Phân loại phương pháp giải đại số giải tích 11

"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, I I , 12", do nhor i tdc gid chu

512.0076

PH121L

JH TRUONG - TRAN VAN THUONG - NGUYEN PHU KHANH HANH KY - NGUYiN M I N H N H I E N - 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 x s m p

•sin

NHA XUAT BAN DAI HOC QUOC GIA HA NOI

N Q i U Y t N I A N bItNQi - t)(J N ( 3 Q C T H U y

(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, I I , 12",

do nhor i tdc gid chuyen todn T H P T 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 T N H H Mpt Thanh Vien - Dich vu V a n hoa Khang V i ? t

71, D i n h Tien Hoang, P Dakao, Quan 1, T P H C M

1 Cac hSng dang thuc;

* sin^ a + cos^ a = 1 voi moi a , , i ' •/j

*tana.cota = l voi moi a — , ,

2 thiic cac cung dac bif t , ,

a Hai ciin^ ddi nhau: a va -a

cos(-a) = cosa sin(-a) = - s i n a tan(-a) = - t a n a , • cot(-a) =-cota

b Hai cung phu nhau: a va — - a

cos( —-a) = sina s m ( — a ) = cos a

2 2 >•

ii'i'-tan(^-a) = cota cot(^-a) = tana

c Hai cung hii nhau: a va n-a ,.t JI i t

sin(7r-a) = sina cos(7t - a) = - c o s a

d Hai cung han kem nhau n :a va n + a

sin(n + a) = - s i n a cos(7t + a) = - c o s a tan (7r + a) = tan a cot(7r + a) = cota

b Cong thiec nhan

sin 2 a = 2 sin a cos a cos2a = cos^ a - sin'^ a - l - 2 s i n ^ a = 2 c o s ^ a - l

sin3a = 3 s i n a - 4 s i n ' ' a cos3a = 4cos"'a - 3cosa * *'

c Cong thiec ha bac

2 l - c o s 2 a 2 l + cos2a ^ 2 l - c o s 2 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 = ^ [ c o s ( a - b ) - c o s ( a + b)]

sina.cosb = ^[sin(a - b) + sin(a + b)] #

e Cong thiec Men dot to'ng thanh tich

D i n h 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

• H a m so y = sinx l a ham so l e nen do thj ham so nhan goc tpa dp O lam

tam doi xung *

• H a m so y = sin x la ham so tuan hoan voi chu ki T = 2:1

• H a m so y = cosx nghjch bie'n tren moi khoang (k27i;7i + k 2 T : ) , dong bie'n

• H a m so y = cos x la ham so chin nen do thi ham so nhan true Oy lam true do'i xung

• H a m so y = cos x la ham so tuan hoan voi chu k i T = 2 7 i

• 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 )

Phdn loai va phuang phdp gidi Dai so - Gidi tick 11

• La ham so tuan hoan voi chu k i T = TT

• H a m nghich bien tren moi khoang (k7r;7r + kn)

• D o t h i nhan m o i d u o n g thang x = kn, keZ lam mpt d u a n g t i ^ m can

B PHLTONG PHAP GIAI TOAN

V a n de 1 Tap xac dinh va tap ^ a tri ciia ham so v

I P H l / O N G P H A P V

• H a m so y = 7f(x) c6 nghla o f(x) > 0 va f(x) ton tai

• H a m so y = —5— c6 nghla o f(x) 0 va f(x) ton tai

Phdn loai va phuang phdp gidi Dai so-Gidi tich 11

Cty TNHH MTV DWH Khang Viet

Bai 3 T i m tap xac dinh cua ham so sau:

sin 3x

1- y =

-sinSx - s i n S x 2. y =

tan 4x cos4x + sin3x

Cho ham so'y = f(x) tuan hoan v6i chu ki T

* De khao sat sy bien thien va ve do thi ciia ham so, ta chi can khao sat va

ve do thi ham so tren m o t d o a n c6 do dai bang T sau do ta tinh tie'n theo cac vec ta k.v (voi v = (T;0), k G Z ) ta dugc toan bo do thj cua ham so

* So'nghiem ciia phuong trinh f(x) = k , (vol k la hang so) chinh bang so

giao diem cua hai do thi y = f(x) va y = k

* Nghiem cua ba't phuong trinh f(x) > 0 la mien x ma do thi ham so

» y = f(x) nam tren true O x

ÍliuH loiii vti f>huintgpilidpgidi Dai sd'—Gidi tich 11

• Ham só f(x) = asinux + bcosvx + c ( v o i u, v E Z ) la ham sótuan hoan

1%

voi chu ki T =

(u,v) ( (u, v) la uoc chung Ian nhát)

Ham sóf(x) = ạtanux + b.cotvx + c (voi u , v e Z ) l a h a m tuan hoan voi

Ta-c6 f(x) = —(cos x + cos2x) => ham sótuan hoan vai chu ki co so TQ = 27t

V i dv 2 Xet tinh tuan hoan va tim chu ki co so (neu co) cua cac ham sósaụ

1 f(x) = cosx + cos (^/ịx) 2 f(x) = sinx^

Giai

1 Gia sir ham so da cho tuan hoan => co sóthyc duong T thoa man

f(x + T) = f(x) <=> cos(x + T) + cos\/3(x + T) = cosx + cosVSx

cosT = 1 Cho x = 0 => cos T + cos \/3T = 2 o •

cos V 3 T = 1

T = 2n7i /~ m m

r- => V3 = — v6 l i , do m , n e Z = > — la so huu tị

v3T = 2mTc n n

Vay ham so da cho khong tuan hoan ' <> ' « f / • f • - " f A )

2 Gia six ham soda cho la ham so tuan hoan ' '''-^

= > 3 T > 0 : f ( x + T) = f(x)c>sin(x + T)^ =sinx2 Vx G #

Cho x = 0 sinT^ = 0 <=> = kn => T = Vkrt f(x + v ^ ) = f(x) Vx e i J ^

Cho x = V2k7t taco: f (A/2k7t) = sin(N/k27i j = sin(k27t) = 0

f(x + 7iai) = sin(Vk27i + >/k7c] =sin(3k7r+2k7r>/2) = ±sin(2k7i\/2) => f(x + Vior) ^ 0

Vay ham so da cho khong phai la ham so tuan hoan

Cty TNmi M T V P V V H Khan^ Viet

Vi du 3 Cho a,b,c,d la cac so thuc khac 0 Chung minh rang ham so

f(x) = asincx + bcosdx la ham so tuan hoan khi va chi khi — la so hCru tị

d

" , Giai • • •

* Gia su f(x) la ham so tuan hoan => 3T > 0 : f(x + T) = f(x) Vx

asincT + bcosdT = b |cosdT = l

UẤ^ Giai

Vi — la so hiJu t i nen ton tai hai so nguyen m,n;n ; t 0 sao cho: , , ,;

