Toán học và tuổi trẻ Số 136 (2/1984)

Chia sẻ: Physical Funny | Ngày: | Loại File: PDF | Số trang:16

Thêm vào BST

Báo xấu

78
lượt xem 7
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Sau đây là tài liệu Toán học và tuổi trẻ Số 136 (2/1984), mời các bạn tham khảo để hiểu rõ hơn về câu chuyện hình học giả Ơ-Clit; việc so sánh một số với nghiệm của tam thức bậc hai; đẳng thức có điều kiện trong lượng giác; một lớp bài toán hình học.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Toán học và tuổi trẻ Số 136 (2/1984)

'p#--- fi.- ie'" -' f: r'. a .. :,: rrEr KEoa Eqc ,Yr8T NAlx nQr roir xgc YIST NAM E' I It. t ff it !* .r,il.i, :i:}1i-,rn$ffi_$,: ..,gli tSKSil*:W I ! 't II B-i{' Hi HAr 1'tiA\{l .\l0T fi\' ir, r I Cinh Toan -t'r'ir Huog Dai,, tin N6i i: I'bng 6trt g6p . .r{guY6n Trrr si, 1iJ t)0 i rrd:i 5:'B : i L. F ?hd TEng. bidn l{p; Ngd tlaE' T{r s t- I F f, N6i ehuyQn **ircdre iran trir v0* to6n :1-r.'. , .1 ' . r" i' 'r': , : ;l i i,t.,,t i I i. i;1 i"': I N{iLYflr Lli\rl 'r0Ari : '' 'TnoNC bio I'oan'hoc iit trriii tril -,0i tlil c6 < r.ring tlirn dilh lrLr,'rng cir tiirtt 1rt'ri'i i) )i tiri dip so LrQ gioi thii'rr r'0i bau,lr:u !iinh a rlrt uttg th:iug )'-li rrrr)l triir isl'rp til rir ,':i,' \ i,lll' lrsc gii t)-cfit ducri dang irinl: hcc ci'c vr)ng tron clinh hu,'rrrg-r ! ltr vrii urr)t I'rirg ilr)n r]inlr trr)n dfnh hudng c6 t{nr ir€n mdt rltt
.-::r',r.tn€-1ll .i-. , .,.. 1i l '-E- :lr,\ - th6 nio' Cho rrtn truric hdt' c6 }F ptrii aglriin (( citu i'6 g6c r1[. Nirrng .-' tli-i' xStt^ ilqllg it$n gdc n6i t.'hiiirg n:i\ clti rmii iiLrqg ri$n )) gi': ', tr0ng- N0n ' tEr c6 th.bl-larn tiriii tliii[-, r iirr rli: rr gr';c rr c6 th] ft gn[ {{)c Dtng {'i{t ,''ir:ltll l;r,llr \(i:t.r t''hrrrltr lltl.r(ir t'tr-tt tlC itlttt sntr : r';i , 'Ciin ctr \':\o (l), tl r:dltirb qqhi rilng tron.q iririli hr,rl lrir.r'. iliic il-rmg cho tam giirc'i tiiing li tirrli .nfr ltnt'' piii'rnng :t4nir hrrj'0n Jrirng liir''tr l;illr ' lllt-v. rtrji htdt rirch 1;irircrug llai Er?nh gdc vtr0ng', . .: Lhec : iinh phtrnlg"cauh hrrf ir:n bii:g li'rru binh , ,, ' I t inti plltr.oriq Atn ; ttirrL "':ii- ie lhi.i' lhoit rlir''vir t "O < g/re )} iri lrin aii' ir',j thlnir : lghii'n citu cllc rldng ilin tiinh hri6lg -{' B' C' (trong .trlrng cila ba t16 hoirc ,1, r,i. jJ irit{r: A ','i i.' iihtiirq tri ti6 1r lu.r'C:ti' tf'tf;f, ll,tt, i"'i i-ri:"rtg i'i ti ''5c {ong trr)n)'' "f,";g sac chrr . HJnh] . . .t.)'i 'i;r : i- ,t: ".\lt i:" r1-... "t{l .l lFlr lrinir i l:t 6'.. ' .), t j.=iepn 42-r2 (tlr,n. r{u lrr': ,if; lrrrit,l*r, ir:r) - UL { ! l::--ilr -= ,.,u, ' t':. ., t:z - b2, ''*"=(t:1 l'i-t"^'tl "il" '). i'';- - \-iv dIr trLlri! i-i ;.lrr-rt' hin[r h'-.rc lit hei g6c d, p 1,iri nlrrLr: ,? o d:"l ^ -- {rI -ai)' --|Lz - fl\)' ' - a*F=5(i2' ' :\lJ -2 Nhu r'lir', ndu ta hiAu riiSn ba 1'orrg tri)n ;. Ii, dli,=Jct' ir t)2 -(cz' ut)z C theo thti tLt' Jr&i c;ic rlitnr A', B', C', theo thir . or, !1, cl, aii br. c2 pltai'qtran ir.Q vtri nhntl sao clttr tlr c(r io:r dd l-)€.cnl tl0tg thtrting lir (01, az), {bt, bil, (d, ce) rrhrr .ir lr:inh 1', thi hei clrr0ng f' I'rit -,ri)l - (rz:a2il? 12) thing -{?' vi,f i-' tto rtsn nrOi gdrc cii m6t phin o gihc sorrrl sollg v,ir Plr;'r n qi,ic llrri'rrhit. '- hnt t .li6l rlair lltn .diiv t1,,,,, i,i'i-i sinh ra l mrii.: Ta ' ,t , ,, -- eill -'i- i,; - lbtcr * ci'2 *' b*-l ?iric2 biEu dii'ru nr6i r-r.;nS Lrirn ilinir Lrr6ng t-o lirnt tr'0n D cri tr;rr riq rr., rl) LiLng di}.irr c6 l.oa ri
