B-i Du-ng K- Nang Gi-i Toan

S GIO DC V O TO

TRNG PH THNG TRUNG HC CHUYN VNH PHC

SNG KIN KINH NGHIM

BI DNG T DUY GII TON

CHO HC SINH THNG QUA

CC PHP BIN I LNG GIC

Ngi thc hin : o Ch Thanh

T : Ton Tin

M : 55

S in thoi : 0985 852 684

Email : thanhtoan@vinhphuc,edu.vn

Nm 2012- 2013

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MC LC

Trang

M u 3

PHN I : S DNG LNG GIC GII MT S BI

TON I S

Dng 1 : Mt s bi tp i s s dng h thc lng c bn 6

Bi tp t luyn 11

Dng 2 : S dng cc cng thc cng cung 12

Bi tp t luyn 15

Dng 3: S dng cc kt qu bit ca tam gic lng 16

Bi tp t luyn 20

Dng 4:Gii phng trnh, h phng trnh s dng lng gic 21

Bi tp t luyn 23

PHN II - KT LUN V KIN XUT

25

Ti liu tham kho 27

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M U

1. L do chn ti:

Trong giai on hin nay, vic cp bch trnh t nc c nguy c tt hu

v kinh t, khoa hc k thut l phi nng cao cht lng gio dc, thay i cn

bn phng php dy hc.Hc sinh phi pht huy tnh tch cc, t gic ch ng t

duy sng to, bi dng phng php t hc hc sinh.

Bn cnh , hm s lng gic v phng trnh lng gic l khi nim

kh, tru tng i vi hc sinh THPT, phn phi thi gian ging dy v hc tp

chim thi gian rt t v vy gii cc bi tp lng gic i vi nhiu hc sinh l

kh kh khn.

V vy nng cao cht lng dy v hc ca hc sinh i vi mn ton,

gip cc em thy c cc mi lin quan gia cc phn c hc trong b mn

ton vi nhau ti tng hp , phn loi mt s bi ton i s c th gii bng cc

kin thc lng gic nhm gip cc em c cch nhn mi , phng php mi

gii mt s bi tp i s. Mt khc nhm gip cc em n luyn cc kin thc

hc chng hm s v phng trnh lng gic

2. Mc ch nghin cu

Mc ch ca bn sng kin kinh nghim ny nghin cu mt s bi ton i

s c gii bng phng php khc nhm gp phn rn luyn yu t t duy sng

to cho hc sinh .

3. Gi thuyt khoa hc

S dng cc kin thc lng gic gii mt s bi tp i s nhm bi

dng t duy sng to cho hc sinh , gp phn i mi phng php dy hc trong

giai on hin nay v nng cao cht lng dy hc ton trng ph thng trung

hc

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4. Nhim v nghin cu

Xy dng v khai thc h thng bi tp i s ph hp vi s pht trin t

duy sng to cho hc sinh.

5. Phng php nghin cu

- D gi, quan st vic dy hc ca gio vin v vic hc ca hc sinh trong

qu trnh khai thc cc bi tp sch gio khoa, cc bi tp nng cao.

- Tin hnh thc nghim s phm vi lp hc thc nghim v lp hc i

chng trn cng mt lp i tng.

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S DNG LNG GIC GII MT S BI TON I S

Dng 1: Mt s bi tp i s s dng h thc lng c bn

Ta bit mt s h thc lng c bn hc sinh dc hc t lp 9, song

vn dng cc kin thc ny cn hn ch. thy c vai tr ca cc h thc c

bn ca lng gic trong ton hc ti phn loi ra mt s bi tp sau.