T2

' = — r r > nT| = mTj = T Khi do f(x + T) = f(x + nT,) = f(x) va g(x + T) = g(x + mTj) = g(x) Suy ra f(x + T) ± g(x + T) = f(x) ± g(x) va f(x + T).g(x + T) = f(x).g(x),

^ ^ ^ ^ - i ^ = TCi-do ta CO dieu phai chung minh f ' | ;

Phan loai va phucntg phdp gini Dai so- Gidi tick 11

VP(1) = sin — = 1 => ( 1 ) khong xay ra voi moi xe E

Vay ham so da cho tuan hoan voi chu ki co so TQ = 2:1

Cho x = 0 => VT(2) = tan2T 5i 0 , con VP(2) = 0 =^ (2) khong xay ra vai moi x e I

Vay ham so da cho tuan hoan voi chu ki co so TQ = ^ / /:

• H a m so' y = 2sin x la ham sole

• H a m so y = 2sin x la ham tuan hoan voi chu k i T = 27i

• Cac d u o n g tiem can: ^ ~ ^ '^^ •

• Do thi ham socti quan cac diem ( ^ ; 0 )

o

1 3

V i d u 2 Xet su bie'n thien va ve do thj ham so'sau y = 1 + 2cos^ x

• H a m so y = 2 + cos2x la ham tuan hoan v o i chu k i T = TI

H a m so dong bien tren moi khoang

, nghich bien tren

• Do thi ham so d i quan cac diem ( — ( 7 i + k7i;3)

2 2

II C A C B A I T O A N L U Y E N T A P

Bai 1. Xet su bien thien va ve do thi cac ham so y = sin2x

Giai • ' 'A

Do t h i ham so: y = sin2x ; | I ' ^

Bai 2 Xet su bien thien va ve do thi cac ham so y = 2|cosx

Giai

Do thi ham so: y = 2 cosx

Van de 4 Gia tri Idn nhat va nho nhat cua ham so

V i dy 2 T i m tap gia tr| Ion nhat, gia trj nho nhat ciia cac ham so sau

1 y = 6cos^x + cos^2x 2 y = ( 4 s i n x - 3 c o s x ) ^ - 4(4sinx - 3cosx) + 1

Giai 1- Ta c6: y = 6cos^ x + (2cos^ X - 1 ) ^ = 4cos'* X + 2cos^ x + 1

D$t t = cos^ X => t e K h i d o y = 4 t ^ + 2 t + l = f(t) '

f(t)

1 Vay m i n y = 1 dat duoc k h i cos x = 0 <=> x = — + kn

2 Dat t = 4 s l n x - 3 c o s x = : i - 5 < t < 5 V x e i ?

K h i d o : y = t ^ - 4 t + l = : ( t - 2 r - 3

V i t e [ - 5 ; 5 ] = > - 7 < t - 2 < 3 = > 0 < ( t - 2 ) ^ < 4 9 D o d 6 - 3 < y < 4 6 ,

V a y m i n y = -3; m a x y = 46. ; j 'f;Hi j f v

-V i dv 3 T i m ta't ca cac gia trj cua tham so m de ham so sau chi nhan gia t n

• d u o n g : y = (3sin x - 4cosx)^ - 6sin x + 8cosx + 2 m - 1

Giai ' •»

Dat t = 3 s i n x - 4 c o s x = > - 5 < t < 5 ' ' ' - f

Ta c6: y = t ^ 2 t + 2m - 1 = (t -1)^ + 2m - 2

D o - 5 < t < 5 = > 0 < ( t - l ) ^ < 3 6 = : > y > 2 m - 2 = > m i n y = 2 m - 2

H a m so chi nhan gia tri duong <=>y>0 Vxeji^Ci>miny>0 o 2 m - 2 > 0 < = > m > l

Vay m > 1 la gia trj can t i m

V i du 4 T i m m de ham so y = •y/2sin^ x + 4sinxcosx - ( 3 + 2m)cos^ x + 2 xac

siny > sin I2 - X J = cosx

> sin X cos y + sin y cos x = sin(x + y)

71

/ \ 7:

X

12

= cosy

= cosx Suy ra: sin^x + sin'^ y = sinx.sinx + s i n y s i n y

< sin X cos y + siny cosx = sin(x + y)

M a u thuan v o i (*)

• Ne'u x + y = => (*) dung Vay (*) o x + y = ^

V i du 6. T i m G T L N va G T N N ciia cac ham sau: / ! 1

1. y = 3sinx + 4cosx + 5 2 y = sin X + 2cosx + 1

sinx + cosx + 2

Giai

1. Xet p h u o n g trinh: y = 3sin x + 4cos x + 5

o 3sin X + 4cosx + 5 - y = 0 => p h u o n g trinh c6 nghiem

« 3^ + 4^ > (5 - y)^ « y^ - lOy < 0 <=> 0 < y < 10 Vay m i n y = 0 ; m a x y = 10

Phdn loai va phuomgphapgiai Dai so -Giai tick 11

2 Do sin X + cos x +2> 0 \/xe M => ham so xac djnh voi Vx i

1 T a c o l < 2 s i n x + 3 < 5 r : > l < y < N / 5

Vay gia tri Ion nha't ciia ham so bang N/S , dat dugc khi sinx = l<=>x = ^ + k27r

Gia tri nho nha't bSng 1, dat dugc khi x = -^ + k2n

Bai 2 Tim tap gia trj Ion nha't, gia trj nho nha't cua cac ham so sau:

1 y = 2sin^ X + cos'^ 2x 2 y = 3sinx + 4cosx + 1 ,jvy,i ,

Cty TNHH MTV D W H Kliawj; Viet

2 A p dung B D T (ac + bd)^ < (c^ + d^ )(a2 + )

„ • , , I • a b Dang thuc xay ra khi - = — •,

T a c o : (3sin x + 4cosx)'^ <(3^ + 4^)(sin''x + cos''x) = 25 -5 < 3sinx + 4cosx < 5 = > - 4 < y < 6

Vay maxy = 6, d a t d u g c k h i tanx = - :;<;twh kij^, 0 - v n

min y = - 4 , dat dugc khi tan x = ( - ' ^ ^ ' ^ C^M y; Voi each lam tuong tu ta c6 dugc ke't qua tong quat sau:

max(asinx + bcosx) = \la^ + , min(asinx + bcosx) = -yja^ +

Tiic la: -7a^ + b^ < asinx + bcosx < 7a^ + b^ Bai 3 Chung minh dang thuc sau: asinx + bcosx = 7a^ + b^ sin(x + a)

Trongdo a e 0;2n va a,b khong dong thoi bang 0 i

Khi do: asinx + bcosx = 7a^ + b^ (sinxcosa + cosx sin a )

•'ill = Va^ +b^ sin(x + a )