\fl *r efr \}\J s,&-hlffi l*S,r $#, v{JE ,, ,A rumruffim g+*a 'TAF* Tffir"sfl m&c PIr,\l{ }:ji'r'{l g1i1"rt ? F+ flf l,;p I {hi i(, ni.r:,i. s,il i.}:i i'oc clr }i0 thrie f'1 di, 1. iuc d;r:/l ;,:!i) pit:rrt,;{ irlinlt U, 1.'i-c!.' ,i ,l,r:L r,, 1q1 ri,,ni. .,:1 rl1i1i c[t' lu'- I tf z " !''|, :':iI i 'E '-' tt i./ rilhi'r, ,.:: 1,1....r" i:iiiir ;:iit i,::i. 'ltlvQt * tlai tir sd -i.i6r sirl, ri.:iL j;:11! iitr iirit,l l5.rr ki6n ti it :tltitt rrlr rg/ridrn X z- 1. ihr!,' j,.r 'll'.i:rl; .: i.,1. fir.ii. Liil "{ ;.i ]!;r, ;rirlrunr,. trinl: iritr lr& ii' l,',p irl ilti: '!.J niir:r). i.:, (ii' ii:r.ii t( iiilji f i' il$c vi: dliu ,:rj:r-'ir,:i.: iirir'c l::ri: iltii::. iir,.irS cliuiIlrll (11 -l- 11.,''; -2r"1-{.r' - l.i) t) * {r. ( tt'irrl: rilrn{ c:r,r i,r ,':' ii;: r l:c iri i:r..' i:, r'31 ding. c6 .-,-i'r,,:r cii* : (,io s.i.lh rrdi s'i vr'ri ;,'';) liir ia eln :i,'rc ,"iinir 3' r1E (1) r:i.r ii nh:it nr{t nghi6nr cira tan tlirc b6c l:a,i;0. n!liri!\rn r ); t. \:i ii tl6. ni:r.,iig r;iirr ilii-n ilil-r. & niriil-u iru,ri'rgr 'ii;rr U= - l, l!': i il) lrii Ii,,'rrrh - 2u"..'l*{r. hoe siirir rhrgc h 0, ti','c ii r:6u y )> -- !, thi phuor]$ - c6 ci, irai nElri'r)m:tr#n lior #, I trinh c6 it nh{t ,r:r6t r:ghi{m r:iunng (r'i n0u eh - cd it *h{t rif ngi:i,.:m lut, }ir-;n f;.i l:ai nghifrn { n t,iri S rr;: - co.cl hai ;rgi-:ii:;r lrit: ho;r "" r,it irhir ilirn Fi -( c6 ii.,nir{t r:r-\i i:11}:ii:;-,t liiti i;l,n cs i'a nhii il)';;;,,.S vi ciir riiic !;r:, t l,irii i'-:en iib r-Jt lr- ,, '... e m5'i Iog"i h:1i lo{D i-iiit. .)::::;;1 circ ilrii-i i,1n iir- ,,_ ;.*-_4 V_-J \2tl. ,. rlt)i-ti6i giii lii; gil;ir rri.: ru biit i;irL.'ing .ti'inir' ' tl+l : thuring phei 1:ip l;ilr;, .;i '..rn..rl;J. ;'it rjrt ldi ni;rr ii, bli t"rn's., sjrr:l: lr::! -,.1 r,ii crLc rrglliim, \ a)' .:i-t r]{u, rrgl!rgllt tiil: hri, }'1r')ng iiii,l nliirr Trli itr{1' riuit t}Ai la rh'i .li,'r-J r':ic l-i' thn; \ri-6t rl illr; cf,l bii i,riur. iti.^:-r iri tifi, hcril c rhi n g pirtrori,; p l; ir ;r
:r =l-f--i- i- t lliuir rlE (,h6 qir:di dnng Gidi. ViEt. phrri;ng , ?.r +-i .? -1 thi ]rii loin iCr'ilrtrg trroug tt-r v,.;ri bii toAn -t 'lr'rlrg vi du 2 Xdt lilrn g: -- JJr l ;n6c6 r'* 1 iiao: h* rn 'ljr**2r*3 -! lt - (f," - r)* t D / llttt s: - vQ'r, n6 r:6 bing bi€in thiEn.- 11 btttl / \ , a, Oil thi tri:rr hinh \e (lru Ilil.t\ l9'n-(, '] -- . , Il, ..:: ti -i-:? ^5 tric Il 3r.'i { .! .\ :, fi drr 5. i , nr trt ,l? i', !:r r a:t g tthet, re r:hi ciin liip h&ur: bi6n thiin viii 0 (.r: ' t;i l'' l- .i I lli :' ( t). fu'bing brin thi6u ur-r- isei cAc g-iir tli cira v'vdi 0 ( is,:< ti, ta sirr: r'a li€t quii piili cit il rthilt rngi ngl:ilrrr r lri'i I inr: nr ) l. .* - '14 !nin - l) -- ls>in *3)(n-4i ffiI B&ffi /a(n-1).-19(5(n*3i ( nlo [ri #*rrrc
