chuuong6 - phuong trinh dang cap

8/14/2019 Chuuong6 - Phuong Trinh Dang Cap

1/7

CHNG VI: PHNG TRNH ANG CAP

2 2a sin u b sin u cos u c cos u d+ + = Cach giai :

( )Tm nghiem u k luc o cos u 0 va sin u 12

= + = =

2Chia hai ve phng trnh cho cos u 0 ta c phng trnh :

( )2 2atg u btgu c d 1 tg u+ + = + at ta co phng trnh :t tgu=

( ) 2a d t bt c d 0 + + =

Giai phng trnh tm c t = tgu

Bai 127 : Giai phng trnh( )2 2cos x 3 sin 2x 1 sin x * = +

V cosx = 0 khong la nghiem nen

Chia hai ve cua (*) cho 2cos 0 ta c

( ) ( )2 2* 1 2 3tgx 1 tg x tg x = + + at t = tgx ta co phng trnh :

22t 2 3t 0+ =

t 0 t 3 = =

Vay ( )*

= = = = + tgx 0 hay tgx 3 x k hay x k , k3

Bai 128 : Giai phng trnh( )3 3 2cos x 4 sin x 3 cos x sin x sin x 0 * + =

Khi x k th cos x 0 va sin x2

= + = = 1

th (*) vo nghiem Do khong la nghiem nen chia hai ve cua (*) cho cos3x=cos x 0

ta co (*) ( )3 2 21 4tg x 3tg x tgx 1 tg x 0 + + =

( ) ( )

+ =

+ =

= =

= + = +

3 2

2

3tg x 3tg x tgx 1 0

tgx 1 3tg x 1 0

3tgx 1 tgx

3

x k x k , k4 6


2/7

Bai 129 : Giai phng trnh

( )4 2 2 43 cos x 4 sin x cos x sin x 0 * + =

Do cosx = 0 khong la nghiem nen chia hai ve cua (*) cho 4cos x 0

Ta co : (*) 2 43 4tg x tg x 0 + =

= =

= = =

= + = +

2 2tg x 1 tg x 3

tgx 1 tg tgx tg4 3

x k x k , k4 3

Bai 130 : Giai phng trnh ( )sin 2x 2tgx 3 *+ =

Chia hai ve cua (*) cho 2cos x 0 ta c

(*) 2 22sin x cos x 2tgx 3cos x cos x cos x + = 2

( ) ( )2 22tgx 2tgx 1 tg x 3 1 tg x + + = +

3 2

t tgx

2t 3t 4t 3 0

=

+ =

( ) ( )

=

+2

t tgx

t 1 2t t 3 0=

=

= +

tgx 1

x k , k4

Bai 131 : Giai phng trnh( )3sin x sin 2 x sin 3 x 6 cos x *+ =

( ) 2 3* 2 sin x cos x 3sin x 4 sin x 6 cos x + = 3

( ) = = Khi cos x 0 ( sin x 1) th * vo nghiem

Chia hai ve phng trnh (*) cho 3cos x 0 ta c

( )*

2 3

2 2

2sin x 3sin x 1 sin x. 4cos x cos x cos x cos x+ 3 6=

( )

( ) ( )

+ + =

+ =

=

= = =

= + = + =

2 2 3

3 2

2

2tg x 3tgx 1 tg x 4tg x 6

tg x 2tg x 3tgx 6 0

tgx 2 tg x 3 0

tgx 2 tg tgx 3

x k x k , k ( vi tg3

2)


3/7

Bai 132 : (e thi tuyen sinh ai hoc khoi A, nam 2003)Giai phng trnh

( )2cos 2x 1

cot gx 1 sin x sin 2x *1 tgx 2 = + +

ieu kien sin 2x 0 va tgx 1

Ta co :( )2 22 2 cosx cos x sin xcos 2x cos x sin x

sinx1 tgx cos x sin x1cosx

= =

+ ++

( ) (= = +cos x cos x sin x do tgx 1 nen, sin x cos x 0)

Do o : ( ) ( )2 2cos x 1

* 1 cos x sin x cos x sin x sin 2xsin x 2

= +

( ) ( )

( )

=

=

= =

2

cos x sin x 1 sin 2xsin x

cos x sin x sin x cos x sin x

cos x sin x 0 hay 1 sin x cos x sin x (**)

( )

( )

= =

2

2

tgx 1 nhan so vi tgx 1

1 sin xtg x do cos x 0

cosxcos x

( )

( )

= +

+ =

= +

2

x k , k

42tg x tgx 1 0 vo nghiem

x k , k nhan do sin 2x 04

Lu y : co the lam cach khac

( ) ( )1 1

* * 1 sin 2x 1 cos 2x2 2

+ =0

= +

= +

3 sin 2 x cos 2x

3 2 sin 2x : vo nghiem4

Bai 133 : Giai phng trnh ( )sin 3x cos 3x 2 cos x 0 *+ + =

( ) ( ) ( )3 3* 3sin x 4 sin x 4 cos x 3 cos x 2 cos x + + 0==

3 33sin x 4 sin x 4 cos x cos x 0 +

V cosx = 0 khong la nghiem nen chia hai ve phng trnh cho tac

3cos x 0

( ) ( ) ( )2 3 2* 3tgx 1 tg x 4tg x 4 1 tg x 0 + + + =


4/7

( ) ( )