Nhdn xet T u ket qua tren, ta c6

• Gia trj nho nha't cua ham so y = a sin x + b cos x bang -Va^ + b^

• Gia tri Ion nha't cua ham so y = a sin x + b cos x bang Va^ +b^

• -Va^ +b^ < asinx + bcosx <\/a^ + b^ Vx e # Bai 4 Tim tap gia trj Ion nha't, gia trj nho nha't ciia cac ham so'sau:

1 y = sinx + 7 2 - s i n 2 x 2 y = tan^ x + cot^ x + 3(tanx + c o t x ) - 1 •

19

Pkan lo^i vd phuomgphApgiai Dai so-Giai tick 11

Khong ton tai max y

Bai 5 T i m m de ham so y = \/5sin4x - 6 c o s 4 x + 2m - 1 xac d j n h v o i m o i x

Giai

H a m so xac djnh v o i m p i X <=> 5 s i n 4 x - 6 c o s 4 x > 1 - 2 m Vx

Do m i n ( 5 s i n 4 x - 6 c o s 4 x ) = - > / 6 1 : = i > - V 6 1 > l - 2 m ••.->m>^'^^

• i " 2 Bai 6 T i m tap gia tri Ian nha't, gia trj nho nhat ciia cac ham so sau

Bai 7 T i m m de cac bat phuong trinh sau diing v o i m o i xeR ,

3sin2x + cos2x , 4sin2x + cos2x +17 ^

sin2x + 2cos2x + 3 ^ " , i>

(Do sin2x + 2cos2x + 3 > 0 V x r : > ham so xac djnh tren M)

<=> (3 - y)sin2x + (1 - 2y)cos2x = 3y Suy ra (3 - yf + (1 - 2y)^ > 9y2 o 2y^ + 5y - 5 < 0

2 T r u d c het ta c6: 3cos2x + sin 2x + m + 1 0 Vx e

* Cac t r u d n g h o p dac biet: x , s v; n!

1 cosx = 1 <=> X = k27i 2 C O S X = -1 o X = 7t + k27i

Chuju: * Ne'u a thoa man n n < a < —

2 2 thi ta viet a = arctan m

X

| u , v n7t

^ I I D A N G 2: P H l / O N G T R I N H B A G N H A T D O I V O I S I N X V A C O S X

La phuxmg trinh CO dang: asinx + bcosx = c (1) ; v 6 i a,b,c e K va a^ + b^ 0

Cach gjai: Chia hai ve cho Va^ + b^ va dat cosa = , ; sina = a b

(1) <=> sinx.cosa + cosx.sina = , _^ o sin(x + a ) = ^

-i2

sinu(x) + b cosu(x) tanu(x) cotu(x)

c - 0

Cach giai: Dat t = ta CO phuong trinh: at^ + bt + c = 0

K h i dat t = , ta C O dieu kien: t € f - l ; !

sinu(x) cosu(x) tanu(x) cotu(x) Giai p h u o n g trinh nay ta t i m duoc t , t u do t i m dugc x

sinu(x) cosu(x)

IV D A N G 4: P H l / O N G T R I N H D A N G C A P

La p h u o n g trinh c6 dang f(sinx,cosx) = 0 trong do luy thua ciia sinx va

C O S X cung chan hoac cung le

Cach giai: Chia hai ve phuong trinh cho cos"^ x ;.i 0 (k la s o m u cao nhat) ta

dugc p h u o n g trinh an la tan x

V D A N G 5: P H l / O N G T R I N H D O I X l / N G ( P H A N D O I X l / N G ) D O I V 6 l SINX V A C O S X

La p h u o n g trinh c6 dang: a(sin x + cos x) + b sin x cos x + c = 0 (3)

De giai p h u o n g trinh tren ta su dung phep dat an phu:

Thay vao (5) ta dugc p h u o n g trinh bac hai theo t

Ngoai ra chiing ta con gap phirong trinh phan doi xung c6 dang

a ( s i n x - c o s x ) + bsinxcosx + c = 0 (3')

De giai p h u o n g trinh nay ta cung dat:

t = sin X - cos X = V2 sin X n

2. P h u o n g trinh o (sin x + sin3x) + sin2x = (cosx + cos3x) + cos2x

o 2sin2xcosx + sin2x = 2cos2xcosx + cos2x

o ( 2 c o s x + l ) ( s i n 2 x - c o s 2 x ) = 0 <=> 1

cosx =

2 sin2x=cos2x

X = ± — + k27I

3

x = + k 8 2 ,

-3. A p d u n g cong thuc ha bac, ta c6:

r,, > , l - c o s 6 x l + cos8x 1 - c o s l O x l + cosl2x Phuong trinh •» • =

-4 Phuong trinh •» (1 + cos6x)cos2x - 1 - cos2x = 0

<=>cos6x.cos2x-1 = 0 <=>cos8x + c o s 4 x - 2 = 0

<=> 2cos^ 4x + c o s 4 x - 3 = 0c:> cos4x = l o x = k - ^ , k e Z

* Nhdn xet:

- 6 c o s 6 x c o s 2 x - 1 = 0 ta c6 the su d u n g cong thuc nhan ba, thay

cos6x = 4 c o s ' ' 2 x - 3 c o s 2 x va chiiyen ve p h u o n g trinh trung p h u o n g doi

voi ham so i u o n g giac cos2x

- Ta cung c6 the su d u n g cac cong thuc nhan ngay tie dau, chuyen p h u o n g

trinh da cho ve p h u o n g trinh chi chua cosx va dat t = cos^ x

Tuy nhien each duoc trinh bay 6 tren la dep hon ca v i chiing ta chi su d u n g

cong thuc ha bac va cong thuc bien d o i tich thanh tong * • ' '•'

V i du 3 Giai cac p h u o n g trinh sau:

<=> sin 3x + \/3 cos 3x = 2 cos 4x <=>cos

V i d u 4 Giai cac p h u o n g trinh sau:

1 cos(7tsinx) = cos(3Tcsinx) 2 tan

2

Cty TNHH MTV DWH Khans Viet

• Xet p h u o n g trinh sin x = k Do k e Z va - 1 < sin x < 1 nen ta c6 cac gia

•^ri cua k : -1,0,1

T u do ta CO cac nghiem: x = m7r, x = — + mn, me Z

2 «

• Xet p h u o n g trinh sin x = ^ Ta c6 cac gia trj cua n la: n = ±2, n = ±1, n = 0

T u do ta t i m dugc cac nghiem la: x = - + ITC, x = k , x = ± - + k , \

Vay nghiem cua phuong trinh da cho la: x = m?:,x = - + mir, x = ± - + mn , m e Z

2 6

2 Phuong trinh <=> ^ ( s i n X + 1) = j + k n

<=> sin x + 1 = 1 + 4k o sin X = 4k <=> sin x = 0 o x = m n , m e Z ,

V i dv 5 Giai cac p h u o n g trinh sau:

1 3sin^x + 5 c o s ^ x - 2 c o s 2 x = 4sin2x 2 sin^ X 71

2 ^ 4 tan^ X -cos^ — = 0

Giai

1. Phuong trinh da cho t u o n g d u o n g voi

3sin^ X + 5cos^ x -2(cos^ x - s i n ^ x) = 8sin xcosx

o 5sin^ X - 8 s i n xcosx + 3cos^ x = 0

• » 5 t a n ^ x 8 t a n x + 3 = 0c:>tanx = l hoac tanx =

-<=> x = — + k7i hoac X = arctan — + k7t; k e Z

2 Dieu kien: cosx;itO<=>x?^ — + k7i

2 Phuong trinh <=> 1 - c o s i x )

Phan loai va phuong phdpgiai Dai so'-Giai tich 11 Cty TNHH MTV DVVH Khang Viet

V i d\i 6 Giai cac p h u o n g trinh sau:

1 sin'' X + cos^ x = sin X - cos x 2 2cos-^ X = sin3x

1 Nhan thay cosx = 0 khong la nghiem cua p h u o n g trinh nen chia-hai ve'ciia

p h u o n g trinh cho cos^ X ta duoc: ,

tan x - Stan x - 6 = 0

l=lan X tanx = - 1 tanx = 6

Do cosx = 0 khong la nghiem cua p h u o n g trinh nen chia hai ve phuong

trinh cho cos^ x ta duoc: " ' •

t=tan X

2tan x - 3 t a n x + l = 0

tanx = 1 tan x = —

Vi d u 8 Giai cac p h u o n g trinh sau:

1 cos3x + cos2x - cosx - 1 = 0 2 2 sin x ( l + cos 2x) + sin 2x = 1 + 2 cos x

, Giai

1 Ta thay trong p h u o n g trinh chiia ba cung x,2x,3x nen ta t i m each dua ve cung mgt cung x

Phuong trinh o 4cos^ x - 3cos x + (2cos'^ x - l ) - c o s x - l = 0

2cos X + cos x - 2cosx - 1 = 0 ' ' )' Iff; J r,',']iinsir i.a

Chii y: Ta c6 the giai bai toan tren theo each sau:

Phuong trinh c:> cos 3x - cos x - (1 - cos2x) = 0

<=> - 2 s i n 2 x s i n x - 2sin^ x = 0 <=> sin^ x(2cosx +1) = 0 sinx = 0

2 Ta chuyen cung 2x ve cung x

Phuong trinh <=> 4 sin x cos^ x + 2sin x cos x = 1 + 2 cos x

V i d u 9 Giai cac p h u o n g trinh sau:

1 4|cos3xcos^ x + sin3xsin'' x j + >/3sin6x = 1 + S^cos^* x - sin'' x|

+ s i n 4 x ^ \ / 3 - 1 - t a n 2 x t a n x = 3 •

sm X + cos X

Phdn loai vd phucntg phdpgidi Dai so'-Giiii tick 11

Giai

1. Taco: 4 cos3xcos^x + sin3xsin''x) = 3cos2x + cos6x va cos''x-sin*x = cos2x nen

Phirong trinh <=> 3cos2x + cos6x + \/3sin6x = 1 + 3cos2x

<=> Vs sin 6x = 1 - cos6x o 2>/3 sin 3x cos3x = 2sin^ 3x

<=> 2sin3x(\/3 cos3x - sin 3x I = 0

Suv ra nghiem can t i m la x = k—;x = — + k —, k e Z

Ta c6: 4 sin"* x + cos'* x j = 4 - 2sin^ 2x = 3 + cos4x

sin2x sinx cos2xcosx + sin2xsinx c o s ( 2 x - x ) i

,^1. Cho tan a, tan p la hai nghi^m cua phuong trinh x^ - 6x - 2 = 0 Tinh gia tri

cua bieu thuc sau P = sin^(a + P) - 5sin(2a + 2P) - 2.cos^(a + p)

2, Cho tan a, tan P la hai nghiem cvia p h u o n g trinh x ^ + b x + c = 0 {c*\)

T i n h gia trj cua bieu thuc P = a.sin^(a + P) + bsin(2a + 2p) + c.cos^(a + P)

l + tan^(a + p) 1 + 4 5 ' j

2. Theo djnh l i Viet ta c6: t a n a + tanp = - b , tana.tanp = c ' >

„ ^ / o \n a + tan p - b Suy ra t a n ( a + P) = •

1 - t a n a tan p 1 - c Taco: P(l + tan2(a + p)) = = atan^Ca + P) + 2btan(a + p) + c ,

cos^(a + p)

_ p _ a t a n ^ ( a + P) + 2btan(a + P) + c_ "(1-c)^ J-c ^ _ ab^-2b^(l-c) + c(l-c)^

l + tan2(a+p) ~ b^ ~ (l-cf+b^

II C A C B A I T O A N L U Y E N T A P Bai 1. Giai cac p h u o n g trinh sau:

1 Phuong trinh sin2x = 2cos2x <=> tan2x = 2

<=>2x = arctan2 + kn <=> x = —arctan2 + — , keZ

1 Phuang trinh cos x-2sinxcosx = 0

<=> cos x(cos x - 2 si n x) = 0 <=> cosx = 0

2sinx = cosx

cosx = 0

1 « tan X = -

1 Dieu kien: sin 2x 0 <=> x ?t kTt

Phuang trinh cot2x = 0

Phuong trinh <=> cot8x = tanSx = cot — 5x

2 o x = — + m —

26 13 Ket hop dieu kien ta c6 nghiem ciia phuong trinh: x = — + — , m ;t 13n + 6

4 Dieu kien: sin 2x ?i 0 « x ;<!^ ^ Ta c6: tan^ x + cot^ x > 2 > 1 + cos'^

Nen phuong trinh <=>

tan^ X = cot^ x sin 3x + -4 = 0

<=> X = — + k7i la nghiem cua phuang trinh da cho

Bai 5 Giai cac phuong trinh sau:

Bai 6 Giai cac p h u a n g trinh sau:

1 >/3(sin2x + cos7x) = sin7x - cos2x 2 c o s x - 2 s i n x c o s x

2 Dieu k i ^ n : 2cos^ x + sin x - 1 ^ 0

• Phuong trinh <=> cos x - sin 2x = Vs cos 2x + N/S sin x

Cty TNHH MTV D W H Khang Viet

Bai 7 Giai cac phuong trinh sau: *

1 sin2x + 4 ( s i n x - c o s x ) = 4 2 V2(sin x + cosx) = tanx + cotx

Phuong trinh <=> V2(sinx + cosx) = — - — <=> sin2x(sinx + cosx) = ^fl

^ sin2x

Dat t = sinx + cosx,t6 -^/2;^^l =^sin2x = t^ - 1

Thay vao phuang trinh ta c6 duoc:

Dat t = cos X - sin X = V2 cos X + — n

V 4 , ,te Sin xcosx = •

Phdn loai va phuang phap gidi Dai so - Gidi tich 11

2. Phuong trinh <=> 2cos4x = 1 + cos6x <=> 4cos^ 2x - 4cos^ 2x - 3cosx + 3 = 0

cos2x = 1

cos^ 2x =

-X = krc cos4x = —

2

X = kTI

x = ± — + —

12 2 Bai 9 Giai cac p h u o n g trinh sau:

1 2cos^ X + 6 s i n x c o s x + 6sin^ X = 1 • 2 2V2(sinx + cosx)cosx = 3 + 2cos^ :

3. tanx + cotx = 2(sin2x + cos2x)

3 Dieu kien: 2cos^ x + s i n x - W O o cos2x + sinx ;t 0

p h u o n g trinh <=> cosx - sin2x = V3cos2x + s/Ssinx

V 4 , [sin2x = t 2 - l

Phdn loai vd phuantg phdp giai Dai so- Gidi tich 11

Bai 12 Giai cac phuong trinh sau:

1 cos^x-\/3sin2x = l + sin^x 2. 2N/2(sinx + cosx)cosx = 3 + 2cos^x

3 2cos^x = sin3x 4 4sin''x+3co6-\-3sinx-sin^xcosx=0

o 3tan^ X 2 \ / 2 tanx + 52\/2 = 0 phuong trinh v6 nghiem

-3 Phuong trinh c:>2cosx = 3sinx-4sin-x

<=> 2 = Stan X ^1 + tan^ x j - 4 tan^ x <=> tan'' x-3tanx + 2 = 0

Cty TNHH MTV DWH Khang Viet

4, Ta thay cosx = 0 khong la nghi§m cua phuong trinh

Nen phuong trinh <=> 4 tan''x + 3 - 3tan x(l + tan^ x)-tan^ x = 0

1 2sin^ X - 3sinx + 1 = 0 2 3cos4x - sin^ 2x + cos2x - 2 = 0

2 Phuong trinh da cho tuong duong voi:

3(2cos^ 2x -1) - (1 - cos^ 2x) + cos2x - 1 = 0

o 7cos^ 2x + cos2x - 6 = 0 <=>

cos2x = -1 cos2x = —

Bai 14 Giai cac phuong trinh sau:

1 4cosx.cos2x + l =0 2 16(sin^x + cos*'x) = 17cos^2x

Giai

1 Phuong trinh-»4cosx(2cos x - l ) + l = 0 o8cos^x-4cosx + l = 0o(2cosx-l)(4cos^ x + 2cosx-l) = 0

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 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 cac phuong trinh sau:

1 2tan2x + 3 = — 3 2 sin 2x+— -Boos ( 7n] X

cos X I 2j I 2 j =l+2sinx , o 2 3x „ 4x

<=»2 \^ + 1 = Q o c o s x = l o x = k27i,k6Z cos^x tosx ,

x + — cot - X = tan X cot X

I 3 , J v6 J = 1 Phuong trinh <=> 1 -^sin^ 2x = 1 + sin2x <r> sin2x = 0

» X = k7t, la nghiem cua phuong trinh

Bai 16 Giai cac phuong trinh sau:

1 l + 3tanx = 2sin2x 2 cot x - tan x + 4 sin 2x = sin2x

Giai

1 Dieu kien: cos x 9t 0 <=> x — + k7t

Phuong trinh « 1 + 3 ^'"^ = 4 sinx cosx <=> cosx + 3sinx - 4sinxcos^ x cosx Day la phuong trinh dang cap bac ba nen ta chia hai ve ciia phuong trinh

cho cos^ X (do cos x ^ 0 ), ta dugc phuong trinh:

1 tan — + 3 — = 4tanx <=> 1 + tan^ x + 3tanx(l + tan^ x) = 4tanx X '' ' ' '

oStan'' X + tan^ x - tanx +1 = 0<=> tanx = -1 <=> x = -— + kTi thoa man dieu kien

4

* Nhan xet: De giai phuang trinh nay ngay tu dau ta c6 the chia hai ve cua

phuong trinh cho cos^ x hoac su dung cong thuc:

^ 2 sinx cosx 2 tanx sin2x = • sin^x+cos^x 1 + tan^x

phuong trinh chi chua ham tan nhu tren

2 Dieu kifn: sin2x 5^ 0 x ^ k^ ''' ' ^

r>u - u cos X sin X ^ - 1

1 huong trinh <=> — + 4 sin 2x = —

va chuyen phuong trinh ban dau ve

1 X':}::' sTi));'; 'i(ji)h;rj.,;.r ;

Sinx cosx sinx cosx ,

O cos^ X - sin^ X + 4 sin 2x sin X cos X = 1

o cos2x + 2sin^ 2x - 1 = 0 <=> 2cos^ 2x - cos2x - 1 = 0

ocos2x = - ^ (do sin2x^0<=>cos2x^±l)<=>x = ± ^ + k7i, k e Z

Phdn loai va phtfangphdpgiai Dai so-Giai tick 11

2 3cot^x + 2V2sin^x = (2 + 3\/2)cosx

3 2sin2x - cos2x = 7sinx + 2cosx - 4

8 n cos4t l + cos2t

s i n ( ^ + 4 t )

<=> (1 - 4cos2t)(l + cos2t) = 6(1 -cos2t)(2cos^ 2t - 1 )

<=> 12cos^2t- 16cos^ 2t-9cos2t + 7 = 0 o ( 2 c o s 2 t -l)(6cos^ 2t -5co,s2t - 7 ) = 0

< b 3cos^ X - 3\/2 sin^ x.cosx + 2\/2 sin"* x - 2 s i n ^ xcosx =tl"°

<=> (cosx - \/2sin^ x)(3cosx - 2sin^ x) = 0

<=> V2ccs^x + cosx-\/2=0 2co6" X + 3cosx-2 = 0

Bai 18 Giai cac phuong trinh sau:

2 cotx + tanx = sinx + cosx

= sin X cos X (sin X + cos x)

o (cos X + sin x)(cos x - sin x cos x - s i n x ) = 0 cosx + sinx = 0

cosx - sin x cosx - sin x = 0

3 Dieu kien: cosx ^ 0

Phuong trinh o 3sinx + 2cosx = 2cosx + 3sinx

Bai 19. Giai cac p h u o n g trinh sau:

1.2cosx + tanx = 1 + 2sin2x ,; ;>

2 cosx - 2cos3x = 1 + sinx

46 • • :