Cc h thc c bn v h qu:

1/ 2 2sin cos 1

2/ sin

tgcos

3/ cos

cotgsin

4/ 22

11 tg

cos

5/ 22

11 cotg

sin

6/ tg .cotg 1

Sau y l mt s bi tp minh ha

Bi 1 :

Cho a2 + b2 = c2 +d2 = 1 Chng minh rng : 1ac bd

Bi gii :

Do a2 + b2 = 1 nn t sin = a; cos = b;

Do c2 + d2 = 1 nn t sin = c; cos = d;

Thay vo ac + bd th ta c sin .sin+cos .cos = cos( - )

Li c cos 1x nn ta c 1ac bd

Bi 2 : Cho x;y tha mn 2 23 3 3x x y y (1).Tnh x + y Bi gii

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T (1) chia hai v cho 3 ta c

2 23 3

133 3 3

x x y y

T biu thc cho ta thy x>0,y>0

Vi a 0;2

nn ta c th t:

2 2

2 2

( ) 1 tan ( ) 1 tan (2)3 3 3 3

( ) 1 cot ( ) 1 cot (3)3 3 3 3

x x x xa a

y y y ya a

Bnh phng hai v ca (2) v (3) ta c

2 22

2 22

1 tan 2 tan3 3 3

1 cot 2 cot3 3 3

x x xa a

y y ya a

Hay

22

22

tan 11 tan 2 tan 2 tan cot (4)

tan3 3

cot 11 cot 2 cot 2 cot tan (5)

cot3 3

x x aa a a a

a

y y aa a a a

a

Cng (4) v (5) ta c: x+ y = 0 (pcm).

Bi 3: Cho x;y;z i mt khc nhau tha mn : (x+ z)(z + y) = 1

Chng minh rng :

2 2 2

1 1 14

( ) ( ) ( )x y z x z y

Bi gii : Do (x+ z)(z + y) = 1 Vi 4

a k

ta t : x + y = tan a; y + z = cot a nn x y = tan a cot a.

Do ta cn chng minh :

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2 22 2 2 2 2

1 1 1 14 tan cot 4

(tan cot ) t an cot tan cot 2a a

a a a a a a

2 2

2 2

1tan cot 2 2 (2)

tan cot 2a a

a a

Ta thy (2) ng theo bt ng thc Cauchy.

Bi 4: Cho 0 < x;y;z < 1 tha mn : xyz = (1 x )(1 y )(1 z ) (1)

Chng minh rng : 2 2 23

4x y z (2)

Bi gii :

T gi thit : xyz = (1 x )(1 y )(1 z ) ta c : 1 1 1

. . 1x y z

x y z

Vi a;b;c 0;4

Ta t :

1 1tan . cot 1 .tan cot

1 tan cot

1 1cot . cot 1 .cot cot

1 cot cot

1 1tan . 1 .tan

1 tan

xa b x x a b x

x a b

ya b y y a b y

y a b

zb z z b z

z b

Vy Bt ng thc (2) tng ng:

22 2

1 1 1 3

(1 tan ) 4(1 tan . cot ) (1 cot . cot ) ba b a b

Theo bt ng thc Bunhiacopxki ta c:

2 2

2 21 tan cot (1 tan )(1 cot ) ; 1 cot cot (1 cot )(1 cot )a b a b a b a b

iu phi chng minh tng ng :

22 2

2 2 2 2

2 2 2 2 2

1 1 1

(1 tan )(1 tan . cot ) (1 cot . cot )

2 tan cot 1 1 cot cot cot 1

(1 tan )(1 cot )(1 cot ) (1 tan ) 1 cot 1 cot (1 cot )

ba b a b

a a b b b

a a b b b b b

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Ta chng minh :

22 2 2

2

cot cot 1 34(cot cot 1) 3(1 cot ) (1 cot ) 0

(1 cot ) 4

b bb b b b

b

ng

Bi 5 : Cho : a2 + b2 2a 4b+ 4 = 0 (*)

Chng minh rng :

2 2 2 3 2(1 2 3) (4 2 3) 4 3 3 2A a b ab a b

Bi gii :

T (*) ta c ( a 1 )2 + (b 2 )2 =1 Ta t a = 1+ sin x; b = 2 + cos x

Thay vo A ta c :

2 2sin cos 2 3sin .cos 3sin 2 cos2 2A x x x x x x (pcm)

Bi 6 : Chng minh rng :

2 23 2 3 2

3 12 2

x x x

Bi gii :T K bi ton ta c 1 sin2 2

x x a a

Thay vo :

2 2 2

0

1 cos2 13 1 3sin sin .cos 3 sin 2

2 2

3 1 3( 3 cos2 sin 2 ) cos(30 2 )

