GII PHNG TRNH BNG PHNG PHP T N PH V PHNG PHP A V PHNG TRNH TCHI- PHNG PHP T N PH 1- Phng php - i vi mt s phng trnh lng gic, ta c th biu din tt c cc s hng bng mt hm s lng gic duy nht, l n s ca phng trnh. Khi phng trnh lng gic c quy v phng trnh i s qua php t n ph. a) t t = sinx ; t = cos x (

t 1) ; t =

t = tanx ; t = cotx (t R ) - Nu phng trnh khng thay i khi th: + x bi - x, chn n cosx + x bi - x, chn n sinx + x bi + x, chn n tanx VD1: Gii phng trnh: -iu kin: Cosx Pt

1 1 ,t= ( t 1) sin x cosx

1 2 2 5 tan x + =0 2 cos x 2 2+ k (k Z)

0 x

1 1 2 5 ( 2 x - 1) - cos x + 2 =0 2 cos 1 Dat t= ( ( t 1) cos x 1 5 1 Ptbd (t2-1)-2t+ =0 t2-2t+2=0 2 2 2Vay pt co 2 ho nghiem VD2:gii phng trnh: cotx=tanx+2tan2x

t+2

1 1 =2 cosx= 2 cos x

x=

+2k (k Z) 3

sin x 0 iu kin: cos x 0 cos 2 x 0 t t = tanx cotx =

sin 2 x 0 cos 2 x 0

sin4 x 0 4x

k

x

k 4

(k Z).

*Cch 1: S dugj phng php t n ph:

1 t

v tan2x =

2t . Khi phng trnh c dng: 1-t 2

1 t

=t+

4t 1- t2= t2(1-t2)+4t2 t4-6t2+1=0 1-t 2

t 2 -2t-1=0 t 2 +2t-1=0

t=1 2 t=-1 2

tanx=1- 2 =tan1 tanx=1- 2 =tan2 tanx=-1- 2 =tan3 tanx=-1+ 2 =tan4

x=1+k x=2+k x=3+k x=4+k

(k Z)

Vy phng trnh c 4 h ngim. *Cch 2: S dng phng php lun h s phn tch:

pt cotx-tan2x=tanx+tan2x

cosx sin2x sinx cos2x-

=

sin3x cosx.cos2x+ k x=

(cosxcos2x-sin2xsinx)cosx=sin3xsinx

cos3xcosx-sin3xsinx=0 cos4x=0 4x=

2

8

+

k 4

(k Z)

Vay *Nhn xt: Thng qua 2 cch gii trn, ta thy vi nhiu PTLG cch gii chnh quy i khi khng t ra hiu qu.

2x= +k VD3: Gii phng trnh: tan2x+cotx=8cos x dk: 2 x=k 2

x= + k 4 2 (k Z) x=k (1)

*Cch 1: Xt x

+k v tha mn dk (1). 2 2tanx 8 1 pt 2 x + tanx = 1+tan 2 x 1-tan 2t 1 8 8 1 t4+8t3+2t2-8t+1=0 t2+8t+2- + t t=tanx ( t 0 va t 1)ta c: + = =0 t t2 1-t 2 t 1+t 2 1 1 1 1 (t- )2+8(t- ) +4=0 t- =-4 2 3 m t- = tanx-cotx=-2cot2x t t t t cot2x=tan =cot 5 2x= 5 +k x= 5 + k cot2x=2- 3 24 2 12 12 12 Vy suy ra: (k Z) cot2x=2+ 3 cot2x= 1 =cot 2x= +k x= + k 24 2 12 2- 3 12 Cc h nghim ny u tha mn dk. Vy *Cch 2: pt

sin2x cos2x

+

cosx sinx

=8cos2x sin2xsinx+cos2xcosx=8cos2xsinxcos2x

cosx=8cos xsinxcos2x 2

cosx=0 cosx=0 8sinxcosxcos2x=1 sin4x= 1 2

x= +k 2 4x= +k2 6 5 4x= +k2 6 +k (k Z)

x= +k 2 x= + k (k Z) 24 2 5 k x= + 24 2

*Nhn xt: Cch 2 ngn gn hn, cch 1 lm mt 1 h nghim x= *Ch : Khi t n ph t=tanx, cn th xem x=

2

2

+k (k Z) c l nghim ca phng trnh khng. Nu

khng lm nh vy c th xy ra vic mt nghim (mt nghim thng xy ra khi trong s cc iu kin phng c ngha khng c iu kin x= b) t t = cos2x (t [ 1;1] ):

2

+k (k Z) ).