I .. . :, f*+c=o {^{ '['rr I h5'r Nhu vi:', db tirn r:gtrigrl ngu-1'6n cira phuo*g r'5 ryr a t4++ s Yz{' - trinh, ta plilin lir:h 1964zu ilrunh ba srj ngtrytl; :189 (rrrotl 1Utr0) 172 lrorre ii6 , ,i hlti s,i hiin! nlritr vi' eo tirrh t h$li ' '' ' ' 16 khlcr'sd ttrLr 3, Vi khi ptr*n tieh 19ti4 ra thriu 1;{ * b21. (motl 1000) , 4'[! ++ ' etu 4,. i;8 ,-.t41 irriorl 1000) ;. 1904 = 1.2.491 1?16 481 (lnott lt-too) s fu, = Lr.t f P-t ' n6rrchi c6 tlrb phAn lich 19643u th6u mirr 36ir = ,\ )72o , 6ol (rrrorJ i000)' cau lrCn Ihco ciLc clc]r siru rrFn: l7l1l0h j i (rrr,,d iflt:0) .' 1r\ fi) r-Fu = zto.4ot* 2m 1734 - . Itl.(rrlocl tC00) {F +t= l2o - Jg (2) "*i . n? = 0,. 1..,, Jll 'l rr {'1) .r :r (2 ) su v lr l nn ; - t : (:i n: t-l "y+ -'.2'" -4'.t1"' 2000 elfri .. l=19i20-2- *l m = 0. !....,10 I 984 .-'1i'- ( (3i i s+u J4$ l'o .tg8;*"" ' -121 (moit 10{JU). er,r" ,. .' t, ( r - g+1:4'r120-!nr.24c tlrc la uir dlltt so lau {'ltng \:uJ " .,- do ln. 121. ---/ f s(J A t)tft n1:flr: {,.,.,i 0 N.D.V o. f, .., ( r*u=-491n' ---1 \4/ 4e I .,,-r* I =lgl20-2t.240 BAi'4/133. Clto. n s6 tt! rhr'dn oi, a2,.,.: a.n, rl1' -0,,1,...,.i0 (irii c{c hg plrLrorrg lr!rrli lr'0n (la r6ug-vilc l) ail:. 19Ei ed:i,ntoa ii1,2...., rr; q ua dun giirr) t:r si' ' rluoc J4 rrghigrrr ngu.r ilrr l) BSCffN (rit, a;) ]> 1(i*; t'ai net i, j.x 1,.,.,n tltro'ng cira phu'ong trinh tl6 cho,, .. va tf I,' J P.(t G. d .," Chb:ng railfi rdng : ' BAi ;ii I33. ?im lla cirii sd tQn aing'efi.a sd , I.bt et/lt,f'a tle}' r'irrg, ndu .'ti rrgu-v€r e'thda ' ; I t985 ) 1u.:ij/(nr + l), ,i do nr Ia s6 nrarr l'Jb3. rrr b n -rnt t,; ,rr,ieri'"thi co ch;rn 'quiL boi .sd ciraj,o Ii.: o, 2a,..., aru lih0ng vutlt 1983. Vi viy n€u' k-r' hiqrr /r1 ii ;6 .crc si, llorrg ti'p a * loi, t1'83 Lai B{t!i.(cfia ,Vguyin Tiin Dilng, Ao10 DI{TH dB, ... .rn I t iro,rg vu,.,i irtt& 1983/j rir ltin hon,' Hd NQi). \'il. 1985 la. rndl. sd l6 v* chiu }6t uLt.r isas/f j-+ 1), vdi:-; = x,2; .;. 1s82/ ti;i trrlri d6 tLeo 5 nerr tr6rr ta c6 i j,. ,: 2000 -. - : i'I * 2fte"1- .., -{- jAi , {. '/ + .-' 1 986 li sfr c'ir.c sai la Lr6i cira it nhi I rndl s6, lliuoc tf p. 5(2ir'l t)(k € JY). : hgp A rrra kh6ng vrrol rltri 1C83. \lrirrrg do diEr-r' Suv ra : I
,t.:..,-lt .. i r,iL chir l' kr * l:r * -. * lrj [ri (,nr rili Iy' (*)l ( [rr {r)l x ta c6: q r: l/ioI l.r.-rl + l/'r-r)lx ol ur) 19S3 l + I:t'-;c *"4,. l BA1" gif i* rhirlg nr.infu r: d gS:" 'lilrlrc r[1- t...-..".-..-i sE gi&siir cri p sii thu6c.{ * } ct,....rrn lr-i i i,:.,..,99.1t lzl ,^t Kl:i tlo si cu p btir sii r-ri:a eire sd nay thugc -r iJ.' .-rl , f J -.1 ,i:-1'
&;t:.:' T: jj1s=ffi] ::i. _ri*t!i- :1 L ':.:.r [' : I tr i I i Tti P (0) ='3F (l)--- 38 {2) * P (3) ta e6 Phfln ticJ: lroiil toi-n lnoirg tq ia ciing c6: I 3 P(1)-P (o)+ 3P {:i : P (s) = 3 * c x16 . r) A P r (sa Bu .b Sec}r) . (gl 3*cr * r-.2n-3 .: -)a-p(t):a* ilc , .;- C0ng (1), (2), (3) r,d'theo vd ta dunc: Nhrt v{,1' vrii 1 ( {.3, n ta lcri: . .s * p{3) I * lo * ll3.-lt^3* i tr A. EC+ ${5.'C.{ + yC >4S.166, AI} ding thirc x&v ra.lilri vr.r chi khi If Ii tnrc t{lr a3 ,'l t _ o ri,i = ts -..i*q r?= r / "t . r,r^_31 cfia .taur ginc dIJ{1. Yn1 dibu:.cLn tim },I li lrqc \ la - /, (I)l - J-i:-*.- \/1 , ltinr cria 1am giAc .dBC1 t I i) t [r'PiO1! =1r::i .I 1. b) 'l'arn gi6c Afif, ("1) goe tri 14i .4. i I(hi d6"tia ihilc P {c) r,rl lirn e''i c6c i$ rrr'i 4 d.8,C Ia n gh if ,rr ciia lr,' : (,lx + 3 B + (. : 6o? - i2,r *3 {l-l+zB+c -2trE--larl { Cao-I ( r;-r'* t ++ { tl B * 5o= - 'r,rZ 'i- {24 *'l (tt i-:it:u-:r,*-iu*{ {t=,, d tlirrh 'i '' +- {E ll = (- {i,ir 'i ?ri r !tl,;r; (:i) -,tt:' giri: viri .1C viL 3ll'= '1jJ' Gie .l I tn - irr" - 11),'(j. 