512.0076 PH121L JH TRUONG - TRAN VAN THUONG - NGUYEN PHU KHANH HANH KY - NGUYiN MINH NHIEN - NGUYEN TAT THU NGUY6N TAN SIENG - DO NGOC THUY (Nhom giao vien chuyen Toan THPT) ^ 1,-0 z PHUONG PHAP GIA (Tai ban c6 sda chOa bo sung) Danh cho hoc sinh Icip 11 on tap va nang cao kien thifc ^ Bien soan theo npi dung sach giao khoa cua Bp GD&OT Sin a xsm p •sin NHA XUAT BAN DAI HOC QUOC GIA HA NOI NQiUYtN IAN bItNQi - t)(J N(3QC THUy (Nh6m gi4o vien chuyen To^n THPT) PHUONG PHAP GIA DAI Sfi - 6IAI TiGll (Jal ban c6 sda chOa bo sung) e- Danh cho hoc sinh Idp 11 on tap va nang cao kien Mc ©• Bien soan theo noi dung sach giao khoa cua Bo GD&OT ',3 t . ' NHA XUAT BAN DAI HQC QUOC GIA HA NQI Cdc em hoc sinh than men! "Phdn loai vd phttcfng phdp gidi Dai so - Gidi tich 11" Id nipt trong nliQng cuon thuoc bo sdch "Phan loai vd phucfng phdp gidi theo chuyen de: Idp 10, II, 12", do nhor i tdc gid chuyen todn THPT bien soan. Vdi cdch viet khoa hoc vd sinli dgng giup ban doc tiep can vdi man Todn mot cdch tu nhien, khong dp luc. Ban doc trd nen tu tin vd rmng dgng haii, hieu ro bdn dmt, biet cdch pMn tich de tim ra trgng tdm cm vdn de vd biit gidi thich, lap ludn cho ticng bdi todn. Su da dgng cua lie thong bdi tap vd tinh huong giup ban doc ludn hvcng thu khi gidi todn. Tdc gid cliu trgng bien sogn Tilivcng cdu lidi md, ripi dung ca bdn bdm sdt sdch gido klioa vd cdu true de thi dgi hgc, dong tlidi plidn bdi tap tlmnJi cdc dcmg todn c6 Idi gidi chi tiet. Hien nay, dd thi dgi ligc kJiong klio, to hap cua nliieu vdn de don gian, nliung chvca nliieu cdu hoi md neu klidng ndm elide ly thuyct se lung tung trong vice tim Idi gid/ bdi todn. Vdi mot bdi todn, Uidng nen tlioa man ngay vdi mot Idi gidi rmriii tim dugc ma plidi cd gdng tim nliieu cdch gidi nlidt cho bdi todn do. Klii gidi mot bdi todn, thay vi dung tlidi gian de luc Igi tri nlid, thi ta can plidi suy nghi plidn tich de tim ra phuang plidp gidi quyet bai todn dd. Mon Todn ddi hoi plidi kien nlidn vd bin bi ngay tit nliixng bdi tap dan gidn nlidt, nlivcng kien thvcc ea bdn nlidt. Vi chinli nliQng kien thvcc ca bdn yndi giup ban dgc hieu dugc nliUng kien thuc ndng cao sau nay. Gid day, cliung tdi chat rJid tdi cdu noi cua Ludwig Van Beetlioven: "Gigt nudc c6 die lam man tdng dd, klidng plidi vi giot nudc c6 sicc ingnli, ina do nudc clidy lien tuc ngay dem. Chi CO su plidn dau klidng met inoi indi dem Igi tdi ndng. Do do ta c6 die klidng dirJi, klidng nliich ti^ng budc thi khong bao gid c6 thedi xa ngdn ddm". Mac dutdcgiddd ddnli nliieu tdm liuyet cho cudn sdch, song su sai s6t Id dieu klid trdnli klioi. Chung tdi rat mong niidn dugc su plidn bien vd gop y quy bdu cua quy dgc gid de nliiing Ian tdi bdn sau cuon sdch dugc hodnthienhmi. Thay mat nhom bien soan Chu bien: Nguyen Phu Khdnh Nhd sdch Khang Viet xin trdn trong gi&i thieu t&i Quy doc gid vd xin long nghe moi y kien dong gop, decuon sdch ngay cdng hay Hon, botch horn. Thttxinguive: Cty TNHH Mpt Thanh