2 2 2

ax x x a a a a

a a a

Ta c - 1 cos(300 + 2a ) 1 nn 2 23 2 3 2

3 12 2

x x x

Bi7: Cho 1 a 3 Chng minh rng : 3 24 24 45 26 1S a a a

Bi gii : Ta c : -1 a 2 1 Nn ta t : a 2 = cos x (0 x )

Thay vo biu thc S ta c

3 24(2 cos ) 24(2 cos ) 45(2 cos ) 26 cos3 1S x x x x (pcm)

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Bi 8: Chng minh rng :

2 1 3

2 ( ; 1)a

A a aa

Bi gii : t 1

cosa

x ( 0 ;

2x x

)

Ta c : 2 21

1 3cos 1 cos 3.cos sin 3.cos 2

1

cos

xA x x x x

x

Bi 9 : Chng minh rng :

3

32 2

3 41 ( )

1 1

x xS x

x x

Bi gii : t x = tan a ( 2 2

a

) Khi

33 3

2 2 3

3

3tan tan4 3tan .cos 4.tan .cos

1 tan (1 tan )

3sin 4sin sin3 1

a aS a a a a

a a

a a a

Bi10 : Chng minh rng : 2 2( )(1 ) 1

( ; )(1 )(1 ) 2

a b aba b

a b

Bi gii :t a = tan x; b = tan y vi x; y ;2 2

th ta c

2 2 2 2

2 2

2 2

( )(1 ) (tan tan )(1 tan .tan )

(1 )(1 ) (1 tan )(1 tan )

sin( ).cos( )cos .cos .

cos .cos

1 1sin( ).cos( ) sin 2( ) ( ; )

2 2

a b ab x y x y

a b x y

x y x yx y

x y

x y x y x y x y

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Mt s bi tp t luyn

Bi 1/Chng minh rng :

2 21 1) 1 ( , 1; 1)

.

) ( )( ) ( ; ; ; 0)

a ba a b a b

a b

b ab cd a c b d a b c d

Bi 2 : Cho a2 +b2 = c2 + d2 =1 Chng minh rng: 2 a(c d) b(c d) 2

Bi 3 :Cho a2 + b2 = 1 : Chng minh rng: 2 2

2 2

2 2

1 1 25a b

a b 2

Bi 4 :Chng minh rng: 2 23 2 3 2

3x x 1 x2 2

( 1 x -1)

Bi 5:

Chng minh rng: 3 32 21 1 a 1 a 1 a 2 2 2 2a (1 a -1

Bi 6 : Chng minh rng: 22a a 3a 3 2 ( 2 a 0 )

Bi 7 : Chng minh rng: 3 24a 24a 45a 26 1 a 1;3

Bi 8 : Cho x2 + y2 = 1 . CMR : 5 5 3 316( ) 20( ) 5( ) 2x y x y x y

Bi 9

Cho 0 < x;y;z < 1 tha mn : xyz = (1 x )(1 y )(1 z )

Chng minh rng : 2 2 2

1 1 112

x y z

Bi 10 Cho x;y tha mn 2 24 4 4x x y y .Tnh x + y

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Dng 2 : S dng cc cng thc cng cung

Mt s bi ton s dng cc cng thc cng cung v cc cng thc bin i khc.

Ta nu li cc cng thc hc sau :