'Kil vutllg I = A vi .${ r:iim troitil ianr giAc '481C sir 31'khitL: Yly cla;Lhirc I'(:r) = cr2 + Br -i- C t'ti A' ii,C (ndu irhringlh€ ta kii ai;' -[* dl]' r'rc' : c(l li clr{'irg x{c djnh bi}i (3) sd kit.u ciro trii rrir ti.c s'..r ti'nlg n-rinh lrtctrg tr.l'.)' (*,r cleu'nlrr'r !rurr l. D.I}, K, ' -.'- -;='. "/_\ + MB }-, J16' cs''si.isgi'-+ cB > cB" va Yn5,-ll;t i){: 'l $ll}. {.1 4- i,tc . AB> MA"ts'C '. -l. . Bai 7/133. Cfto inrn tlfu ABC, ?"im di?m ,lf ri , lrong larn giat sao iJio &J.{,r'j{,- + rUJ.iC + MC'-'til AC.;dE'+ tli:.ili3' i 'u,r.'. l4;l' lr'Lruug irqp a) ilqt gtd b! nhi nhdt. ffA.J.l,C +" Hr.. AC + ff{) .,{IJ, > Lo't gidt (eiia Yd Iiia ?-tttii-- irrl: 1)E, PT(-:1'' 4 S.\B-C : 2.:',/l' . :lC - 2'18 ' AC' Qur \hori). Iiv ireng tlul;ilg hrril ri:iv 6|i|111 cir;r 1in: p'rll '11 1a a) Tam giiic ABC c6 l:a aoc tiltort h.q-v la itiu ,iiiil, .+ riiLL g itA( RtI 't t:Ii) Tu'gii thidt suY. ra' ' hay tr{:t . $C> 2 (SlnM + s-rCrr) (1) Y a1 = (b + a .", q) 5in(,.112)
I -! a t) d'= n /l - SlSr Liti Ei&i.. 1) Tr c,r | ' l: '::'. 51= 3sBr. -6rCr. {,'rAr/(4 ft)r ' S =;iB . 8r:. Cll(4R), si ArBr BtCt ctJt I AI} BL CA D6 rhd-v l:ngt u! A,ElAil Uinh 5 ArBt iBr le+ ---_.- = :- a* Tuong trr ta ctng e6 i h,t) t)-, ni-=(h*c-a)-s in2 (..fl:) AB IA' 1" , =(b+c-a)1tt- cos lli2 BTCT ICl CLAT IAt ,:. r | ,/ b2+cz ^\ BC, IB CA IC =_!i = -.(b+c-o)"t 1----------- I Zbc I -:5--*, \Iiry la cti .I slis = Ittl te. IEtlIB. ICtitC 1bc ^ + /\Cllz: ;/_\ 2) fa c6 ICBI * G BrlC + IBL = BtC. I =_,__ lbz_(o _ _6 _ orr I I-ai cti LQnrct can va ,t,{tri}:G tbc "a)l_I*: . ,/-\ /.\ //__\ X6t 41.41C c6 CIA| : (A +.C)/2 = A1CI. llAt t,it'lr lroan lorlrr iir.urrg li.r I;.r (.() : ' V+y AtAtC cfin, l6n l-rF= ? .), -'=+ A "o 4 BtcloBt = It: !IAt b; ( o,'/ 1^ , j .- -:- D,r/ t. t'AtC,.ut L',stot; +) Idrllii= IClBet: - ICIIBt (2) hlri do: Chir -i' Arrlc ut: fslr{B . ( , I, * \cl/ { }-), \t,1 i [-:- \c;,/ \' ,,,, il:t4L V ').2 z al bl cl .'I'ti (2) va {il) ia TCIIBT=IBIICT c6 (3) : I B.tC z I.Bt.ICie /e. r-41 - (t.rriZtt2 ca -_- rq - ,ro ". l! /--;T? -_.-_-.: l') : n! IA. rB. ICi\tAt. IBt. IC|= fit6/5r (ttreo- U i&c/4) (ar! t) (abl4\ . phdn l) , I.9.7' *. Sai 9i t33 i'pe1;"9 tam gidc ABC, cde dtbnq vi L4 .IAt:fi2 -1d2. ncn Rz - a2 = ,12. sls{ phfrn q:dc cdt nhau tai I ud t'dt du'b ng trdn ngloai o :,n /1 - s/s idpcm). tilp L.AB:: lb.n last tci ,{1, ltt, C1 Got,S, g1 p6i T ;\ lreln ret : b4n l{gugdn 7 hanh Qudn ( l2P, trudng dttr*ngtng lddidn tich A.a.BC" diCntich AdrEirCr t;l'f il vua hgc vua ldln Thri Eric, TP Hb Chi uA khod.rig cdch tdi t d?n tdm drrlng {rdn ngo{i Minh) c6 l,ri *iai tdt. f rap a,rsc r.v,T.. rdac : ,/ :.. t'Br ICt Sni t0llSlj hinh chcip tr': gitic d|u S,dBCD. C^tro tt) IC Trin cqnh SC tdg dibm Ct. ' Cltring ll inh-rit,g trong.cqe tht'it di|n {qo fftdnrt 'l:c.' d.a m(t pft8ng dt qua ,.tCt.'cdt .sB, SI,, liti trtitl. drfn song sarig udi BD'td lhtAt d$n.nki nhdt. 7'inh tli€n tich thllt dt|n nhi nhd,t il6 n6u_c0nll clitg ltinh cltrip bdirg 1,, cctnh be n bdng '/5- ,* (,;,- |* :i/S7:j. Ldr qir.i. Goi .8i rir Ot tuong ung li giao cria phing cli qua.dC r.di .SB vi .tD. SfI ld dudag nrq,t cao c,ia hinh ch6p la f/t Ia giao cria , S'f;l'vdi thict c1iQn. Dc chri'ng rnintr atutc ndu lfiitit'di+n
