k'w
N KHOA HQC
vr$r nru
i.BN KHOA
t ang
Ph6
Didn
t ang
tdp: Nguy6n C&nh
btC.n t|p: Plian Srlc
ToAn
Ghinh
't
'Irr.r sr! i 70 I'rhu' I{ung E6o, Hn NQr
' " D&y u6i : 588 25
tsAo EA Hai TFTANG MST *Y
&t& ITeeffi eeffiffi rffiffiffi'ffiffi p$$ &r $$ Tffit
EDIIUONG phAp rlb thi Ih m.Qt phuong phip rdt
Y cd hiQu h.rc tlB gi&i rnQt loat bAi to6n : c6
nh&ng l:iii to6n cb diEn trong chuong trinh
PT'IIJ vI nhBng l:Ai to6n kh6ng miu mge trong
phln 156" hinh & cAc i6p PT chuy€o toAn.
Cf,e bqn dang hgc & c6c 16p PTTH th0ng thudug
c&n chd f tl6n c6c l:&i to6n cb dihn n€u trong
c6.c rl du 1, 2,3 du6i d6y : chring t6i dlc bigt
chgn nhfrng bAi toAn li€n quan tl6u'viQc biQn
lu{u tharn s6, c6c kib-u to6n uAy thudng g{p
trong dB thi tuy&n v*.o Dg.i hgc vA. Cao il&ng.
Yi d$ l. T'c'i nhirng Sid fri nao cfia m thl
phuong trinh sau ddy c6 nghiQm
.or4..*nrsinlo+280?
GIAi. D{t f = cos?r (0 -( i .={ i; ,n, phuong
trinh trdn tr& thAnh.
P mt{m*2=0' (U
Nhu vgy ta'c[n x{c dinh m db phuong trinh
(ti c6 ii'nfrAlrngt ngh{Qrn { vili 0 ( r ( r.
Kinh ng;hiQm gi&ng dqy ciia t6i cho ha-r, riing
{tgi de sd cic bqn hgc siuh thur}ng gi*i bAi todn
PHAN EI}C CHINT{
tr6n bi.ng c6ch so s5.nb cAc sd 0 t i I v6i eie
nghir)m cria (l), thet te chu kj'. C6 e6ch "giii
trong s6.ng hilh nhu sau.
Ilibn nhi€n- t = 1 khOirg ph&i li rghiQrn crta
(ll. \ri v{-v (l) tuong duong v6.i
rn = (i? + Dl(t* I) =/+ I +s/(r - t),
khi ddbdi tc6n d6 chotr& thiin&. |irr, .r, ttl duiing
th&ng g * m edt db {lrf crio ,rdrn
v:r+1+s/(r-r) (2)
tat it ntrdr mfi dibm udi hod.nh dO 0 < t < r.
Eb gi&i, kh0ng ch.n vE db thi hAm (2), te ehl
viQc ldp b&ng l;i6n thiOn cria nd nhu sau
&J
I
1
q- -
, "rf _
+(f
\"
:'fL+t
s
\iru va.r' llr,ng Lho.r.lk it). ,),.lrArrr (2r ln nghiclr
bi6n, r"i llt-r.rg kliriurru tlo. nri l.i-y rrroi gi*."tri g
=.( - :. 'l'irAlth ihr'r drip sd r:ia l.i*.i toiln li rr
Yi dB 2. l,tli nlrirng'gr'{i trl ndr> eiia a,tht btil
phxongtrirth 2c + X> a (V1-a. -l- 1)
td ttghi{-nt?
6,dt. Eit t : l'{*" (, > 0). tlri r = 1 -- tr, i}flt
'- 2f -r 3.'} u (, + t)"
l'a chi xdt cic l)0, val- nO [u<.ing duong y-6r
(-:r2+srr (i+ t))a, . " !l:
vd l:ii,tcr,iin tlr} thnnir : ade dinft o dE hrim sS
t, * (-zjs -i- 3)l it + l) = - z, +- 2 41/(r + r) (3)
cd'pjr4n dA lftj ri'n.q pdi t ,;, t) nilm trdn dtbag
d$a '"'$o rl5, ta r'* rlirge .iii rhi
v \r
NghiEn e*u il6 thi vir c5.c tiArn cfin xier crla
n6, tathdyringyOu ciu eria l:iri.to6n rluqc th6a
rnEn ntsu rlu&*g th&ng l) n&n'r trolg miEn chf,nr (hinh
ve), girli hqn b&i circ dnring thfrrg'g = l/2.r'l g:
2* +- l!2, vi1.v r:rii:,gilJ tr-i i;i:iri. tirn rfia m ii
u{m(2. '
Sau iIAy. li hai vi dq nti'r.r cho c6c b4n chuv6n
to 6,n.