1 Cng thc cng - tr:

1/ sin a b sin a.cos b sin b.cos a

2/ sin a b sin a.cos b sin b.cos a

3/ cos a b cos a.cos b sin a.sin b

4/ cos a b cos a.cos b sin a.sin b

5/ tga tgb

tg a b1 tga.tgb

6/

tga tgbtg a b

1 tga.tgb

7/ cotga.cotgb 1

cotg a bcotga cotgb

cotga cotgb 18 / cotg a b

cotga cotgb

2. Cng thc gc nhn i:

1/ 2 2

sin 2a 2 sin a.cos a sin a cos a 1 1 sin a cos a

2/ 2 2 2 2cos2a cos a sin a 2 cos a 1 1 2 sin a

3/ 2

2tgatg2a

1 tg a

4/

2cot g a 1cotg2a

2 cotga

3. Cng thc gc nhn ba:

1/ 3sin 3a 3 sin a 4 sin a 2/ 3cos3a 4 cos a 3 cosa

3/ 3

2

3tga tg atg3a

1 3tg a

4/

3

2

cot g a 3 cotgacotg3a

3 cotg a 1

4. Cng thc h bc hai:

1/ 2

2

2

1 cos2a tg asin a

2 1 tg a

2/

22

2

1 cos2a cotg acos a

2 1 cotg a

3/ 21 cos2a

tg a1 cos2a

4/

1sin a cos a sin 2a

2

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5. Cng thc h bc ba:

1/ 3 1sin a 3 sin a s in3a4

2/ 3 1cos a 3 cos a cos 3a4

6. Cng thc biu din sin x, cos x, tgx qua x

t tan2

:

1/ 2

2tsin x

1 t

2/

2

2

1 tcos x

1 t

3/ 2

2ttgx

1 t

4/

21 tcotgx

2t

7. Cng thc bin i tch thnh tng:

1/ 1cos a.cos b cos a b cos a b2

2/ 1sin a.sin b cos a b cos a b2

3/ 1sin a.cos b sin a b sin a b2

8. Cng thc bin i tng thnh tch:

1/ a b a b

cos a cos b 2 cos .cos2 2

2/ a b a b

cos a cos b 2 sin .sin2 2

3/ a b a b

sin a sin b 2 sin .cos2 2

4/ a b a b

sin a sin b 2 cos .sin2 2

Sau y l mt s bi tp minh ha.

Bi 1:

Cho x+ y + z = xyz ,vi K mu s khc khng Chng minh rng

3 3 3 3 3 3

2 2 2 2 2 2

3 3 3 3 3 3. .

1 3 1 3 1 3 1 3 1 3 1 3

x x y y z z x x y y z z

x y z x y z

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Bi gii :T gi thit x+ y + z = xyz v biu thc

3

2

3

1 3

x x

x

ta thy n tng

t nh cng thc nhn ba, nn ta t x = tan a, y = tan b; z = tan c

vi a; b;c ; \2 2 6

thay vao gi thit ta c

tana. + tan b + tan c =tan a.tan b.tan c

Theo kt qu bit th a + b + c = k (k nguyn)

Li c : 3 3

2 2

3 3 tan tantan 3

1 3 1 3tan

x x a aa

x a

3 3

2 2

3 3 tan tantan 3

1 3 1 3tan

y y b bb

y b

3 3

2 2

3 3tan tantan 3

1 3 1 3tan

z z c cc

z c

V :3a +3 b +3 c = 3k nn tan 3a+ tan 3b +tan 3c = tan3a.tan3b.tan3c

Bi 2 : Chng minh rng : 2 21 ( )(1 ) 1

: ,2 (1 )(1 ) 2

x y xyx y

x y

Bi gii : t x=tan a;y= tanb ( ;2 2

a b

) Ta c

2 2 2 2

2 2

sin( )cos( )( )(1 ) (tan a tan )(1 t ana.tan ) 1cos .cos .cos .cos sin 2( )

1(1 )(1 ) (1 tan ).(1 tan ) 2cos .cos

a b a bx y xy b b a b a bVT a b

x y a ba b

Vy : 2 21 ( )(1 ) 1

: ,2 (1 )(1 ) 2

x y xyx y

x y

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Bi3: Cho a,b > 0 Chng minh rng : 2 2

11

1 1

ab

a b

Bi gii : t a=tan x;b= tany ( ;2 2

x y

) Ta c

2 2 2 2 2 2

2 2 2 2

1 1

1 1 1 1 1 1

1 t anx.tan

1 t an x 1 tan 1 t an x 1 tan

cos .cos sin .sin cos( ) 1

ab ab

a b a b a b

y

y y

x y x y x y

Mt s bi tp t luyn:

Bi 1: Cho (xy +1)(yz +1)(zx + 1) 0 Chng minh rng

. .1 1 1 1 1 1

x y y z z x x y y z z x

xy zy xz xy zy xz

Bi 2 : Cho x+ y + z = xyz . Chng minh rng

2 2 2 2 2 2( 1)( 1) ( 1)( 1) ( 1)( 1) 4x y z y x z z y x xyz

Bi 3 : Cho xy +yz + xz = 1 Chng minh rng

2 2 2 2 2 2

4

1 1 1 (1 )(1 )(1 )

x y z xyz

x y z x y z

Bi 4 : Cho x; y; z l ba s thc bt k .Chng minh rng

2 2 2 2 2 21 1 1 1 1 1

x y y z z x

x y z y x z

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Dng 3 S dng cc kt qu bit ca tam gic lng:

Mt s bi ton s dng cc kt qu ca tam gic lng :

Ta c mt s kt qu sau

Kt qu 1 : Vi a;b;c l ba s thc dng tha mn ab + bc + ca = 1 th tn ti

ABC c cc gc tha mn a = tan2

A; b = tan

2

B; c = tan

2

C

Gii :Do a;b > 0 nn tn ti hai gc 0 ;2 2 2

A B sao cho a = tan

2

A; b = tan

2

B

T gi thit:t 0 ( )2 2 2 2 2

C A B th 0< A,B.C < 1800 v A+ B +C = 1800

Nn tn ti ABC c cc gc tha mn a = tan2

A; b = tan

2

B; c = tan

2

C

Kt qu 2:Vi a;b;c l ba s thc dng tha mn ab + bc + ca = 1;

abc +a+b +c < 2 Khi th tn ti ABC nhn c cc gc tha mn

a = tan2

A; b = tan

2

B; c = tan

2

C

Gii : Theo trn ta c th tn ti ABC c cc gc tha mn :

a = tan2

A; b = tan

2

B; c = tan

2

C. Ta c/m ABC nhn hay cosA.cosB.cosC > 0

Ta c

2 2 2

2 2 2

1 1 1. . 0 (1 )(1 )(1 ) 0

1 1 1

1 ( ) ( ) 0

2 : 1

a b ca b c

a b c

a b c ab bc ca abc

abc a b c do ab bc ca

Kt qu 3 : Trong ABC c a = tan2

A.Chng minh rng

2 2

1sin ; cos

2 21 1

A a A

a a

(C/m n gin )

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Kt qu 4: Vi a;b;c l ba s thc dng tha mn ab + bc + ca = abc th tn ti

ABC c cc gc tha mn 1

a = tan

2

A;

1

b = tan

2

B;

1

c = tan

2

C

C/m: T ab + bc + ca = abc ta c 1 1 1

1ab bc ca

theo kt qu 1 ta c pcm

Kt qu 5 : Vi a;b;c l ba s thc dng tha mn ab + bc + ca = abc;

1 +ab+bc +ca < 2abc. Khi th tn ti ABC nhn c cc gc tha mn

1

a = tan

2

A;

1

b = tan

2

B;

1

c = tan

2

C (C/m tng t nh kt qu 4).

Kt qu 6:Trong ABC c 1

a= tan

2

A.Chng minh rng

2

2 2 2 2

2 1 1sin ; cos ; sin ; cos

1 1 2 21 1

a a A A aA A

a a a a

Nh cc kt qu ny ta c mt s ng thc; bt ng thc i s c gii bi cc

bt ng thc; ng thc lng gic.

Bi1: Vi a;b;c l ba s thc dng tha mn ab + bc + ca = 1 :

Chng minh rng:

2 2 2

2 2 2 2 2 2

1 1 1 41

1 1 1 (1 ).(1 ).(1 )

a b c abc

a b c a b c

Theo ng thc :cos cos cos 1 4sin .sin .sin2 2 2

A B CA B C v kt qu 6 ta c

pcm.T : sin2A + sin2B + sin2C = 4sinA.sinB.sinC v kt qu 6 ta c

Bi2: Vi a;b;c l ba s thc dng tha mn ab + bc + ca = 1

Chng minh rng:

2 2 2

2 2 2 2 2 22 2 2

(1 ). (1 ). (1 ). 8

(1 ).(1 ).(1 )1 1 1

a a b b c c abc

a b ca b c

Bi3:Cho x, y, z > 0 , tha mn : xy +yz + zx = 1

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Tnh : 2 2 2 2 2 2

2 2 2

(1 )(1 ) (1 )(1 ) (1 )(1 )