Vien - Dich vu Van hoa Khang Vi?t 71, Dinh Tien Hoang, P. Dakao, Quan 1, TP. HCM Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880 Hoac Email: khangvietbookstore@yahoo.com.vn CHd DE 1: HAM SO LUONG GIAC VA PHUONG TRINH LUONG GlAC CHl/ONG 1: , HAM s6 LUOMG GIAC i A. TOM TAT LI THUYET I. CAC CONG THLfC Ll/ONG GIAC 1. Cac hSng dang thuc; * sin^ a + cos^ a = 1 voi moi a , . , i ' •/j *tana.cota = l voi moi a — , , 2 ' ' , * 1 + tan^ a = — voi moi a ^ k2n cos a ' * 1 + cot'^ a = —\ voi moi a^kn sin' a 2. thiic cac cung dac bif t , , a. Hai ciin^ ddi nhau: a va -a cos(-a) = cosa sin(-a) =-sina tan(-a) =-tana , • cot(-a) =-cota b. Hai cung phu nhau: a va — -a cos( —-a) = sina sm(—a) = cos a 2 2 . >• ii'i'- tan(^-a) = cota cot(^-a) = tana c. Hai cung hii nhau: a va n-a ,.t JI it sin(7r-a) = sina cos(7t - a) =-cosa tan(7i - a) = - tan a cot(7i-a) =-cota d. Hai cung han kem nhau n :a va n + a sin(n + a) =-sina cos(7t + a) =-cosa tan(7r + a) = tan a cot(7r + a) = cota 3. Cdc cdng thuclugng gidc "" ' • a. Cong thuc cdng ^ cos (a ± b) = cosa.cosb ± sina.sinb sin(a ± b) = sin a. cos b ± cos a. sin b tan a ± tan b tan(a±b): 1 ± tan a. tan b b. Cong thiec nhan sin 2a = 2 sin a cos a cos2a = cos^ a - sin'^ a - l-2sin^a= 2cos^a-l sin3a = 3sina-4sin''a cos3a = 4cos"'a - 3cosa * *' c. Cong thiec ha bac . 2 l-cos2a 2 l + cos2a ^ 2 l-cos2a sin a = cos a = tan^ a = 2 2 l + cos2a d. Cong thiec bie'n dot tich thanh to'ng j') ' cosa.cosb = —[cos(a - b)+ cos(a + b)] sina.sinb = ^[cos(a-b)-cos(a + b)] sina.cosb = ^[sin(a - b) + sin(a + b)]. # e. Cong thiec Men dot to'ng thanh tich a + b a-b , ^.a + b.a-b cosa + cosb = 2cos .cos cosa -cosb = -2sin .sin 2 2 2 2 .• a + b a-b . ., „ a + b.a-b sina + sin b - 2sin .cos sina - sinb = 2cos .sm 2 2 2 2 . i , sin(a + b) , sin(a - b) tana + tanb = !^ f- tan a-tan b= ^ ' . cosacosb cosacosb II. TINH TUAN HOAN CUA HAM SO , Dinh nghia: Ham so y = f(x) xac djnh tren tap D duoc gpi la ham so tuan hoan ne'u c6 so T ?i 0 sao cho vol moi x e D ta c6: x ± T e D va f{x + T) = f(x). Neu CO so T duomg nho nhd't thoa man cac dieu kien tren thi ham so do dupe gpi la ham so tudn hoan vai chu ki T. III. CAC HAM SO Ll/QNG GIAC * 1. Ham so y = sinx ^* , . • y. T^;:*- ^ , 1 ^^^^ • Tapxacdjnh: D = R '' . • Tapgiactrj: [-l;l],tucla -l<sinx<] Vx e R '' • Ham so dong bien tren moi khoang ("^ + k27i;^ + k27i), nghjch bie'n tren moikhoang (^ + k27i;Y^ + k27t). ^ • Ham so y = sinx la ham so le nen do thj ham so nhan goc tpa dp O lam tam doi xung. * • Ham so y = sin x la ham so tuan hoan voi chu ki T = 2:1. hi \ Cty TNHH MTV DWH Khang Vtgt • Do thi ham so y = sin x . 2. Ham so y = cosx ; • Tapxacdjnh: D = R ' ! ' ; _ ^''*|-' • Tapgiactrj: [-l;l],tuc la-1< cosx <1 Vx e R ^ • Ham so y = cosx nghjch bie'n tren moi khoang (k27i;7i + k2T:), dong bie'n tren moi khoang (-71 +k27t;k27r). % • Ham so y = cos x la ham so chin nen do thi ham so nhan true Oy lam true do'i xung. • Ham so y = cos x la ham so tuan hoan voi chu ki T = 27i. • Do thj ham so' y = cos x. - Do thj ham so y = cos x bang each tinh tien do thj ham so y = sin x theo vee to V = (-—;0). 