t, song song .vai,bfi thi tallr gi6c '.SBtI]a cd diQn tich nh6 nh{l, vi. do tl6 ch6p S.{,B1Clla'c6 tht tlch nlr6 thdt. Trong khi d6,duirdg ca
WT ,-T:;,rj.:-* .r{i{:::."S-:, i::tF':::!= -,.,'-.. . .-.f: . !} H0C SI\t':rr Tirt II'rr 'nAr rrr'a E&i 1{i/136, {.,$o larrr giiic $lt{"' e6 c6'c c-4rrh a, }, c' i-hirlg rninlt' q'j-r* aoc ( :--v1t ----\' -{ AIJC \ ,'r.,lg.t i coiu8 * rotgC I tc__tg "2 2 2 \- ' l, ' Ld Van Nirr Hfni .:. (13E chu5'0n Toin Virrlr Phii) Tim hi6'u s$u t}lOm to6u h9* Xthb thdrrg Pfi.rq* .$E{tI* E*'ffIflU KIEN T RONC LIJSNG GIAC N0uYf;N Bf,uc 'ruuiu , Td {,-,i drir,,r iJriie c('t {ti|u liirt lron'i tuEng gidc Du r1o, ta co: E ll nl;ii,:r1 rlinr: iL*c g..iirn titc hirtrr .sd lugng s,inl, = sinu/nr, c,tsb : .ntosa,/m, giac citrce tlt,,:r lirirl rr' j tri 'L ''ii rrlrir*q 5i'' t.r'i t'lrra ' sin2l., + ,ror?t, = siu2a/rnZ *n?""5?orlml. nhdn ilttce r:,ia i15i sJ tiiing thcii ehin tliilu hiQn ring bu6c lio iio. l-"1::i:rg i::3t, cot'gcr + 1-oJil&-j" S u-r' t'a '" cotgc : cotgat:otg&colgr r-oi a * b * o*t0o' t'a ort: 1 * ao.?L + I,Ze.rs?o thuo.ug g{,p nhiing dang- bii tip saLt : m? * 1: (rtt* 1) .cs2a. , 1. Chq f,-tr{.io "*ir ttin hd g:rr'.r r?'ii s4 itd r7')i hut cos?, * (,rz - l/: inl -t)' phdixtic iinft srr tigi li| giffa "fre.m si itt')i1t gidtc Do d6, mrit tt'otrg .circ giic 1ri cila cosa iA cfra cac d ni s6 do /;'z: 1): ;F - Dt. 2. Cito trnrre' i.r:6{ sr lida Jrd.gi$a h&ni sd !it:;riE gidc eria ddi .($ ,ri rioi ltii lirr: n&&riE str iidn ftii r''i;'rc Yi. da 3' Thinh lip' he thirt'giira ba goc irlrrln |i,rn na* sd irt'1r*g gtitr: tia t'rit tti ettc tlfri sd 't6- a, ir, e bi6t l'i,rg: eu.lgo ccrtgb * cotga cotgc * : eolgb loioc = 1, 3. Cho lrurtt sry iidrr if t'i$c i:iinl s'# irr-c,rg gi'{c Gtd.,i.: lihrln ea1 hai r'6 ,1&ng thut dE cho vfi crln ddj sri un li*i sr,r lictl i:9 gir?* cili;r.1 r:h&,:6 tl6l sotlit u.u... tgtrlgbtgc =F 0, ta eo: l'i dtt 7.ilhrir,g tittnl, tit-"i ndu ( i * r'l=;i I ihi tge $ igl + tgri = tga tgb tgc (t + rge) (r 1 tga) = :. ''.. 4 igc + tgl, stga (tgb tgc - 1) - * -tga =(tgcltgb): (1 itg&tgi) ' Cidt. Eb cho lS tr{i ciia d3.lg thirc cin chirrrg '= + tg (-0) : tg (l-, * c) minh c6 irghia ihi :,, I; * fii: + E3t. \'i { a * b) =+ a * b * c = l;Jt. Nhung vi tr, b, clliSgr'rc n1t9u n6ry0 { e* b { c tg (a,* b) = I "'* (igti {- tg})/(1-tEaigb) = 1 { errl:, vi ]a c6 c * b "l- ' '* ig# -1- tgt/ :l- tgrrtg}$= 1 "::, _ .^ Sau dfiv,, lir rrF-u rir. nidt stI i'i dr1 r'B mQt lodi + {1 _r ts{l) {i * lgc} :2. d&ng ttrrlc ilii:u kiQo (I:Lc bi6t irc)ng lirr.)ng,gi6c' Yi dt.r,2. Chir;rg nrirrlr liirrg rli;u sitra = tnsittlr i'I'a'goi cho gon loai tlirug thuc dd Lit.drt-nii thizc vir tga: n tgO fiii m$t gli,tli cira cosc s€ iii *on|' tum gtic, D6 li nhltng tlang thuc gilta ciic "l/lun: phen t& ho{e efic hdrrt -*.6 lngng girie cira g6c = 1r: (n - r) irong tam gi6c,:' Chia yd vtri ve c[{c ding tiruc. dh cho , Yi de 4- ilrir,,g minh i{ug' trdtt A' B'',C 11 ,ba ' c6c trons ntdl Iilru giire tlroa urAn di'Cu liiQn \ i:.. 'ta c6. cosd = {m/n) coso. 3l;o);;:i" ia;, I "i''t.i:i=r)iir,,r i*s = c = 6uo \t0 \ \