Yi dq 4, a, b, c, h td, b6a s6 tfrorrg ' el'ra trtse,
x, g, z td ba sd thge tkay .dbt, rdng budc b&i dVa
kie n
at*'bg**ra&
Tim gld iri nftd rthdl crin
q l-
5=a}h'-t-,r'Ib1l
(Bni foin ni;r tiy sirrh ra tii ntiri irii tr.rirn
hinh lroc ciia l*.e giir tle drir:g trot.rg liii TI1 ri
'l' l' , sti 5 r {i- 19311).
Gidi. 'l'r0n ni{t ph:ing tqq dQ OXY, xet cfic
dibm
.{ {a,t; err), B (({i + }) }I o.t *}!),
C ((c * b t r:)h; u.r *bg * cr,r.
{"ldi th3,th6-r ,rga;r. riing
or = a d,lfi-F. dB: b/;-rilq
uC = * i/;'' +;1
IAI S Iil" dn <{ai during gdp kh,ic O,^1 Bu :
nhf,t ndu duong grilr kliric d;'lrurrg r-oi
a
,t
a-
lbo
"*.
Iftdng r7 = 61.
I{irr {3,r cri rltto 'irirn :
2--l/tt+{)?<CI
rt' = -
vAl'116 nghieilbidn tr8n cii-r khorlng (- *a; - l),
(*'1 ; * o" ). Vdi {:0, la ro.u=3, tu d6 srrr-'
ra .[i6t, qui p'hi.i tim: a { 3.
, Yi dt] 3.. firn eae gia lri nr soo cho phao*g
trl nh
..r.+14.,!- l-r"ns*I!
ed clrin5r frt; nglligi:r .
^.r" rfi
Girii. Ta phrii xirr dinh ar s*c etrr&ug th&*g
D e6 plrrrtrng trinh
{, = mx i" l/2
O. J ! ,r .
eat (lo llt! cu& lrailr 9()
.- rf ,-t
' lJ =St v{r--l (1)
t4i lrai tli'6r1. Iiunng tlring l) e6 h€ sr5 96c .:t
v&. di qua dihm (0; tiz). 1)6 thi r:ria hr\m_r{) c6
tiQnr cAu .xii.n. rB, i.riu phiLi.y ;.?r- rA 1i6nr cin
xri'n vi: lri;r irri j .\ E *:r" iiirtr i-{) drrr.rt' r:,r ri!nlr
r'(rl f :ij .? tl:l i ii co d.,ro lranr
g, E 1,+ 4nl {4"rs*i ,
! .d
\r6.y hieri nhi6n y'>' O qi r ) t/:. Voi e
< - ,/2, dB -{' rHng lz"l > {;f :, ,,,i1'g'{0
lu rlri str-r rn lting l.,i,1rr llrir:tr
(t ca a5nlt.
DtPu Iftrle
'+y*+c1'h?-
Yi
S utri
(; _
r, - "i+ I
S2 :aii-l -Fai+)
53 =qi+t *ni+2*oi+J
.i
".*'. r- p4
{ei+&it);i
doan tli&ng Of icio. C lA rliBm cd dilir), diEu niy
x&y ra khi
g:-1"1q -. *si-+_f'/as
SSg = rri+t f ri+2 *.., .l ai-1
S1OO = fli+t -iat+"2 + .,. * o; 1 -l-c
dEu kh1ng fim.
6idi. Ndu n ld mtt sd rg nhien ) t$0, thi n*
108m * r v6.i 0- _(. ( 99n vd" ta dEt
^d.t = 0r n6u r., )> 0, co = ot00 lidu r = 0.
Ch&ng lian cl?z - 6?p, 6200 - otO-g, Yirr & l.i mQt
nguy0n khdng am! ra il5.r
f fil= 0, /(k) = {:rl + ,i + ... -+. ,y.
Tr0n miLt ph&ng toa cL), xem cac rl;irn .lr (4.;
/ (&)) (lt=0, 1"...) Ndi cnc dibm u{r -,-oi lr.r-r (k:
0: ,,...), ta drrgc mQt dirdng g6plihrhc, iuAnhodn
vdi chu kJr t{)0 (hinh vE). X€t mttrt chu ki. ch8ng
irql ru O dgn rO0. Ggi I ld gie rri uuo ".to'1ti)
nh6 nh{t: db thi cfra [(a) chung t6 r{ng iI6
Ii ehl sd i ph&i tim icii rlrb c6. nbiEu c_hi sd i
nhu vqy).
tlE
tr'rc ii
l\hr do
cri ch-? bh u&.*"lrlr *c}
r.= il = ;* lil(ts * 6 * c.).
