COSB|.|COSC| < m o i tam giac la c6: => cosAcosBcosC > - (1) cosA + cosB + cosC > cosAcosBcosC > - Mat khac xet ho tam giac A B C v d i A = 180" - a ; B = C = a vcti a > du be sinA + sinB + sinC > Khi do: cosAcosBcosC = cos( 180" - 2a)cos^a = - c o s a c o s ' a A B , C sm — s m — s m — > 2 c A B C , sm — + sm — + s i n — > 2 A B C 2 - cos — + C O S — + C O S — > Giai B C YsmYsm- l i m ( c o s A c o s B c o s C ) = - l i m (cos2a)(cos^a) = - Vay ba'l dang ihurc (1) khong the lam " l o t h d n " => dpcm 3.4 Chiang minh va cac ba'l dang thuTc trcn khong the lam " l o t h d n " A Ta co: cos A + cosB + cosC = + 4sm Tirdo CD tu'rJng liT Vi 0, do liT (1) siiy v d i A ' = : ? ^ ; B ' ^ ; C ' = A ± 2 cosA + cosB + cosC > Ro rang A ' > 0, B ' > 0, O A B C A Ta c o : < sm — s m — s m — < sm — 2 2 coi A', B', C la ba goc cua mot tam giac A ' B ' C khac M a t khac: l i m A->() A') sm — = : • l i m 2y A B C sm—sin—sm — = 2 Vay bat dang Ihufc tren kh6ng the l a m lot hdn va A ' + B ' + C = A + B + C = 180" nen co the Theo phan 1/ suy cosA' + cosB' + c o s C > ^' ' ^ A B C , , =!> sin — + sin — + sin — > => dpcm 2 * Xet ho lam giac A B C A = 180" - a ; B = C = a v6i a > du be B6I duang h ^ c sinh g l o l Lupng A B 2 glAc - rhan Huy Khal Cty C cos a + 2sin — = 27 A B C , Vay ba't dang thuTc s i n y + s i n y + sin y > khong the lam "tot hdn" sin — + sin—I- sin — = lim lim a->{)'^ Do < cos— < 1, < cos— < 1, < cos— < 2 A B C zA B ^ cos hCOS hCOS—>COS h COS + COS ^ C A,B = 2R,sin " lim A B Thay (3) vao (2) va c6: AA, = — R s i n — a (4) Lap luan lu'dng tiTcd: BB, =~-2R > (Iheophan 1/) 2 2 Cho ho tam giac ABC vdi A ^ 180" - 2a, B = C = a, vdi a > du be suy Timn A B C •> ' Bat dang thiJc c o s y + cosy + '^osy > khong the lam "tot hdn" • A B C AA,.BB,.CC, = (h + 16R-r =:>dpcm dang Ihtfc khong co dang lu'dng giac nhu'ng chiJng minh no thi hoan toan di/a Dau bang xay r a o b = c = a o ABC la tam giac deu (10) vao phcp bien ddi cac he ihiJc lu'dng giac tam giac Cac ba't dang tMc NHn gan chat vdi mot bai toan hinh hoc phang lu'dng iJng Ke't qua noi tren van dung ne'u thay ba du'dng phan giac bdi ba difdng Irung Nhif vay de giai xet: chiing Ihi CO siT phoi hdp giffa lu'dng giac va hinh hoc phang (no phan bict trifc, tiJc thay tam difdng Iron noi tiep I bang tam O diTdng tron ngoai tiep vdi Khi AA| = BB| = C C , =2R Ccic bai loan thuan luy ve bat dcing thufc lu'dng giac ma vc mat hinh thifc CO the thay cac bai loan ay khong he c6 mot hinh ve nao ca) Vay AA|.BB|.CC| = R \ B a i Cho lam giac ABC Ba diTdng phan giac cua cac goc A, B, C laa lu'dt c5t diTdng Iron ngoai ticp tam giac ABC AA,.BB|.CC| > D o R > r = ^ R ' > 16R'r => AA|.BB,.CC| > tai A | , B | , C| Chijrng minh Da'u