VD4: Gii phng trnh : sin6x+cos6x=sin4x+cos4x

pt (

1-t 2

)3+(

1+t 2

)3=(

1-t 2

)2+(

1+t 2

)2 1+3t2=2(1+t2) t2=1 t= 1(nhn)

(k Z) x= k x= +k 2 2 c) t t = asinx + bcosx ( t a 2 + b 2 ) cos2x= 1

x=k

t = cotx + tanx ( VD5: Gii phng trnh: *iu kin :

t 2) 1 5 2 2 x +cot x- 2 (tanx+cotx)+2=0 cos

sin x 0 k sin2x 0 x (k Z) 2 cos x 05 2(tanx-cotx)+2=0 (tanx+cotx)2-

pt 1+ tan2x+ cot2x-

5 2

(tanx+cotx) + 1=0

( t 2 ), ta dc: sinxcosx sin2x t = 2 5 1 2 t - t+1=0 1 (loi t = ) t = 2 2 2 2 Vay =2 sin2x=1 2x = +k2 x= + k (k Z) sin2x 2 4 dat t= tanx + cotx = = VD6:Gii phng trnh: cos2x+5=2(2-cosx)(sinx-cosx) pt cos2x-sin2x+5=2(2-cosx)(sinx-cosx) (cosx-sinx)

1

2

[ ( cosx+sinx ) +2(2-cosx)+5] =0 (cosx-sinx) [ 4 ( cosx-sinx ) ] +5=0 4

Dat t= cosx-sinx ( t

t = -1 ( loi nghimx=5) 2 ). Khi do pt tro thanh t(t-4)+5= t2-4t+5=0 t = 5 cosx-sinx=-1 2sin(x ) =1

x- = +2k x= +2k 2 4 4 2 sin(x - ) = (k Z) 3 4 2 x- = +2k x=+2k 4 4 Vy phng trnh c 2 h nghim. *Nhn xt: Cch t n ph to thun li chuyn PTLG v dng i s, t d dng gii v kt lun. d) t t = tan

x (t R) 2

VD7: Gii phng trnh: sinx-cosx=1

*Cch 1: dat t=tan *Cch 2:

x 2

ta dc pt:

2t 1-t 2 x =1 t=1 tan =1 x= +k2 (k Z). 2 2 2 2 1+t 1+t

2 pt ( sinxcosx)= 2 2 2

2

2

x 2 sin(x - )=1 sin(x - )= 2 4 4 x 1

4

+ k2 4 3 = + k2 4 4

=

x = + k2 (k Z) 2 x = + k2* Vy phng trnh c 2 h nghim. * Nhn xt: Cch 2 n gin hn v khng lm mt nghim. *Ch : Trc khi t n ph l t = tan

x 2

, cn kim tra trc tip xem x=(2k+1) c phi l nghim khng

trnh nhng trng hp c th mt nghim. e) Mt s phng php khc VD 8: Gii phng trnh: 2( Dk: cosx 0 x Dat t =

4 2 + cos 2 x) + 9( - cosx) = 1 cosx cos 2 x

2

+ k (k Z) 4 + cos 2 x =t2+4. Khi phng trnh tr thnh: 2x cos

2 cosx

- cosx

2 - cosx = -1 t = - 1 cosx 2(t2+4)+9t-1=0 2t2+9t+7=0 7 t = 2 - cosx = - 1 2 cosx 2 cosx = -1 cosx = 2 x = + k2 cos 2 x - cosx - 2 = 0 (k Z) 2 1 x = + k2 cosx = 2cos 2 x - 7cosx - 4 = 0 3 2 cosx = 4Vy phng trnh c 3 h nghim. VD 9: Gii phng trnh: 4tan x + Dk: cosx 0 x Ta c pt 4(

2

4 cosx

-3=0

2

+k (k Z)

1 4 4 4 -1)+ -3=0 + -3=0 cosx cosx cos 2 x cos 2 x

t = 1 2 Dat t= ( t 2 ), khi phng trnh tr thnh: t2+2t -3=0 cosx t = - 3 Vy phng trnh c 2 h nghim. VD10: 2- Bi tp v d: VD11: Gii phng trnh: sinx + tan

2 =1 cosx 2 =-3 cosx

cosx=2(v nghim) cosx= 32

2 x=arccos(- ) 3

x 2

=2 (dat t = tan

x 2

)

DS: x=

2

+k2

VD12: tanxsin2x-2sin2x=3(cos2x+sinxcosx) (bin i phng trnh ri t t=tanx) DS: pt c 3 h nghim: x=

3

+ k ; x = -

4

+ k

II- Phng php a v phng trnh tch- Phng php a v phng trnh tch l mt phng php rt hu dng trong gii PTLG. 1- Phng php - Nu pt f(x) = 0 c bin i v dng f1(x). . . fn(x) = 0 th tp nghim ca f(x) = 0 l tp hp cc nghim ca cc phng trnh f1(x) = 0, f2(x) = 0, , fn(x) = 0. - t c nhn t chung, cn lu : Sin2x, sin3x, tanx, tan3x, tan2x c nhn t chung sinx. Sin2x, cos3x, tan2x, cot3x, cotx c nhn t chung cosx.

Cos2 2 , cot2 2 , sin2x, tan2x c nhn t chung 1 + cosx. Sin2 2 , tan2 2 , sin2x, tan2x c nhn t chung 1 - cosx. Cos2x, cot2x, 1 + sin2x, 1 + cotx, 1 + tanx, tanx - cotx c nhn t chung sinx + cosx. Cos2x, 1 - sin2x, cot2x, 1 - tanx, 1 - cotx, tanx - cotx c nhn t chung cosx - sinx.x x

x

x

x 2