SminsA{::li j- (o * b -i clsh!
di-t \
tli
HiBu nhi€u trong vi dg ni.y, cfrng uhu ca
trong 4 vi dn trOn, e6 thb dua ra'm0r ldi gi&i
khdng drina tl6n db thi.. Nhung c6c Lryi gl&i AA
u€u dring' phtrong ph6p db th,i ch5.c chfin c6 rli
suc. lhuydt phuc db c6.c ban tn rJrt ra ir6t luirn
chn thidt.
Yi dB 5. 7ii0. s6 fAr.rc ar, i.tl,.... d100 ed tbng bdng
0. Jgir&i {r:r ;ririf tdc sd ?ig theo iilri lir lrln m6t
dt,tuti.1 .tt on.tiir,it lurrrn* (i inr' ,-,, t.htittg minh
ruhq l6n f6i ;ngt clri s,) 1 s*o cjro ldt ra cde tbng
r13
1*,-,{*.--*
it;d #
vE* eesr f&rus v# HAN
SOXC yO L1n (sd hangi I = I * l,lyz + t/rz r... +
H :ln" ... ln o gi,ii h+n cira tb^nj lriu ha rr
Xn = 1 a- 1|Y-f 1/3r *...* lln' khi ,r.-" x.
Tbng v 0 han nay dur.vc gfi.n r,rri tin lubi cira
hai nh&. to{n .}roe Thuv si: I-*-c6p Bec-n.u-li
T+ VAN .r ry
(1054*l?05) vi L6-6-aa l$-ie t1707 - iTBS). I-a.crIp
Bec-nu'ii .trbi ti6ng mQt ph$n do dd p):At minir
ra vi. tinh atucyc gie tri efia nhiBu tbng v6
hqn. Tbng v6 han c;rc ughich dio cr3a c{c s6
binh phucrng tr0n cirng do Ong dD xulit ohu,ng
tlp tim gin tri cirn tiirg !,i) hsir tic, fing ho*n
toan hi bflt il.rcr. Cf.r [[;1 Beo-rnr"li -,'i8t : ( eho
i[dn nayu tOi elii ed g8ng nhiBrr nhrrng r.Sn khsng
tir:l: r;r. "4i..tim drtor-: t':: .l:,i hi*i itti tf,r sin Lr"i-lni
*n v0 *trrg D. LP*fr-:iLr U-le li hr_rc irr) a&n l-$-gan.
Rec*nuili i1S6? *. 1?,tb), errr lrrii t,frt I-*-c$p: i1:)
lrjp lt uog nbriu tlrnr airlir r';ro rirrh qi1, trj cii*
ti\ng l0 irrn itr*n. Sarr i:.hiiu liir: cl:i tinr ilugi:
erie gii,r tri elrr dirng" crrrli erilg irng r.iir tirri r:r
,lirqrc gii tr'!,rli'rng Liinl rtr'i1 i)hrrilng illr[p e$r
li_i tit, Jrno (').
-\{t lilruirng lrirrir r{rri sd i'rir ;r:
eo xo + e 1 .r,1*1 -l- ... + sp*l 3c -J- dG = 0, ao f B
cir :l ngtriiirn q.i, *3r..., itq l..ir:ir,!ilriino, The* .1inh
I i \"il't I :r r./r :
*l stt."" fir, = {--1}" oo'.,o, rlr r.:.i.y 4n +1.6 oU ,o
vi{t *uo.:: .