bang xay R = 2r 16RV ^- Ne'u thay I bcli triTc tam H ciia tam giac nhon ABC Ap dung dinh li Plolcme vdi ti? giac noi ticp Ap dung dinh li ham so sin A A B A i , ta c6: A B A i C taco: AA, =2RsinABAJ = 2Rsin(B + A ^ ) (1); Ta c6: A | B = A.C, nen tif (1) co: _ A | B ( A B + AC) BC i "J^ABC la tam giac deu Giai AA|.BC = AB.A|C + A C A i B 16RV = 2Rsin(B + H A c ) (2) Ap dung dinh l i ham so sin AABA, (vdi chu y ban kinh cua diTdng iron ngoai tiep tam giac cung chinh la R), ta c6: • k flp = 2Rsin(B + 90" - C) =2Rcos(B-C) J t f d n g tir c6: BB, = 2Rcos(C - A ) jf " ^~ VIft B6I duang ln>c sinh gioi I n^ing gUic - fhan Tir d6 A A , B B | C C , Huy Cty Timn nrv Khal = R W ( B - C)cos(C - A)cos(A - , A B C 8sinysin-sin- A B C T i r d o s u y r a A A , B B | C C , < R \ s i n y s i n - s i n - = 16R^r dang thuTc nhi/ng vc)i chieu ngiTrtc l a i B a i G o i I la tam diTcfng tron npi tiep tam giac A B C AT, B I , C I keo dai cat di/5ng tron ngoai tiep A A B C Ian lirm tai A , , B,, C, ;IC = o o o 2R sm—hsm—1-sm — > r 2 2j = R s i n H A C c o s B =4RcosCcosB 2R(cosA + cosB + cosC) > 4R(cosAcosB + cosBcosC + cosCcosA) o cosA + cosB + cosC > 2cosAcosB + 2cosBcosC + 2cosCcosA 1 A sm- -+ • ' B sm~ + < cosA + cosB + cosC < - 2 Hoan toan tiTrtng tiT, ta c6 cac ba't dang thiJc sau IA,.IB,.IC, >IA.IB.IC; — + IA| 2 Cho tam giac A B C noi tiep discing tron tam O G o i I la giao d i e m ciia ba difctng phan giac A M , B N , CP Gia suf A M , B N , CP keo dai cat di/c^ng tron ngoai tiep tai A ' , B', C li/dng i^ng ChuTng minh: A B , C s m — h s m — h s m — = - - sm—hsm—hsm — ~3 sm hsm—hsm — 1 lA' ^ IB' ^ IC' A , B , C H i e n nhien ta l a i c6: I I 1 — + +— IB, IC| ~ l A IB IC < — + —+ — Da'u bang xay hai ba't dang thuTc Ircn A B C la tam giac deu A , B^ C^3 sm—hsm—+ sm— H A , + H B , + HC, A B C A B B C ^ C A , , , s i n y + s i n - + s i n y > s i n y s i n - + s m - s m - + s m - s m y (1) B M TiTdng tir HB = 4RcosCcosA; HC, = 4RcosAcosB „ A B C A B Cl 2R s m — h s m — h s m — > R s m — s m — s m — 2 2 2j A C, Lai C O H A , = H M = H C s i n H C M V a y l A , + I B , + I C , > l A + IB + IC C A X7 TiTdng tir HB = 2RcosB; HC = 2RcosC B C Tifdng l i r l B , = R s i n — ; I C | = R s i n y B vdi dau ngifdc l a i , ti?c la H A + HB + HC > H A , + H B , + H C , HA = 2R sin H C A = 2R cos A • l A , = A,B C sm sm sm A A Laithay A,B = 2RsinY =>IA, = 2Rsiny A Ncu A B C la tam giac nhon va thay bang triTc tam H , thi ta co ba't dang thtj-c nhau, ncn ap dung dinh l i ham so sin A H A C thi Giai B Da'u bang xay A B C la tam giac dcu That vay H A C , H A B , HBC va A B C c6 cac vong tron ngoai tiep bang Chu-ng m i n h : l A , + I B , + IC, > l A + IB + IC ; IB = A — Vay (1) dung => dpcm V a y neu thay I bdi trifc tam H va gia thiet A B C la tam giac nhon la c6 bai De tha'y I A = Vift A B C A B B C C A (3) sm — + s m - + s m - - > sm — sm—h sm —sm hsin — sm — 2 z/ 2 2 2 j A B C^, A B B C C TCr (2) (3) suy s m ^ + sm —+ s m Y > s m — s m — + s m — s m — h s m — s i n 2 2 B) Ta C O the chuTng m i n h diTdc ba't dang ihufc saii day irong m o i tam gi^c nhpn cos(A - B)cos(B - C)cos(C - A ) < P VVn Kbang 2J (2) >1 R Giai Ta c6: B I A ' = I B A ' l A ' = A'B Lap luan nhu'cac bai tren c6: l A ' = 2Rsin B6I duding hgc ainb gtdl Lugng glic - Phan Huy Khal B C TiTdng t i r l B ' = R s i n — ; I C = R s i n Y pgfu bang xay A B C la tam giac deu fjeu thay cac phan giac bang cac trung tuyen va I thay bang tam cua 1 tam gi^' ' ^ h ' '^'^ dang thiJc sau: 1 1 2R A sin sin OA' : >— B• + " R sin sin 2 1 sin sin— (1) MA.MB' = M B M C = =>MA' = f A , B , C sin—hsin—hsin — > C 2 sin B sin G C - R Theo bat dang thiJc Cosi c6: A sin GB' Xa CO theo he thiTc lu'dng du'cfng tron 4MA m GA ^ Tu" (2) (3) suy (1) dung Da'u bang xay o • 2 (3) GA' dpcm GA ^ GB GA' GB' M GC ^ ( b ^ + c ^ - a - ) + ( a ^ + c ^ - b ^ ) + ( a ^ + b ^ - c ^ ) GC' a^+b^+c^ (6) =>6=M:+M:+^,2R^ / GA' H C =4RcosAcosB GB' GC' ' IGA' GB' 1 cosCcosA cosAcosB Dau bang xay o 4R V ( c o s A c o s B c o s C ) A B C la tam giac deu (41 diTcfng tron ngoai tiep tam giac tiTdng tfng tai A ' , B ' , C • /' BB'= /|;,CC= co: < cosAcosBcosC < - 1- H HB' /' /„ L /, Giai (5) 'a CO A A B M ^ ^ j>— ^ =>dpcm HC' ~ R /' ChiJng minh ^ + - ^ + ^ T h c o ba't dang thuTc cd ban tam giac (va A B C la tam giac nhon), i ' HA' GC' ' ^ ^ i Cho tam giac A B C Ba du-dng phan giac A M , B N , CP k c o dai cSt &at A A ' = Tir (4) (5) suy + • „ , , , , — - > — => dpcm GA' GB' GC'~R A' 4R cosBcosC (7) (do (6)) Tu" theo bat dang thifc Cosi, ta co: HC' a^+b^+c^ Do A A ' < 2R; B B ' < 2R; C C < 2R H B ' = 4RcosCcosA; HB' 2b^ + 2c^ - a^ + 3a^ , , AA' BB' CC , GA , GB , GC , Laico: ,+ _ _ + _ _ = i+ _ _ + i+ +i + = GA' GB' GC' " ' GA' GB' GC' H A ' = 4Rco.sBco.sC; HA' 4m,^ + 3a' 2b^+2c^-a^ = /p/ That vay theo cac bai Iren ta co: 2(2b^+2c^-a^) 4m., A HC' " R ' 8mf Tiif suy ra: A B C la tam giac dcu (va goi H la trifc tam) K h i ta co: 1 a' N c u A B C la tam giac nhon va thay cac di/dng phan giac b^ng cac du^cfng cao HB' Nhan xet: HA' 4m., »^ ,GA' = G M + M A ' + - m , + • 4m„ (2) Theo bai toan crtban muc A , thi sin — + sin — + s i n — < - ^ AAA'C AB AM AA' AC c /., be b I, >4 B6I duOng hpc sinh gkil Lugng gUc - rhan Huy Khal Khi (1) m„ + t a 2bccos Do I, = A 2bccos < b+c A I,, 2Vbc nr, /„ < Vbc cos — + —^>4 m, + m„ mu , m, + , mu + i u •+ ^>4 m, I, Ih + ^ > mu (8) m, (2) , /' T i l f ( l ) ( ) d i den ^ > Ta co: M A M A ' = MB.MC hay t„.m, = — (3) - COS m, /' TiTdng lirco: - ^ > L" ^B' h cos I,^ C cos - I IL I Tir(3)(4)suyra ^ + -!i + X > I" Lt) I' t i;os-^ A (4) 4m.; 2b-+2c^-ab-^ TiTitng tir 2a"+2c^-b" 1 eos " cos 2^ (5) m,, Vithc(X)c=> 2(Y + Z ) - X Theo bat d^ng thuTc Co si, la c6: cos A A ,B 2 — I - COS" — h jC cos — 2j 2 1A + ' I B+ COS" - COS" >9 (6) Dat 2a-+2c"-b^ =Y > - 2a-+2b c"=Z>() COS" 2 2i 2A B C + (cosA + cosB + cosC) ^ Do cos — I - cos — h cos — = < —, 2 2 ' /.' /' /' Tif (5) (6) (7) suy + + > => dpcm ci 2b2+2c a'=X>() I, C 2a2+2b c-' — ,^2 _ 2(X + Z ) - Y 2(X + Y)-Z c = (7) Dc thay tiTcac lap luan Iren thi da'u bang xay ABC la lam giac dcii Nhgn xet: Khi lhay ba diTclng phan giac lhanh ba difdng Irung luyen, ta c6 Ni' loan sau: Cho lam giac ABC co dp dai ba trung luyen m,„ mn, m, Goi M,„ Mb, M , la J'' Vlthc-(9)«I±^ +^ + ^ > , Y — + — + —+ I Y xJ i z z XJ + —+ iz z Y >6 C>o X X ) , Y > 0, Z > 0, ncn ihco bat d;1ng thifc Co si Ihi (10) dung, vay (9) dung dpcm bang ,xay dc thay o X = Y = Z dai cua ba dU'dng trung tuycn ay kco dai cho den gap du'(4 m., m^ m^ o ABC la lam giiic deu Giai Goi cac trung tuycn la A M , BN, CP va giii silr A M , BN, CP keo dai tifdng cat diTiIng iron ngoai tiep AABC lao A', B', C Khi ta co: AA' = M „ BB' = Dat t, = M„ - m,; Mh, tb - Cho lam giac ABC noi tiep irong diTcJng iron lam O ChiJng minh OA- ^ O B ^ ^ O C ^ be ca Giai CC = M, = M^ - m^; I , = M , - m, (10) '-'o dinh li ham so sin, la co: ab OC' ab OA^ ^ OB^ 4R^ ca Theo baft dang IhuTc Cosi, thi be I 1 -+ sin B sin C sin C sin A (1) sin A sin B sinBsinC ^ sinCsinA ^ sinAsinB Giai J Neu A|, B | , C| deu nam tam giac ABC Khi ta c6: MB|+MC, MB, MC, ^ (1) ^-^77 ^ = 77r + ^ = smMAB, + s i n M A C • MA MA MA Hien nhien ta c6 (xem bai § I chu'dng 4) sin M A B | + sin MAC, M A B | + M A CI • A < sin = sin — (2) ^ Tir (1) (2) suy (2) ^(sin Asin BsinC)^ Lai theo bat dang ihufc Cosi, ihi V • • D • ^ sinA + sinB + sinC 1 ab ~ ca TiTdng lir ta c6: Neu thay O bang tam di/c(ng tron noi tiep I , thi ta c6: IB^ -+ • ca IC^ • +ab >1 16R^in2^sin2^ > , B' MC, + M A | sin MC (5) B C — — tan —tan — C 2 MA' MB^ (MB,+MC,)- (MC,+MA|)- MC' (MA,+MB, >1 • ^ Sin ^ (6) Chtfng minh (6) liTdng tiT nhtrchiyng minh d trcn (xin danh cho ban doc li'' iiu' lai) vuong gc')c BC, CA, AB ChiJng minh rang: MC^ MB^ MA(MC,+MA|)- -T>3 B ' (4) C sin o M la lam du^dng iron noi licp AABC Theo ba'l dang IhiJc Co si, la co: sin (MA,+MB,) sin 2 2, Dau bhng (4) xay ddng Ihdi co dau bang (1) (2) (3) Bai Cho M la diem nam tam giac ABC Kc M A , , M B , , MC, tiTcIng i^'^' (MB|+MC|)- B M , C M Ian liTdt la cac phi an T i r ( l ) ( ) ( ) s u y ra: Neu thay O bang trifc tam H, thi ta c6: be (3) sin giac eua cac goc B, C B C