3. Ham so y = tan x • Tap xac djnh: D = M\\- + kn, ke Z • Tapgiatrj: X > •'" • La ham sole t - - ' ' ' • La ham so tuan hoan voi chu ki T = 7t f 1 lli '.%h%^' •1,' Ham dong bie'n tren moi khoang — + k7i; —+ kTt 2 2 I • Do thj nhan moi duong thSng x = + k7t, k e Z lam mpt duong tif m can. Phdn loai va phuang phdp gidi Dai so - Gidi tick 11 • Do thi -5JI 2 57t 2 mi 4. Ham so y = cot x • Tap xac djnh: D = M\ k e Z • Tap gia tri: M • La ham so' le • La ham so tuan hoan voi chu ki T = TT • Ham nghich bien tren moi khoang (k7r;7r + kn) • Do thi nhan moi duong thang x = kn, keZ lam mpt duang ti^m can. • Dothj 7t \ \ \ \ 71 \t \ \t 71 \ 2 \ -371 \ 2 \ o \. V B. PHLTONG PHAP GIAI TOAN Van de 1. Tap xac dinh va tap ^a tri ciia ham so v I. PHl/ONG PHAP V • Ham so y = 7f(x) c6 nghla o f(x) > 0 va f(x) ton tai • Ham so y = —5— c6 nghla o f(x) 0 va f(x) ton tai. f(x) • sin u(x) ?t 0 o u(x) = kn, k e Z • cosu(x)?!:0<=>u(x)^- + k7i, k€. • -1 < sinx, cosx < 1 . Cty 7WHH MTV D WH KhangViet U. CACVIDV Vi d\ 1. Tim tap xac dinh cua ham so sau: 1. y = tan(x ) i,:,:"::, !::, 6 ru:.i 2. y = cot2(^-3x) Giai 1. Dieu ki?n: cos(x ^0 x^~+ kn x + kn 6 6 2 3 TXD: D = 271 + k7t, k e Z 2. Dieu kien: sin(— - 3x) ^ 0 <=>'^-3x ^kn <:> -k^ TXD: D = IR \ 271 , 7t , r„ k-, ke£ 9 3 Vi 2. Tim tap xac djnh ciia ham so sau: tan2x 71 1. y = + cot(3x + —) sin X +1 6 1. Dieu ki^n: sinx ^-1 sin 3x + - 6J ^0 Giai x^-— + kin 2 71 nTC X 9t + 18 3 2- y=- tanSx sin4x - cos3x VgyTXD: D = #\ + k27i,-Y^ +y;k,n € Z • 2. Ta c6: sin 4x - cos3x = sin 4x - sin 3x (x f7x Tl\ 3x = 2 cos — + — sin \2 J K2 AJ L 2 4; Dieu ki^n: cos5x ^ 0 cos sin — + — 2 4; ^7x 71 2 4; ^0 • 71 ,71 •,,„;,Q 10 5 . xii — + n27C 2 71 n27t X ^ + 14 7 Vay TXD: D = # \ 71 ^ k7t 7t ^ ^ ^ 2m7i ; n, m e Z L IO^T'2^" '^'"TI^ 7 ' J 14 7 4 [...]... cot3x = cotx 2 cot4x.cot7x = 1 Giai 2 Dieu ki#n: I f kn 4 ; k, n eZ nn < > cot7x = tan 4x = c o t ( - - 4x) = o 7x = — 4x + mn 2 ,m mi „ , 1 T T i k T I x = — + m — • 22 11 0 k + 2 • Ta co: — + m — = — o 2 + 4m = 11 k < > m = 3k = 22 11 4 4 Vi m , k e i J = > - ! ^ = t=>k = 4 t - 2 = ^ m = n t - 6 = — < > 7 + 14m = 22n => 22n - 14m = 7 = 7 Vi 22n - 14m la so chSn con 7 la so le nen phuong trinh nay... arctan — + kn 2 1 4|cos3xcos^ x + s i n 3 x s i n ' ' x j + >/3sin6x = 1 + S^cos^* x - sin'' x| 2 4 • 4 sm 4 X + cos X + sin4x^\/3-1-tan2xtanx =3 • Phdn loai vd phucntg phdpgidi Dai so'-Giiii tick 11 V i dv 11 Giai ,^1 Cho tan a, tan p la hai n g h i ^ m cua p h u o n g t r i n h x^ - 6x - 2 = 0 T i n h gia t r i 1 Taco: 4 cos3xcos^x + sin3xsin''x) = 3cos2x + cos6x va cos''x-sin*x = cos2x nen 2, Cho... cosx) = 0 s i n " sin^ X x o 1 + sinx /-t V, rcosx = l ( I - cos x)(cos x - s i n x ) = 0 tan X = 1 , „ (1 + cos x) = 0 "x = k27i ^ X = — + k7t 4 , k e Z Phan loai va phuong phdpgiai Dai so'-Giai tich 11 Cty