3 sin x + s i n X + — + sin4x = sin 2x 71

4 cos4x + sin2x cos3x + sin3x = 2N/2sin V 4 , X + —

1. 2cosx + tanx = 1 + 2sin2x

Dieu kien: cosx ^ 0

<=> 2cos^ X + sin X = cosx + 4sinxcos^ x <=> cosx(2cosx - 1 ) = sinx^4cos^ ^

<r> (2cosx - l ) ^ c o s x - sin x(2cosx +1)) = 0 <=>

1 ••, , • ^

2cosx - 1 = 0 ,,, J sin X - cos X + 2 sin X cos x = 0 cos x =

(cos x - sin x)^ + (cos x - sin x) - 1 = 0

x = - + 2m7t

3

x = — + 2 m 7 r

3 cosx-sinx = cosx-sinx =

Vai u = cosu t h i t a c o : 2u + 2(4u^ - 3u) = 1 o (2u + l ) ( 4 u ^ - 2 u - l ) = 0

47

Dieu kien: cos 3x + s in3x ^ 0

Ta c6: cos4x + sin2x = cos4x + cos

+ 3

2x —

2 = 2 cos cos 3x + s in3x = cos 3x + cos 3x —

3x — cos X + — 7t

4 j

2J ••••;.5 j ; , ) ' /

CtyTNHH MTV DWH Khans Viet"

Khi do p h u o n g trinh tro thanh: 2cos^ X + — TC = 272 s i n

1. sin^ x.cos3x + cos^ xsin3x = — ,>

4

2 2sin2x + (273 - 3)sin x + (2 - 373)cosx = 6 - ^3

3 sin^" x + cos'° x sin^ x + cos^ x

4cos^ 2x + sin^ 2x

4 cos3x + 72-cos^ 3x = 2^1 + sin^ 2x

Giai

1 A p d u n g cong thuc ha bac ta c6:

Phuong trinh ci> (?, —sin X — sin 3x cos3x + 1 ^

^4 4 - c o s x + — cos3x 4 4 sin3x = — 4

3 3

<=> —(sinxcos3x + sin3xcosx) = <=> sin4x = - 1

Vay ta c6 duoc x = - — + —, ke Z la nghiem cua p h u o n g trinh da cho

8 2

2 Phuong trinh <=> 2sin2x + ^273 - sjsinx + ^2 - 373)cosx = 6 - 73

•» 2sin 2x + 2173 sin x + cos X j - 3^sin X + 73 cos x j = 6 - 73

3 D i e u kien: 4cos2 2x + sin^ 2x ?i: 0 Vx

Ta c6: sin^ x + cos^ x = 1 - - s i n ^ 2x = - ( 4 - 3sin^ 2 x )

4 " 4V / Suy ra sin^ x + cos^ x 1 = — N e n p h u o n g t r i n h <:> sin^'' x + cos^° x = 1 (*)

4 A p d u n g b d t a + b < /2(a^+b^^ ta co:

= 2

<=> X = k 7 t , k e Z ;

cos3x + V2-cos^ 3x < ^2^cos^ 3x + 2-cos^ 3x

Ma 2 f l + s i n 2 2 x ) > 2 nen phuong trinh J c o s 3 x= 7 2 ^ ^ ^jcx>63x=l

o i n ^ Y ^ n sinxcosx=0

V i cosx = 0 => cos3x = 0 ; sinx = 0 :

sin2x=0 cos X = 1 => cos 3x = 1 cos X = - 1 => cos 3x = - 1

D o do (*) o cosx = 1 <=> X = k27i, k e Z

Bai 21 Giai cac p h u o n g trinh sau:

1 1+ sin^ X cosx + ^1+ cos^ x j s i n x = 1+ sin2x

o (sin X + cos x ) ( l - sin x ) ( l - cos x) = 0

Giai ra ta dugc cac nghiem: x = - - ^ + k 7 i , x = ^ + k 2 T t , x = k 2 7 i , k e Z

Phuong t r i n h » sin 2x + 2 cos x - sin x - 1 = 0 • y

<r>(sinx + l ) ( 2 c o s x - l ) = 0 giai p h u o n g t r i n h nay va d o i chieu dieu kien ta

dugc nghiem cua p h u o n g t r i n h la: x = — + k 2 K , k e Z

sin X + —

4

X +

-V 4 , sin x + c o s x cos x + s i n x

sin 3x + —

4 = 2cos sin

Vay nghiem cua p h u o n g t r i n h la: x = — + k::

2. tan^ x + tan^ y + cot^ ( x + y) = 1

<=>tanxcot(x + y) + tanycot(x + y) + tanxtany = 1 ' ' 7 U f J

Ap dung BDT + + c^ > ab + be + ca ta c6: tan^ x + tan^ y + cot^(x + y) > 1

7t

V — \T A. L T T V —

tanx^tany Nen phuong trinh <=>

Cty T N H H M T V D W H Khans Viet

Van de 2 Tim nghiem phufcmg trinh lufong ^ae ecr ban

I , C A C V i D V

Vi du 1 Tim tong cac nghiem trong khoang ( - 7 t ; 7 i ) ciia phuong trinh:

1 sin(3x + ^ ) = c o s ( 2 x - j ) 2 sin2 2x = c o s ^ ( 3 x - - )

8 Giai

1 Phuong trinh <=> sin 3x + - = sin f37l ,2x 1

Phan loai vA phuangphapgidi Dai so'- Gidi ttch 11

Nghiem d u o n g nho nhat ciia p h u o n g trinh x = -^^^^ ,ke Z la x = i

N g h i e m d u o n g nho nhat ciia p h u o n g t r i n h x^ + x - k = 0 la: x =

Vay X = i la nghi?m can t i m '•^i • '

<=>•

16k x>-

9 x = 2 4 k - 4 0 - 25

3k+5

25

Theo bai toan suy ra: — e Z => k € {O, -2, -10

T h u lai ta c6 cac n g h i e m nguyen ciia p h u a n g t r i n h :

- + k —

D o X € (0;27:) nen p h u o n g t r i n h co cac nghiem la: — ; — ; — ; —

V|y tong cac n g h i f m can t i n h la: ST: , ' (

C h i i y: Ta c6 the giai theo each khac n h u sau:

P h u a n g t r i n h <=> Vs sin x + cos x + N/S cos x - sin x = l4l sin 2x

o sin(x + - ) + cos(x + - ) = N/2 sin 2x <=> sin(x + 7^) = sin 2x

6 6 12 Tiep tuc giai ta duoc ke't qua n h u tren

55 I

FflUn will m pmmg pHap* guu Um so - Giat tich 11

V i X e (-7t; n) nen: * V o l x = k27i ta chi chgn duoc k = 0 => x = 0

* V o i X = — + k27i ta chi chpn dugc k = 0 => x = —

Vay tong cac nghiem bang 2n

' ' 3 x - \ / 9 x 2 - 1 6 x - 8 0 ^ Phuong trinh <=:>-

Ket hgp dieu kien, ta c6 x = 4,x =^ 12 la nhCrng gia tri can t i m

Bai S.Tim nghiem nguyen ducmg cua p h u o n g trinh: cos7i(3- N/3 + 2 X - X 2 ) - - 1