1 1 1

y z x z y xS x y z

x y z

Bi gii :

T gi thit xy +yz + zx = 1 ;x, y, z > 0 ta thy biu thc trn tng t nh

ng thc trong ABC : tan .tan tan .tan tan .tan 12 2 2 2 2 2

A B B C C A

V vy t tan ; tan ; tan2 2 2

A B Cx y z th ta c

2 2 2 2 2 2

2 2 2

(1 )(1 ) (1 )(1 ) (1 )(1 )1

1 1 1

y z x z y xS x y z

x y z

Bi4: Cho ba s : x;y;z tha mn : xyz = x+ y+ z chng minh rng

2 2 2

3 3

21 1 1

x y z

x y z

Bi gii : t x = tanA; y = tanB; z = tan C t xyz = x+ y+ z ta c

tanA + tanB + tan C = tanA.tanB. tan C nn A + B + C =

Thay vo bt ng thc ta c :

2 2 2

3 3sin sin sin

21 1 1

x y zA B C

x y z

Ta cn chng minh : 3 3

sin sin sin2

A B C

Vi ; 0 ; )A B A B Ta c :

sin sin 2sin .cos 2sin . ( 0 cos 1)2 2 2 2

A B A B A B A BA B Do

Vy

23 3sin sin sin sin 2sin 2sin 4sin 4sin3 2 2 2 3

C A B CA B

A B C

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Vy 3 3

sin sin sin2

A B C (pcm).

Bi 5 : T bt ng thc : cos cos 2sin2

CA B (*) v kt qu 6 ta c bi

ton sau : Cho a;b c dng tha mn ab + bc + ca = 1 .Cmr :

2 2

2 2 2

1 1 2

1 1 1

a b c

a b c

Bi 6: Cho a;b c dng tha mn ab + bc + ca = 1 .Cmr :

2 2 2

2 2 2 2 2 2

1 1 1

1 1 1 1 1 1

a b c a b c

a b c a b c

Bi gii :

Vi a;b c dng tha mn ab + bc + ca = 1 th tn ti ABC c

1

a= tan

2

A;

1

b= tan

2

B;1

c= tan

2

C nn VT = cos cos cosA B C

VP = sin sin sin2 2 2

A B C

Hay ta cn chng minh : cos cos cos sin sin sin2 2 2

A B CA B C y l bt

ng thc d (C/m da vo (*)) Nn ta c pcm

Bi 7: Vi a;b;c l ba s thc dng tha mn

ab + bc + ca = 1; abc +a+b +c < 2 :

Chng minh rng 2 2 2

2 2 23 3

1 1 1

a b c

a b b

Bi gii :

Vi a;b;c l ba s thc dng tha mn ab + bc + ca = 1; abc +a+b +c < 2

Khi th tn ti ABC nhn c cc gc tha mn a = tan2

A; b = tan

2

B;

c = tan2

C Nn tan A =

2

2

1

a

a; tanB =

2

2

1

b

b; tan C =

2

2

1

c

c

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Vy pcm tng ng vi : tanA + tanB + tan C 3 3 y l BT c bn

Mt s bi tp t luyn:

Bi 1 Vi a;b;c l ba s thc dng tha mn ab + bc + ca =1.

Chng minh rng: 2 2 2

2 2 2

1 1 1 1 1 1. . 4

1 1 1

a b c

a b c

Bi 2:

2 2 2

2 2 2

1/ , , 0 1.

2 3. . . 1 ( _1999)

2/ , , 0 1 4 .

1 1 13

1 1 13/ , , 1 2.