TNHH MTV DVVH Khang Viet Vi d u 8 Giai cac p h u o n g t r i n h sau: V i d\i 6 Giai cac p h u o n g t r i n h sau: 2 2cos-^ X = sin3x 1 sin'' X + cos^ x = sin X - cos x 1 cos3x + cos2x - cosx...Phdn loai va phuomgphapgiai Dai so -Giai Cty TNHH MTV D W H Kliawj; Viet tick 11 2 Do sin X + cos x +2> 0 \/xe M => ham so xac djnh voi Vx i , , s i n x + 2cosx + l Xet phuong trmh: y = 2 A p dung B D T (ac + bd)^ < (c^ + d^ )(a2 + „ • , , I • a b Dang thuc xay ra khi - = — »... Giai e Z 1 P h u o n g t r i n h 5 tan^ x + 2 tan x - 5 = 0 - I + V26 tan X = ^ -l±^/26 , X = arctan + kn 3x Cty TNHH MTV DWH Kliaux Vict Phdn loai va phuang phap gidi Dai so - Gidi tich 11 2 P h u o n g t r i n h 2cos4x = 1 + cos6x 4cos^ 2x - 4cos^ 2x - 3cosx + 3 = 0 cos2x = 1 X = krc X = kTI cos4x = — 2 cos^ 2x = - x = 7T - a - arcsin x =±— +— 12 2 1 2cos^ X + 6 s i n x c o... + 3cosx = - 3 ,keZ 1 3 2 P h u o n g t r i n h o 3 3 cosx - s i n x + 2 s i n 2 x = 1 Giai 3) 18 1 2 s i n 2 x - ( s i n x + cosx) + l = 0 ,k€Z • = ^3 1 Phuong trinh cos 2 x - ^ + k27i k27: + Bai 11 Giai cac p h u o n g t r i n h sau: 1 \/3sin2x + cos2x = 2cos x + s i n x - 1 '5 4 Phuong trinh < > 4 1 - —sin^ 2x + \/3sin4x = 2 = 2 Bai 10 Giai cac p h u o n g t r i n h sau: 3 2' + k2K 71 X = Ket... + cos x = T i sin 2x : sina = — 5 Dat t = sin X + cos x = \/2 cos ^ V X 7t^ 4, [sin2x = t 2 - l - r^ ji " + k27i, k e! '" ' Cty TNHH MTV DWH Khang Viet Phdn loai vd phuantg phdp giai Dai so- Gidi tich 11 Taco: t = N/2(t^ - l ) A 4, Ta thay cosx = 0 khong la nghi§m cua phuong trinh v^t^ - t - %/2 = 0 /2,t = - 4 ' « Nen phuong trinh < > 4 tan''x + 3 - 3tan x(l + tan^ x)-tan^ x = 0 = J: tan X... [-1;!], ta c6 phuong trinh:2t^ - 3t + 1 = 0 < > t = l;t = - = •:t + cos x = V2 cos = l sin^2x 2 1 — sin 8 4T 2x ,keZ -1±V5 8 + rUTi Cty TNHH MTV DVVH l^hiin loai vii phuinig phiijt giiii Dai so- Giai tick 11 Nen dat t = sin^ 2x, 0 < t < 1 ta dirge phuong trinh 16 2 4 Dieu kien: 2t^ = 17(1 - t) « 2t^ +1 - 1 = 0 c= t = ^ sin^ 2x = - 1 -2sin^ 2x = 0 cos4x = 0 o x = — + k—, ke Z 2 8 4 Bai 15 Giai... sin^ X + 4 sin 2x sin X cos X = 1 o cos2x + 2sin^ 2x - 1 = 0 2cos^ 2x - cos2x - 1 = 0 o c o s 2 x = - ^ (do sin2x^0cos2x^±l)x = ± ^ + k7i, k e Z Phdn loai va phtfangphdpgiai Dai so-Giai tick 11 -T+N/3 V2ccs^x + cosx-\/2=0 * CM v: Ta can l u u y den cong thirc: tan x + cot x = sm2x va cot x - tan x M Bai 17 Giai cac p h u o n g trinh sau: 1 < X = ± - + i . 71, Dinh Tien Hoang, P. Dakao, Quan 1, TP. HCM Tel: (08) 3 9115 694 - 3 9111 969 - 3 9111 968 - 39105797 - Fax: (08) 3 9110 880 Hoac Email: khangvietbookstore@yahoo.com.vn CHd DE 1:. PHAP GIA DAI Sfi - 6IAI TiGll (Jal ban c6 sda chOa bo sung) e- Danh cho hoc sinh Idp 11 on tap va nang cao kien Mc ©• Bien soan theo noi dung sach giao khoa cua Bo GD&OT ',3. HA NQI Cdc em hoc sinh than men! "Phdn loai vd phttcfng phdp gidi Dai so - Gidi tich 11& quot; Id nipt trong nliQng cuon thuoc bo sdch "Phan loai vd phucfng phdp gidi theo chuyen