&Ar. Ta cd tu d&ug thirc dd chol: Ta bjdn ribi srn.{ g srnlt J- sinC 1l A-B rt+B\ = sin-l * siu* f s:n (rt + Il) .2 \ -tcos 2' -cos*_-_-lX . 2 t : ,Lrrr .,, + 1, ,. .. ,--- -B .1 I :, 1,i., J x Sln *=-l 28 '',\ 'L B -i.'jnl- I .o. e-a - sin c\ sin C1, .? \ , -J 24 -& -t B n D{t t = sin (C/2), ra c6 ptr"uong triah bge hai d6i voi l: _g dB ,'1 .2 I -COS 'l* "-'.. = 0 2 ,4 'l'i bi6n _rl,ii sirr 2.{ i si.n ls -{- iin 2a; ^ r A.-B l. ,=2sinlccs-t ,+ 2s;n($+{jcos(g - C) i '2 = fs.rr^,1lrrs.i l. f .ir:-ll.os;i.,..C) i_ ,d I a.-s * 2sin.1 [co,,,t -i- crrs{Ji - C)} t vi cos l- (rnenA{0, : Iisit.4[ : r:rsi s -.1_{.,i-j:,:.,s{g *..c)] l2 : 2sin,.t . ]s :r i:j.s {- = .isi l.,i si n Bs i illl' ; ! r': rr6 cho ptruong trinh Li: lrri c6 n;Jiriim ihi ,7 ii Jl A_ A e0 tric ld cos _*' A_ D -B- : 1 {cos__+_1 i:1::: 2 2 l: {.' '.I"ndu bing - I thi A-.8.= lS.Eriiu nir;,'hLi:ng ' slE --"- t:|:i --""- thb c6, vi ,4 vir B ld go, eria nrdt lrrn giac;. 2, Khi do, La cbtt2 - t + l/.tr = 0 ruc (r*t/l)e = 0, Do do'i!) 'ruotg rlrrorig rii i , v+y :' I = sin tCl2) = tl2 /1 4l.t Jt /,1. . ,l 15 r;- a1'., "._ 4:-ts (: \, - :2t'' -- vi co-s --J_ : 1 n€n ;l*j? 0" : 2, .: ){ * ii;iri -.1- gi;: --B rilr ..- c\ f i, 2l f: g. iri'c ,1=3, \ : '.2 Nh* vAy, inm giAc -t$r) la t*rir giic cir: I'i c6 '3/jC j-f- m0t g6c bing-lE13 n6h no la tam gi4, .li.r. Ta iiiri;,, r'i:rg .1 iros-i ..: Llrs 0 vi ::':1,1 Vi d,l 5. Chung minir rXng, mdt tam gi;Lc.4BC li.,d}u khi r'4 chi i
-i : : f I :::,:. l'.-.4 t:, :.i:..4.1:::! :: !!1's Gldi. Ta goi ba canh cirn tanr giic lA a. a*d, ,2) .o.8,r..* ,'or2b.* cos?c a*?d. (rl D ul t n gdc ddi di(rr r:rt i canh (a.Fri) ' = 2i,os(rcosb""i". l{ B. 'lirco d!nh lf huur sd t.osinr. ta e6 : Z. ihri'ng' rni{rh- r,ing .n6u tga e f ig} thi siu(a*6) = 3sin(n*b). . (oiale =,o1 + (o"+ :a)z .-. ta(o * 1d)cosr ".. 3. Bi0t ring sin(4u * Ir) : ?sinb,. cor, * 0, ,2 +z *d, + d2 :o2 + o. + Jdz + 4ad.-- 2(o ?+.:ad)co sE gos(a-t-D) * O, i,tu tirrr srr ti..ri h-o giirri tg(a1b) t, -" '" va lga. ._ t.'rr**rid*3ds oz +2od+idl ,, . coso*cosb Vi ,2 at2rd } 1 nerr co eb U2 thi cd mot gi.1 tr'1 cir;r'ig(.r.12) ld tj(di:t,)rg(bi2). 5.'Thinh l4p c{e hq rhuc git, lru gr:c nhon - . ,d6: cosB )lcos0t)o. f"uf,.o e ( Otto. Do a, .b, c, b;6t r{ng, i l-i tltt 7. Cfing tninh r.dng, d6i vrri inOt tanr 1) sirr2a*siird5 J 11;26 ?sin'asinisiq:re, giirc, ndu tbng cfc nghlch dio sd. do r.i'ra hai \l = ' 2) cairh blng nghich dio s6 ilo drrcrng pl.r{n'-gi6c *D) sinc cosc. tg(ri = cfia g6c r'en^giira hai canh trOn thi.gle xen g-ita 6. Tipr 'srr lii'u lr0 girirt c'irc gric ,'ti l ii b, bi6t ! t'ang sin{rr - b) - sin?o -siu2l,. 7. Chung minh ring: t:rnr gi6c .{8L' la ein; khi ta c6 rliQt'trong cic..hQ thirc sau d?r-r,: 1i sirrC/sirrB = 2cos,l. l) 2uin,4sin 8/sinC = t'o tg(C/l). B) sin (.{/2)co.sir/2):'rrntrl2)cos3(tizr. . r.8. Chtrug milh ring {anr gi6c .lBC ln'iuOng l
\ r" tu$r}'t Lffi$- ffiAg T*-:AN S{tfxiI 6gt'tt PHAM :'NU NA Lrio r1l'r.,. gi6:i t!rii'a v6i brin cloc rndt phu'rruc Iiitl,ng dicrr"tich ciu 9800 Iirr curr i.r5 ng gS00.7f .1r lffio' phirp cri icir ciro rri;ri uLiiiu ],ii toin r irl:c :981)$3f (..rn*) nlr6 hnl disq tich cii:, hiuh rron Itinh hoc ihrrr)rrq iltrr.rc r.o i li lilritrt rniu linn kinh Ugcrrr (=gSitlgfutnr). Dr, 11, 1ire,t ngurCn mrc v{ hav g;i1r llorq cic li.\' thi ciru1.€u to*n lli 1 phiri tdn tai di?inr I nirnr tlong hinh trdn I r"i. hoi: siuii liiri. iliru.kinh 99cm r.il nim ngo:ii nroi lAn ei.n. R6 l Trirrrt tiOu chrhlg ta hir,y linr quen -vrii rri6t rlng hinL trdn tirn r ]rrln Jilnh lr.m hi lrinil tr0n I kliii riiirn mlri : riin e&n hdn Linft r' ciia mOt ftinjr I phfif1 Id idp hgp cac rtilnr ,fr6i.r ar4f pftrlng c