G i a i

Phuong trinh « 7 t f 3 - \ / 3 + 2 x ^ l = n + k27i, k e Z o 2 - 2 k = \/3 +2x - x ^

Ta c6: 0 < ^ 4 - ( l - x ) 2 < 2 va 2 - 2k la so chan nen ta c6 cac nghiem la:

X = - 1 , X = 3, x = 1 ,

57

Phihi loai vaphucmg phdpgiai Dai so - Gidi tick 11

Bai 4 Tim x e [0; 4n] nghiem diing phuong trinh: cos3x-4cos2x + 3cosx-4 = 0

Giai Phuong trinh <:=>4cos x-3cosx-4(2cos x-1) + 3cosx - 4 = 0

<=> 4 CDS''x - 8 cos^ x = 0 o cos x = 0 ci> X = — + k 7 i , k 6 Z

2

Vi X G 10; 47il =>x = — , x = — , x = — , x = —

2 2 2 2 Bai 5 Tim nghiem tren khoang (-7r;7t) ciia phuong trinh:

2(sinx + l)(sin^ 2x - 3sinx +1) = sin4x.cosx

Giai

Ta CO phuong trinh da cho tuong duong vol: ,

1 -cos4x

f 1

2(sinx + l ) •-3sinx + l = sin4x.cosx

o (sinx +1)(3 - 6sinx - cos4x) = sin4x.cosx

o (sinx +1)(3 - 6sinx) - sinx.cos4x - cos4x = sin4x.cosx

o 3(1 - 2sin^x) - 3sinx = sinSx + cos4x

o 3.2.cos(— + -).cos( ) = 2.cos( ).cos( )

-1; / Dieu kien: cos2x ;t 1 <=> 2x ^ k27r o x ^ k7i

2 cos 2x sinx Phuong trinh <::>

<:i cos2x = cos 2 x

^ ^ ^ 16 16 16 16

2lTt

X = •

Van de S Phurcmg phap loai n^iem khi ^ a i phufc/n^ trinh

lurcmg ^ae eo dieu kien

Phirang phap 1: Bieu dien cac nghiem va dieu kien len duong tron lugng giac

Ta loai di nhCrng diem bieu dien ciia nghiem ma triing voi diem bieu dien cua dieu kien

Voi each nay chiing ta can ghi nho: rj ahi j^viv

• Diem bieu dien cung a va a + k27r, k e Z triing nhau " ' '* • '

• De bieu dien cung a + len duong tron lugng giac ta cho k nhan n

n

gia tri (thuong chon k = 0,l,2, ,n - 1 ) nen ta c6 dugc n diem phan bi^t

each deu nhau tren duong tron tao thanh mgt da giac deu n canh noi tie'p duong tron

Phuang phap 2: Su dung phuong trinh nghiem nguyen

kTt

+

m,n e Z da biet, con k , l e Z la cac chi so

chav-Gia su ta can doi chieu hai ho nghiem a + — va P + —/ trong do

n m

59

Phan loai va phucnigphdpgiai Dai so'-Giai tich 11

Ta xet phuong trinh: a + — = p + — <x> ak + bl = c (*)

n m Voi a,b,c la cac so nguyen i '/ <

Trong truong hop nay ta quy ve giai phuong trinh nghiem nguyen:

ax + by = c (1) ^

De giai phuong trinh (1) ta can chu y ket qua sau:

• Phuong trinh (1) c6 nghiem <=> d = (a,b) la uoc cua c '

• Neu phuong trinh (1) c6 nghiem (X(,;y()) thi (1) co v6 so nghiem

X = x„ + — t

,t€

Phuong phap 3: T h u true tiep

Phuong phap nay la ta di giai phuong trinh tim nghiem roi thay nghiem

vao dieu kien de kiem tra ; > • ct,,

Phuong phap 4: Bieu dien dieu kien va nghiem thong qua mot ham so'luong giac:

Gia su ta co dieu kien la u(x)9!:0 (u(x) > 0,u(x) < 0 ), ta bien doi phuong

trinh da cho ve phuong trinh chua u(x) va giai phuong trinh de tim u ( x )

II C A C V i D U

V i du 1 Giai cac phuong trinh sau:

1 cot3x = cotx 2 cot4x.cot7x = 1

Giai

1 Dieu kien: x^k —

3 Phuong trinh <=> 3x = x + nn o x = — , n G

Loai nghiem: De loai nghiem ciia

phuong trinh ta co cac each sau:

* Cach 1: Bieu dien cac diem cuoi

ciia cung k - ^ ta co cac diem

Vay nghiem cua phuong trinh da cho la: x = — + m 7 i , me Z

<_iy iiurm mi v uv vii Knuilg Vll>l

, nrt kTt 2k k = 3t ,, ,

* C a c h 2 : T a c o - = - « n = - ^ ^ ^ ^ ^

Do do ta can loai nhung gia tri n chan

Vay nghiem cua phuong trinh la: x = - + mn, m e iT

Vi 22n - 14m la so chSn con 7 la so le nen phuong trinh nay v6 nghiem

Vay nghiem ciia phuang trinh da cho la: r n ^ H ^ ^ ^ ' t e Z

V i du 2 Giai phuang trinh sau: ^illii5£l^ _ ]

Dieu kien: sinSx

<-> sin6x - s i n 4 x sinl4x - s i n 4 x o sinl4x = sin6x

Nghiem x = ^ bi loai khi va chi khi mot trong hai phirong trinh sau c6

18k-10m = l ta tha'y ca hai phuong trinh nay v6 nghiem

Vay nghiem ciia phuong trinh da cho la: x = — + — , x = — + — , k e Z

De tha'y nghiem (2) khong thoa man (*)

Bieu dien nghiem (1) len duong

tron luong giac ta dugc cac diem

A j , A 3 Trong do chi c6 hai

diem A p A j nam phia tren Ox

Hai diem nay ung voi cac cung

^ 1 r , - 571 , ^

x = — + k27t va x = — + k 2 7 t

6 6

• Voi sin X < 0 (**) thi phuong

trinh da cho tuong duong voi:

Cty TNHHMTVDWHKhans Viet

nam duoi Ox (sin x < 0) } l ,

Hai diem do ung voi cung:

7t , „ N 5K , „

x = — + k27t va x = - — + k27i

6 6 Vay nghiem ciia phuong trinh da cho la: x = + — + , k e Z '

6 3 ,'\

Bai 2 Giai phuong trinh: cos3x tan4x = sin5x '

Giai Dieu kien: cos4x ;^ 0 :; • ' ; Phuong trinh <=> sin 4x cos3x = sin 5x cos4x ,

<=> sin 7x + sin x = sin 9x + sin x <=> sin 9x = sin 7x

* 0 diing voi moi k

Vay nghiem cua phuong trinh la: x = kr:, x = — + — , k e Z

V a y n g h i e m cua p h u o n g t r i n h da cho la: x = — + 2n7r va x = + 2n7i