- .ir , t-l
uu.t- "l- sl#--'' +,,, -i- fr;1 x {!6{.r * c1},."(c - cn)
* (qolce) (r*c11 ... {,c*{x.} = olo
(x*at) ... {;r -(n)
{-- L}o at n! .," un
= c* (l .- .rlctl) {l * lelc.r} .., (J -- ;riso} {l}
BXng pbrong phsp dbng rrhi.t hQ sd, so s{nh
e6c hQ s6 t'ng voi *, ta ed
€a-l = -"*i! (tjdrt -|. 11012 +-... +i,.fiE) i2)
Lai xdt irhurrng trinh .hl!c 2rr d4ng :
$o * Iru-trt +... "+ (-l)n6.:rea e 0
'ei ?n ngtli+m khirc l.hfing sl, -e[!, &p" *ds,.,,,.crr
-a*. Thd thi, trrong *r.r{1} tn cc:
&n - Dn-l *2 -f ... + ( - l)tr .1.;.eE =
* $a {l * ri 11) (t --:rlg"r}... {; -- rl*s} t, i -F s/e*}
;= {:n (t - iBl"I} i I - :rzi q.i)...{ i - r,r2i s3}, ($}
ri tirnnu tr,r {2:1 {a cii:
[rr*t = oo(rlclf '+ tia|+ ... i riaii t4']
BAy gii-r' Sle xet phuong t,;inir sin r=0 hit-v
fi clgng khai tril* theo ir$e eii* c li. pht:,:ng trin;.
tla i sii lrie r'0 lran :
xlt'. - r3/,1 I . i.-'is 1-.r.?i 7,r + ... = u. (5)
l',16 r:6 vd irSn ngtri$nr 0" :{, -tf, 2fr, -2:Il,...
Sau "khi b6 nghir)rn 0 ili r-i chir'r r-6 trai cid {i)
cho ,{ i:r c6 phtrtrng irinh :
t - J"'16-F,r* 5 | -.16,'7 ! -l-... =0
vdi vO ir+n nghionr 3[, --rY,2]E, -2:f.,.. Din
tlty Ole .ilp rlilng trong tr.r !;-tjt 'r1u&. nhrr tt.irrirg
hrip him hq.n, thr:o {:l).c1,:
I --rt6-ira,'5!-16 -i i t.,.- o
i
t rr lheu (*i cd :
I _l , t I
*- 5i *-. f ---"-- '-
(j tr *frj. nt.,tTl
lrat :
Iei t t;'2t +, I.r32 -+ ..,-!- !rlt i."... =.Jl:rri (6)
Sau lilri tinr r4 kil c;ufi, nrir rlir cr'i r,hiEu cc
*g .18 tiri v*o ph&t nrinli r:ii:r rninh nbung 6ng
r.*.n th;iy kdt lufrn eiia.milih li t/ru l:ao li thi6u
co sfr ilir;r bfr.o c{c kft qu& t3} vir (4) trong
. tlu&ng irr,rp uhuong tr.init brie h&u han i.nn tlring
*hc truonl;,hEp phrrong trinir. bis vS han {l).
YE sau.-ioiin hoe caa r:i{p ii} rirc r}h*rr iinh dring
cl{n ilria l.;i:t iu4n (6), r:hung tui l.:iirr lhtr khdag
sq efip. D*t1i d[y sF chirrrg n1irrh k*.it qui. i6)
h*lg phunng phfp dr: hitsu rir r.rii ki&r thirc
$+ cl6 p.
Sfr dung cdng thf,e Moavro (.i = .lT{7 r.i khai
triln nhi thirc Niutsn ta cri :
cos ^ri* "{- l*in rr& a {e*s€t * isina)r :
| .i n-j
CJ iisinc) eos 'r.
n
= (ce$r" - *: I****3 esin2a +
e j "o*
o*d,* sin4 c * ...)
-F, (Cj Qos Ir*1c sina-Ci .**'*3 q siasq * ...)
T'ri' dd suy r&
sin n r = c,i.o'n* !,r*in, *c3 *nr t-'n.,nr* o
. ,,S u-5 q
+ t. (.os :t Sjri- 0t - ... (i)
n
Sr"r dgn6 (fi inav n hdng (?n +"1). ra co:
sin (3n * 1) a sa
^I 2n-. -. ^3 2t't-2
Cirr+t crrc'"{x sin cr - l,Zn f t "ou 't'-' o *in8fl
. ..5 2u -{
-,-1..,,. eos 6rirt54 -.,,
Jn i'{
. iln*l ,-1
:sin"''' a (c.',.,,, eotggna - trin+,.orgznl2a
lnrl
. ,.5' 2n*"4
* tzr+, t:ol€-- -ct *,...1. $,
Ta th51' n Si* tri sr=i?fl{!a + t)" i E 1,,.., r
l* nghigru cfia phrbng trinh sin (:n+1;s.=0 nhung
hlr0ng c6 gi6 tr! nto trong dd lS ngh{iqm efla
phuong trinh sin a s 0, ntsa tt (8i ta c6 n gi{
tri ni 1191 eflng l& n€htfnr cfrr phucng lrinh:
.i=0.
ciu+ r cotg2"a * al**, .,otg2u*t or .i,
^5 2n *,i
uin + I colg-* -ct - ".. *' I
har'* giir tri ts * ,,otg'-$, i*i,.,.,n Ia ugbifi4
2n*l
eria phuarig {rinh €4"i $ii biic n:
-l u .,3 t*l
tlrr{-t' - L2n+i t +
^5 ti-?