I r$l - ;rd lhrrg dinn liclr lhir cin cfra I00 rla qiic sE.b6 han IOtr .4f.r- lrli,jf ',.rlrr) .( tOC. +, "S,: = t:gO . < i'lr0 : i1s'{e 1r,) li tii*rrrtiih hinh r,-uOng. ftrec, ngulin Ij 1 :ririi i',', rl;i),:r'1 ii*;ii tror:!.trini r;uOngl C4nlt llt,.'nt t'r, ;l:: ,,r ;tl;:'a, i.C iC l:i;: gg1.1", .), LIO Cliitlt It t;fux eri:: hi:ri trriri pi:iri tim. Erii io;ft 3, i -,,,;; Irir,:.rr r!{,, nlii.lt kiclt thrcoc .tAy.l{} c$ i3? d::r,ir iiii,";g dd sdI .?. Cilrirgr";ninh r&ng ha,s gJn r:,iri*;r- iiri: &,ti,3.€ 2 d:r.Em n&nt trin 2 rtuun l:iiti,- itJt:tr, ,..,':,,,i1;4 ytii.lt i.!.iir;tq uttot qui l. a;i.ir.,\{it ii:ic liir: c:in J_:ri;r lqinl..i 1lt ifra nr6i filnli 3 ([olt tl ll']:Lll'l ! | l"'1), r-ir .-i:Eil'{i ddi sirng nhau grra Az, d,o t}ri diQu iich . .r4-**_**f\* 1,1 ir r..,r; ,"'rr i
r-6rn= r F.rt ': k1 . . -*: : ?rin'D*y l{inh. Vd Thnnh Nam, (tte); Nguy6n I{oa Rilrh), Nguy6n.Duy Quy.Ot (88 Blch Thugng. HoAnh li.rdng (12I); (Trung Yu'qng, Qui .\hon). Duy fit1n. LL\ narn Niuh). L€ Thanh Ht, Tiuong I-8 Thif.n T'lrsnh,(10C), l{guyEn AEh ll\i:in {ttCl), Vin Cirnls. 'rl'rinh, Ydn fii*u, FC Tro*g Vinh, (11 (An Nhcvn,l). l'hau Dri'e lliro (li lJ An Nhr"rn 2i. $T' l.am':itin, T,ir*nh }kit), Ng,r-r'r1 ir ViQt Anh Tlinh Diro (ltri€ir, LB Ouy Can, (r\n Nhcrn). tr{uirnh tio cr i:han tl1)i,(t eol, lliry6n Th6ng'l'rLing- (1i"{ IIi?u Ptrrirl ( li.\[ (.)uxng Tlulrg. Q'ri Nh,vn). Nqttltrn Ngu3'In \iirn 'l'rii, C"en LSc), (Xgi,g 'i'ir:h). I".0 'h.t iioring.TLrdn (11119 .l rnrr Qiiilc Tuirt,.QLrfrng trgli)' Lri . (lZC I' Quii,' ho,, l{rri), }iqtrri'n ilr}n (12:\ })ai Ngu;6n I{inh Son' (1tC I}hri }Ii' 1)' Pl'"snr lluv i,i'c, QuJrn,l N,rnr lli \iug). -"'qrilI'.rr "linr An 'l oin {SE LA Ltri; Qtri Ihtrn). \'ir 'l':in }'hdt' {t2I}, l-rung lrr{n ii}r.4.i (1?H}" (Phtn tlti. Cheu' 'i'::in i3A -Phan 'iirn (Jilfl Pleiku {lflf' r;ui }ihrru). NgLr-r'ln t}irc .(.li,Su .Tiridi)' ii ti i'huot:g 1)' phanr).. Ngrr.r,6n Birir ljg1rc i(hlil {{}B 116i Thrrong 2, flri fi.lrdnh: \gu1 6n \lilh \uftn, Trrin Xrifi.n (118 fleiliLr), ((iia l-ri K0rg 'l-um). Flr:lrl *inh l'larn \li{ru. Dii l)ul liiriinh' Ilu.i lh Vln 'l'hinh,.(lOC't l; Dic XIin), Bt'r i 't'lr il*a {12\i I}Lrr-irt }la T'hu{t)' ()iqtvFn \:ir 1&i, }iiia '{J}i.. li"). Ph4m '{)}ri Tliiliin 'I (ts i.'L), --l'riin 'Iif n \guy[n i)uc i{ir1' t It:C,i) ; :Bia,t ti':*, Ildng Nai), frangi. Dir:g'l rrrrrg 1''trq ( tiil Iirri'nlr'i'hiir: HLl ng' 1'h:\iih itittr, {Xgo Qrrt'd'n, Va n N in.h). Lhur l)rii ii0 R Thii tr)iiir .lr{c)i}, 1,0 Ii6nr;'l Son Cdc ttinh .(lrdc: \gtr-r.in }finh f irrin (iC CT), (tO,fs Ti{ (t*r ini rli, {Song R{}: Lt}: Ngiit' i)ng Phqn\ Cto I{hirn {i2 C'l'), ( i'hrii Plritln. 1l:ii Plrr)ng)' ti; :;rl'fr'r.q Nintr). L0 Xlir'!itrrlng' i'Jgu3'€n Nguy6r Viru Phdrc'(12 A .l'iou Yin, Quirrg liitlr)' . \qoc An, li{iir-il IIi "l'hlrnir, ill CT I;oir("X-tr}-An' i', i:litrr 'l irr:, ili) ;)'ic Tl'ri' {10'\2) Piram tr{anh Crrtingl (11 A.\ti+l ll'ri), Dii ilti Iiirsnir .11 ti;ieng). iq',,', 'r!ra't D,),' i;i1lir I\i.'r. Cal i,:1,v. 