IC' A B TifdngtiTco: = tan —tan —; —— = lan —tan— ca 2 ab 2 Tir suy (5) diing ^ +^ > , ca ab A n A, Dau bang (2), (3) tu-dng tfng xay IB- ^ ^ n (2) c• MA, + M B B C B 16R sin sin - cos cos 2 2 M MB B C That vay theo cac bai tren ihi IA = 4Rsin —sin y ^ /+Y\ A M la phan giac ciia goc A Nhgn xet: lA^ be (1) :.sin Dau bang (1) xay Do la dpcm Dc thay dau b^ng xay o ABC la tam giac deu be > A A sin" B sin" C sin A sin 2 B sin 2 C• (5) Thco ba't dang ihiyc liTdng giac cd ban ihi sin^ —sin-^ —sin^ — < 2 Tur (4) (5) (6) suy dpcm irong iriTcfng hdp (6) Da'u bang xay o ABC la lam giac dcu va M la lam cua lam giac a'y Ncu A | , B | , C| CO diem nam ngoai lam giac ABC (chang han A, nam ngo^j lam giac ABC) Khi la c6 cac danh gia sau: MB| + M C MA - a)' + 4b(p - b)- + 4c(p - c)" < R ' ( R - r) c> a(b + c - a)' + b(a + c - b)' + c(a + b - c)' < 6N/3 R ^ ( R - r) c> 4abc - (b + c - a)(a + c - b)(a + b - c) < ^ R ' ( R - r) ' Ap dung cac cong ihiJc: ,;• , abc S = p r : S = 4R —; (1) — = (p-a)(p-b)(p-c) _ (h + c - a)(a + c - b)(a + b - c) ' 8R' o cos(A - B)cos(B - C)cos(C - A) > 8cosAcosBcosC o cos(A - B)cos(B - C)co.s(C - A) > -8cos(B + C)cos(C + A)cos(A + B) Vi the ket hdp vdi sin-Asin'Bsin-C > suy OA'.OB'.OC > 8R' ^ cos(A - B)cos(B - C)cos(C - A) cos(A + B)cos(B + C)cos(C + A) sin A,sin B,sin B sin C sin C sin A ~ sin A,sin B.sin B sin C sin C sin A (1 + tanAtanB)(l + tanBtanC)(l + lanClanA) > -8( - lanAtanB)( - tanBtanC)( - lanCtanA) (5) E)at X = lanA, y = lanB, z = lanC va chii y rang moi tam giac khong vuong, ta co: lanA + lanB + lanC = tanAlanBlanC ' ' ' ' =^x + y + z = xya (6) Khi ( ) ( +xy)(l +yz)(l + zx) > 8(xy - l)(yz- l)(zx- 1) ^ j ' ^ : o (z + xyz)(x + xyz)(y + xyz) > 8(xyz - z)(xyz - x)(xyz - y) (7) :, R, = Theo bat dang thtfc lu-dng giac cd ban suy dpcm A, Bai 11 Cho lam giac ABC khong vuong noi tiep du-dng Iron tam O ban kinh R DiTdng thang AO, BO, CO tiTdng rfng cat cac diTdng Iron ngoai ticp cac tam giac OBC, COA, AOB lai A', B', C ChiJng minh: OA'.OB'.OC > m\ Giai Gpi RI la ban kinh diTdng Iron ngoai tiep tam gidc BCA', thi theo dinh ly han' so' sin, ta co: Vib ... 2c^ - b^ _ (2a^ + 2c^ - b^) + 2a^ ' , 2S 2S _ 4BM^+2a^ _ B M ^ + a ^ 2S ~ a 2, sin BOC 2sin2A TiTdng liTco: Thco bat dang thi?c Cosi, ta thay 2BM- + a^ > 2% /2 BM.a (3) 2V2BM.a 2^ /2 TCr (2) (3) suy... Dat 2a-+2c"-b^ =Y > - 2a-+2b c"=Z>() COS" 2 2i 2A B C + (cosA + cosB + cosC) ^ Do cos — I - cos — h cos — = < —, 2 2 ' /.' /' /' Tif (5) (6) (7) suy + + > => dpcm ci 2b2+2c a'=X>() I, C 2a2+2b... N /2 BAC ABC ACB^ + OBCOS + OCcos >p 2 A b =• 2sin2B 2sin2C Khi r, + r: + r, > 3R a o o Bai 18 O la diem bat ki nilm tam giac ABC Chi?ng minh: OAcos a , S o b c 2sin2A 2sin2B 2sin2C 2R,sinA 2RsinB