C^-,,r" --','xi],
zE-r t
. Vi thd lirer: riixir i.i V;et te ca:
cotg' - lr* -i".-c,t:1r -lI- + .,. -.- .otnt.i{*
2n*l 2n*l '' 2n1t
r'
"2ir+! n(?n * i)
= *- " ffi- -..- " (gl
,A
f-l 6
-2n-ri
Khi dLp d+ng" c&*g
tri {S} ta r:{i:
nl .-x
:#
6 I Jt
I;l 5I]' .%
2n'r1
thGp t * cotgzn = l/siote.
* z ifi , 'n (?n*lr ?e(n*li
n-r 4 ,l,riij-*- * =,: =.-€(10)
i=:l 2a:1 3 3
L4i c6 : 0 { sina { a .( tga hay l/sina}11s5
cotgalE, & do o(. {d/?. nen voi a:i3ri(?n*1},
t -1,..., n ta c6 :
r1r:*e .iT- .>(rlt, )r>*o,n, irt . r)o vir:
?a-f i \ r3r j - ln*l
.a iEl
s1fi Itr*I
>l(3:rLllI"n,s"*g-'
i=r\ i?t l-;:, '2n+i
I
Iteo ($), (to) ra r-t:
Znin*r) > lrj';11)'$ r- , nt2n .- r]
3 \ Ji l,i=1 i" - 3
h-a ]'
in(n*t] 3fe. .n(2n-l) 'ta
;-::j:*' .:r . * "-_.,,.,,)_.,5_+_ . _ fit)
r2n = l)? 3 , (?o.* 1}. 3
'fu !lJ), iireo rliuh i.r ;:iii hqn, la e6
*,i
il
T - lirn i'r = **-
n+? 6
T6i d*3 , ci.ch giii nri.t tlrac sc i.irtr r ua ldag
nhilu -bqn trt' -vru tu,in. $E ed tluoc e6ch giii
d6. Ailu tl&u tiill r-uiii phirt- iu f nghi 'r'E bili
toAn iid I& : mtii. dir Bcl-rli-li ci,u gF.p.lihr'r lilrH.n
m{c ittl lrii t*{n 116 dttr.vc giil birng:litrur.rog ph6p
hhOng s<r r.:{p... niirrrig v6n t16m lirll vr\ lin tu6ng
r&ng sC tilrr iirro'e phuong plf,p n:dii tfjt hon vA
ph* hqp ilii r'6i kriiin iirirr eila rninh.
ilS lf, tu luil,tg ti$1 rriug trolg hqc t{.p!
't
.--.-**
(') ffuai n&rFr rau kni cd dip $uag lql vdn db
ndg, O'le dd tim dztyc mgt phtong ph&p ttn cQg
tll tinh ,6ns ud han d6.
-r*
11
L
?a-
mg ffieffi r,i:rj';:t:"J; ;1': ",r1'3#1;Ht';
#$ r-rwr*
'Bni Ii 130. .?$i nlgt guitg &dn uda pidag
plrlm eri &ria Jt*i losf Dd. Gid m6i qa5pln e6 {d
sd nguyln rlBng. C rnruc .f6 qay*n t*, S nrna ?{i
qug?n v& vd C mua,?8 gnyln u&. ,IIdt ngnat plld,i
lrd 56 dbng: Btttt {Eg tim gid nz6i gug}n u&:ci6i
Iogf.
"LAl gi8f (eria fr&a i!unr 8f;rg - IECI phaa 6hu
Trirh, !! NSag; Yfi Jvgpe !'.In ktrrr * 2/?E ,trEn
Cao Vln, OrI Nflng vi nrrlt s6 b3n kh{c),
€
ha.*
cs*gtl0-q)=56
-:r8 * g{16.* &) = 5s.
*e*y(3{i*d)*56
{e*g}e+1&y=$6
(s*g)&*I$u=56'
(s*y)e*169=5S
Yi c* y)0 net tu (3) ct 2tigd56+ y(3.
l) 9 = 1: Tti (1) rn (?) cho
(c-yla=<6
(r-u)&={o
-) arh =23/t0 + o i?g. 16 I,v r-i a ( to.
(r i
C.)\
(5)