'l'iilr: i;iairli- lig-Lrfin iii-Cr rr,",g ,itfo,isin {vinti i'iii'l- }r'grt5-0. N1o:tir (i iii-r', iliiri 'i'::':i' 1'rLrin11 dlinir Can (Ji tli't,'l'lr*n Di llgii (1?'I)i {Ngd Si Lien' Yr-Ln 11:' 'l 11& 'll:r.u ti iang)' tli Bir:). Lp.u16n Vd.n Cieng (11 E [6ng ngirif p ?' ThOr:g i 12'Ciin ho Bgn dgc. ritn tdi ? rvE*T M# #rqm F$B€# B&i to6n'6i 1ti6 t:irrr lliro 'lt'irn lroc lI irii)i E,ii ciiiiig lirirr: i:iitt chirirg mitlil trrol,q ilr"rring vtri tr6 li: I n --;---- \ , -=-:*-***- (.hring ntinh ring vui ntoi si'l tr.r nhi€n- a ln c6 ,'- t.t 1/ li --'-- I i n , U (ot + l.'{nX * !i...!rrr l l) I G-.F-iil >r+ il;i { lj Ll113 1O1 glal cfrrl 'l'rin Dirl'' Li&n -- toA.f:n'ltt ItI Ndi, cldng !r0l blio itl9, t0i tir.n thAm. tiu
1.:.rt:11.r., '. 1 -=.:"F' '-,.il R r :' e _.' . .i i >..i ,,. . t, r rI r-qng.{#') i ttreo vE trt drrg,:. li6f quir ' *.., li) i:1"2,.,.n. Sau i16 ta cQng theo v€ vl {*# t, cdli c.htlg minh. ' i bidn dbi'tuong drorlg, ta drgc b{t tl&ng thric . '- fay gii;: tr .b{t, d&ng thirc (:) ndu thay sd (5) vA'crng 1{ tj{t ding thilc tbnglqu{t nhf,ti 1 bing s,: b bf,r i,j, {rr } u) ta th{,v bdt tting t}iug .n n vil dirng,, l','l'1 l:-, ro: n. n n IIai* l! n . i=.1 a l-l -U nl V.n.(u1 *b)'>b+ v i-II I'arj (:) .nD .1.----t- l:! -1/ ,, r t.l fl ,rrr+...+ l/V fI ri. (5) n li * . (Chriri!,- urirrh hctir& toirn tuorrg trr). i:l i=I ' Bo-" gicr niirr r"r lej tr;ti ciia (*) vri-(*o)la thav Btr gio tu rlXl.: . 1 bing it, bg,'..$n,.li )- u. tr cti cric.bf,t dirig lh&cr cO'si rorrg Iron.. nn,n n !; * .,,. "Iib,=yr..,., /4 = -1r er".. 4e_ _::--.:-- . vi bi€n bd't dfing thuc (5) r'd tlang V (arlbr).. {rrnibn) n,7\.-; t n Dlr .n,,i tgi n ...rijt .t: - , rt {-Dt an*tbo I t.+ I \ n n \i' ln tr j ft .r'i + fl gi + ...* rl :i l. t0l t/./ \i=,1 i=l f=l i * \6u D : 2 thi ta ilusc b{l ding thtre Brrn- . n brac() l)xKr: itl l/ r...-l' ----lir-_ j(+r:r.+*) t , .r r\l t 'r r\ \..-l n\arlLr an+ b; / (,i * s;+...*=il\ *i +. a;+...+';1., lo (-dng (**+ ) r'i, (+**o) i Ireo vd r :r bi6n .}bi, ta duqc ) (ilrz +'vttiz + ..', { zs2)z bf,t tlins thirc t + Ndu n : 3 thi ta rlugr: / J 3 B\,/ 3 3 :l\ /rrr + oi; - ,* + 0,1,2 ' y'rr..r, +1/t,r.-b, (o, * 9.r +...r rtl|, *Z- Uz-:..+ rzJ Btv' giti ndu db y : , ,.1 s 3 B\ x [ 9t,, f 9., -l ...---.,J - ___u _ \- ) r .)l ,3 a-to-f...+t )> (r1 1113 +. U4U2.J3 'r ..' i :l:2:3)- i: ..,I;,', ' ,l rlrg tlrrir.nr& r',int ciru bii s6 5ll2$ trong I *1 } 1o I.iir l,r.rcl .,,i titbi trG sri I nirn I983' ', a+b+...+I, : {11 yi6t n'I.,lil c1&ng: thr'rt: riang (4**) nll t: tt g rlt ,. )i .t Y.'--\ viEl' QUA\G :.: "; d$.1- ai'r&i tirrvf,h?qg lbng ri sri hqing i'ttt': bi. : {.ic.0 $il1'!t tId ;-r$i) \ . cAil s,}.N rlL' T;lI cIiI 'l'o.A.N cF{fi ,1, , e' ,.: ' . ,j.;b.c6 i1l.r 6ii tir6ag Iin r;0:cric !rrn r]r.:. tiii 'fda so+ri cl] n''hi mi,i ban di giri b{ii du thi girl , .l oa scett .{c'lllr'\tlq- tiir .r:irt Ca}- : I .i: :" vi: , " 1. IIo vri t0p Itirini: ihiir t.riou$ grLIv.tr:i 1r. - trtrdrrg .._ r':, btit dalli ghi trong bei giii du thi, '. rrlrt\ . n6ll c6). ' d '! ?. -\,'*t, :. lltti:,;. lir:r: :ir.li .: ;,3:.\rri:r|la..,.iiii.--'.,^.-; ' /1. I.,'rl, 'i lriirn.t, I{ritin.r tr:r:"lr. r'} inir. ihiinlr pirrJ. ' ,''5. Dia r:iri nlrir lii:lg. 1 : ' 1 ,Ta)A SOTN .-i i ,.,d1-!a+**T**:*--: ' I ,.i in, li.iii) l4i'Xi ns,'r:ip irl ?5 I{hng Bii -' Hlil lrii.'Chi.sd: rrrcri t 1288{ Gin : 1dO0 Sd in ?2r l, I Irr xong r n gr]i lutr ehitu th;rng 7/84 t,," ts