Bai 4 Giai p h u o n g t r i n h : t a n 2 x tan 3x tan7x = tan2x + tan 3x + t a n 7 x

Loai n g h i e m : VcVi bai toan nay neu chiing ta su d u n g p h u o n g phap loai

nghiem bang each bieu dien len d u d n g tron l u g n g giac hay p h u o n g phap t h u

true tiep se phai xet nhieu trirong hop Do dcS ta lua chgn p h u o n g phap ctai so

V i cac n g h i e m cua p h u o n g t r i n h p h a i thoa m a n d i e u k i e n (1) nen ta t i m each

b i e u d i l n cac n g h i e m qua sinx , ,

65

Vay p h u a n g t r i n h da cho vo n g h i e m 0 « £ + x f t O O 8 x ' ar^f^.?^

Bai 8 Giai cac p h u a n g t r i n h sau: ^jj D rtia

1 sin X + sin2x + sin 3x _ ^

cos X + cos 2x + cos 3x

6 6

V, j rn.r; ^ • , l i J I i ; - y.^-' • • •

2 Dieu kien: cosx T^O - ysy>-i '

P h u a n g t r i n h <=> 2cos^ x - 1 - tan^ x = 1 - cosx - 1 - tan^ x

4 Dieu kien: cos x 0 <=> sin x ;^ ±1

P h u a n g t r i n h o sin'* X + cos'* X = ( 2 - s i n ^ 2x)sin3x „ •

1

<=> sin 3x = - <=> 3sinl « 3 s i n x - 4 s i n - x = - , n^J « d a t a ; X - 4 sin'^ x = - (*) , K

T a t h a y s i n x = ±l k h o n g thoa man (*) Maihov,

5 Dieu kien: cos5x?^0 mo

cosx = — n it X feon :n§i^ u a i O

j Ket hop dieu kien ta c6 nghiem: x = — + k27r, k e Z

4 7cos2x + V l - s i n 2 x = 2 V s i n x - c o s x

= 8cos^x ,'• 0 , ^ ^ « « j

s X n.'ri

G i a i

1 Dieu kien: sin2x?t0<=> c o s 2 x ^ ± l xni^; + / ^nt«£

P h u o n g t r i n h o 8sin^ x + cos2x = 4sin^ 2x +1

nguyen ta t i m duoc nghiem ciia p h u o n g trinh la:

T h u true tiep ta thay: x = ^ + k27t,x = TI + k27r, k e Z la n g h i e m p h u o n g t r i n h

Van de 4 Phtifcfn^ trinh luiong ^ae chtiia tham so

Phiiii loai va phumig phap giai Dai so - Gini tich 11

;-I ;-I C A C B A ;-I T O A N L U Y E N T A P > ' '

Bai 1 Giai va bien luan cac p h u o n g t r i n h sau:

1 4 s i n 2 x = 2 m + 1 •tin ^nxniM 2 ( m - l)cos^(4x + - ) = 2 m

Phdn loai va phucmg phdp giai Dai so - Gidi ttch 11

Phirang trinh (2) <=> cos 2x +

4 NeU m = 0 ^ p h u o n g trinh v6 nghiem ' ^ ' ' ^ ' '[ ^"^ ^

• N e u m * 0 thi p h u o n g trinh da ch tuong d u o n g v a i

1 - m

m

,keZ

Cty TNHH MTV DWH Khang Viet

2 • Neu m = - => p h u o n g trinh v6 nghiem

• Neu m - thi p h u o n g trinh <=> tan^ 3x = m + 2

2 m l

-1 <; • +) Nell - 2 < m < - => p h u o n g trinh v6 nghiem; ['^ 5 , , , ;

Bai 3. T i m tat ca cac gia trj ciia tham so m de p h u o n g trinh J / -"^.cn x rn •

1. cos2x + cos^x + 3sinx + 2m = 0 c6 nghiem - i h o ^ m mlA •

n

2 c o s 2 x - ( 2 m + l)co&x + m + l.= 0 CO nghi^tntren

v: , ; Giai / r!Bl(!+- ra^^j / •'nsJ'ffii^

1. Phuong trinh <=>3sin^x-3sinx = 2m + 2 ,,^4 x-nr,? rnC)(r - x nr>l mS) 0 Dat t = sinx, t e [ - 1 ; 1] Ta c6 phuong trinh : 3t^ - 3t = 2 m + 2

X e t h a m so f(t) = 3t^ - 3 t , t e [ - l ; l Bang bien thien:

Phau loni im phtfmig phnp f-ini Dai so - Ciiii tick 11

Bai 4 Giai va bien iiian phuang trinli:

sin Xcos^ X = 0 <=> sin 2x = 0 <=> X = — , i< 6 Z ;} t.u't hi n?a 'jm to&ttsU X \i

• Neu m ;t 0 , chia hai ve'phuong trinh cho cos'^ x ?t 0 ta duoc ^^^'^^^

Neu m = 0 thi phirong trinh c6 nghiem x = - y , k e X

Neu m ^0 thi phuoiig trinh c6 nghiem:

+ k27I

Ket luan:

• Neil - 1 -\/2 < m < - 1 + Ny2 =>phuong trinh c6 nghiem = ^ +1^^"' ^ = l ^ ' ^ ' k e Z

• Neu ^ ^ => phuong trinh c6 nghiem: =1 -xnUT + x ^nci rn X

3 Phuong trinh o m — cos2x

sin2x 1 - 3 s i n ^ xcos^ X xni*'

Phihi loai vii phianig phdp gii'ii Bai so - Giiii tich 11

thi p h u a n g trinh da cho c6 nghiem x = — + k —

m < 2 thi phu'ong trinh da cho c6 nghiem x = — + k— '

• m ; t O V i p h u o n g trinh luon c6 4 nghiem tren [0;2n) neu yeu cau bai

toan o p h u o n g trinh m sin x - 1 = 0 v6 nghiem hoac c6 cac nghiem tren

o (*) CO hai nghiem phan biet t , , t2 > 1

1 - m ^ O A' = l + 4 m ( m - l ) > 0 ( t ^ - l ) + ( t 2 - 1 ) > 0 ( t i - l ) ( t 2 - l ) > 0

<=> i

m 7t T, m ^ — t| + t j - 2 > 0 ' ' f j o ' ] •; f i f i n j u n ' i i j f i ' j

1 - m 1 - m + 1 > 0

m ?t l , m / 2m

-Vay m > i la n h u n g g i a trj can t i m noiri) mUi a

4 Phuong trinh <» 2cos^ 2x - 1 = l + cos6x m ( l - c o s 2 x )

o 4cos'' 2x - 4cos^ 2x - 3cos2x + 3 + m ( l - cos2x) = 0 i

• 1

o (cos2x - l)(4cos'' 2x - 3 - m) = 0 <p

cos2x = l - i - • cos 2x =

77