+ + =

=

+ =

= + =

= =

= + = +

3 2

3 2

2

tg x tg x 3tgx 3 0

t tgx

t t 3t 3 0

t tgx

t 1 t 3 0

tgx 1 tgx 3

x k x k , k4 3

Bai 134 : Giai phng trnh ( )35sin4x.cosx

6sin x 2 cos x *2cos2x

=

ieu kien : 2 2cos 2x 0 cos x sin x 0 tgx 1

Ta co : (*)3 10sin2xcos2xcosx6 sin x 2 cos x

2cos2xcos 2x 0

=

36 sin x 2 cos x 5 sin 2x cosx

tgx 1

=

( )3 26 sin x 2 cos x 10 sin x cos x * *

tgx 1

=

Do cosx = 0 khong la nghiem cua (**), chia hai ve phng trnh (**) cho

ta c3cos x

( ) 26tgx

2 10tgx* * cos x

tgx 1

=

( )2t tgx vi t 1

6t 1 t 2 10t

=

+ =

= =

= + + =

3 2

t tgx vi t 1 t tgx vi t 1

3t 2t 1 0 (t 1) (3t 3t 1) 0

= =

t tgx vi t 1 : vo nghiemt 1

Bai 135 : Giai phng trnh ( )3sin x 4 sin x cos x 0 * + =

V cosx = 0 khong la nghiem nen chia hai ve phng trnh cho cos 3x th

( ) ( )2 3 2* tgx 1 tg x 4tg x 1 tg x + + + 0=


5/7

( )

( )

=

+ + + =

=

+ + =

= +

3 2

2

t tgx

3t t t 1 0

t tgx

t 1 3t 2t 1 0

tgx 1

x k , k4

=

Bai 136 : Giai phng trnh ( ) ( )2 2tgx sin x 2 sin x 3 cos 2x sin x cos x * = +

Chia hai ve cua phng trnh (*) cho cos2x

( )

( )2 23 22

3 cos x sin x sin x cos x

* tg x 2tg x cos x

+

=

( ) = +3 2 2tg x 2tg x 3 1 tg x tgx

( ) ( )

+ =

=

+ =

=

+ =

= =

= + = +

3 2

3 2

2

tg x tg x 3tgx 3 0

t tgx

t t 3t 3 0

t tgx

t 1 t 3 0

tgx 1 tgx 3

x k x k , k4 3

Bai 137 : Cho phng trnh( ) ( ) ( ) ( ) ( )3 24 6m sin x 3 2m 1 sin x 2 m 2 sin x cos x 4m 3 cos x 0 * + + =

a/ Giai phng trnh khi m = 2

b/ Tm m e phng trnh (*) co duy nhat mot nghiem tren 0,4

Khi x2= + k th cosx = 0 va sin x 1= nen

(*) thanh : ( ) ( )4 6m 3 2m 1 0 =

1 0 vo nghiem =

chia hai ve (*) cho 3cos x 0 th

( ) ( ) ( ) ( ) ( ) ( ) ( )3 2 2* 4 6m tg x 3 2m 1 tgx 1 tg x 2 m 2 tg x 4m 3 1 tg x 0 + + + + =2

)

( ) ( ) (3 2t tgx

t 2m 1 t 3 2m 1 t 4m 3 0 * *

=

+ + + =


6/7

( ) ( )2t tgx

t 1 t 2mt 4m 3 0

=

+ =

a/ Khi m = 2 th (*) thanh ( ) ( )2t tgx

t 1 t 4t 5 0

= + =

= = + tgx 1 x k , k

4

b/ Ta co : x 0,4

th [ ]tgx t 0,1=

Xet phng trnh : ( )2t 2mt 4m 3 0 2 + =

( )2t 3 2m t 2 =

2t 3 2mt 2

=

(do t = 2 khong la nghiem)

at ( ) ( )2t 3

y f t Ct 2

= =

va (d) y = 2m

Ta co : ( )( )

2

2

t 4ty ' f t

t 2

+= =

3

Do (**) luon co nghiem t = 1 [ ]0,1 tren yeu cau bai toan

( ) ( )

( ) ( )

=

=

d y 2m khong co iem chung vi C

d cat C tai1 iem duy nhat t 1

32m 2m 22 <

3m m

4 < 1

Cach khac :Y C B T f(t) = ( )2t 2mt 4m 3 0 2 + = vo nghiem tren [ .),0 1

Ta co (2) co nghiem [ ]( )

, ( ). ( ) ( )

af

f f hayaf

S

0

0 0

0 1 0 1 0 1 0

0 12


7/7

( ) ( )

m m

mm m hay

m

m

+

>

>

24 3 0

4 3 04 3 2 2 0

2 2 0

0 1

m 3

14

Do o (2) vo nghiem tren [ ), (m hay m hay f ) < >30 1 1 1 04

=

3m m

41 <

BAI TAP

1. Giai cac phng trnh sau :

a/ 3 2cos x sin x 3sin x cos x 0+ =

b/ ( ) ( )2

sin x tgx 1 3 sin x cos x sin x 3+ = +=

c/ 22 cos x cos 2x sin x 0+ +

d/3

23

1 cos xtg x

1 sin x

=

e/ 3 2 2 3sin x 5 sin x cos x 3 sin x cos x 3 cos x 0 + =

f/ 3 2cos x sin x 3 sin x cos x 0+ =

g/ 1 tgx 2 2 sin x+ =

h/ 3 3sin x cos x sin x cos x+ =

k/ 2 23tg x 4tgx 4 cot gx 3 cot g x 2 0+ + + + =

m/ ( sin ) cos ( )cos

x xtg x tgxx

+ + =2 22

3 13 84 2

0

n/sin x cos x

1sin2x

+=

2. Cho phng trnh : ( ) ( )2 2sin x 2 m 1 sin x cos x m 1 cos x m+ + =

a/ Tm m e phng trnh co nghiem

b/ Giai phng trnh khi m = -2 [ ]( )S : m 2,1

Th.S Phm Hng DanhTT luyn thi i hc CLC Vnh Vin

chuuong6 - phuong trinh dang cap

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