1 1 1

Cho a b c sao cho a b c Cmr

a b c abc Poland

Cho a b c sao cho a b c abcCmr

ab cb ac

Cho x y z sao cho Cmrx y z

x y z x y z

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Dng 4 :Gii phng trnh, h phng trnh s dng lng gic

Mt s phng trnh, bt phng trnh trong phn mn i s ngoi cch gii

i s thng thng cn c cch gii lng gic, m nh n bi ton c gii

nhanh gn hn. Sau y l mt s v d

Bi 1: Gii phng trnh sau :

3 24 3 1 : 1x x x Dk x

Bi gii : t cos 0;x t t Ta c phng trnh : 4.cos3 t cost = sin t

hay

83 2

32cos3 cos2 4

3 252

8

t

t t k

t t t

t t k

t

Vy phng trnh c nghim: 5 3

cos ;cos ;cos8 8 4

Bi 2 : Gii phng trnh : 2 21 1 1 2 1x x x

Bi gii

Ta c : k 1x theo ta t : sin ;2 2

x t t

vy phng trnh

31 cos sin (1 2cos ) 2 cos sin sin 2 2.cos 1 2 sin 0

2 2 2

16

2

12

t t tt t t t t

tx

xt

Vy phng trnh c hai nghim x = 1; x =

Bi3 : Gii h sau : 2

2

11 1:

11 3

xx yDK

yy x

Bi gii

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Vi k trn ta t :

cos 0;

sin ;2 2

y b b

x a a

ta c h:

2sin .cos 1sin sin 1 2 2

cos cos 32cos .cos 3

2 2

a b a ba b

a b a ba b

Ta thy cos 02

a b nn

1tan

2 33

sin sin 1 cos 13 6 6

a ba b

N n a a a a

Vy nghim ca h phng trnh :

1sin

1 36 2;

2 23cos

6 2

x

y

Vy (x;y )= 1 3

;2 2

l nghim ca h phng trnh

Bi4 : Gii h phng trnh :

2 2 1

2( )(1 4 ) 3

x y

x y xy

Bi gii :

t

cos 0;2

sin 0;2

y b b

x a a

khi t h ta c phng trnh sau:

6 6

sin cos 1 2sin 2 sin cos 2sin 2 .sin 2sin 2 .cos2 2

132

6 5 6 3sin 3 cos3 cos 3 cos

7 22 4 6

36 3

k

k

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Theo K bi ton ta c 7 31 55 11 35 59

; ; ; ; ;6 6 6 6 6 6

Vy nghim (x; y) = (sin ; cos ) vi 7 31 55 11 35 59

; ; ; ; ;6 6 6 6 6 6

Bi 5 :

Gii h phng trnh sau : 2

3 2

2 (1 )

3 (1 3 )

y x y

x x y x

(HSG _ Qung Bnh)

Bi gii : Ta thy 1

1;3

y x khng l nghim ca h nn h phng trnh

tng ng vi

2

3

2

2

1

3

1 3

yx

y

x xy

x

Khi ta t tan ( ; )2 2

y

T phng trnh (1) ta c : 2

2 tantan 2

1 tanx x

T phng trnh (2) ta c : 3

2

3 tan 2 tan 2tan 6

1 3tan 2y

Nn ta c

tan tan 65

k

Theo iu kin ta c 0; 1; 2k k k

Vy nghim ca h :

2 2 4 2 4 2

0;0 ; tan ; tan ; tan ; tan ; tan ; tan tan ; tan5 5 5 5 5 5 5 5

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Mt s bi tp t luyn

Bi 1: Gii phng trnh sau :

2 4 2

2

2

1/ 8 (2 1)(8 8 1) 1

2 / 2 21

23 / 1 1

3

x x x x

xx

x

x x x x

Bi 2 : Gii cc phng trnh v t sau :

2 3

3 2 3 2

2

22

2

2 2

) 1 4 3

) (1 ) 2(1 )

35)

121

1 2 1) 1 2

2

1) 2 1

2

) 2 1 4 2(8 1)

a x x x

b x x x x

xc x

x

x xd x

xe x

f x x x

Bi3: Gii h sau :

3

2

2 2 1 3 1

2 1 2 1

y x x x y

y x xy x

Nghim ca h :

3 3cos ; 2.sin

10 20

Bi4 : Gii cc h sau :

2 2 2

2 2 2 3 4 2 2

4 3 2 2 2

2 1 2 ( ) 1

1/ 3 3 1 2 / 1 2 2 2

4 4 6 (3 1) 2 ( 1)

z xyz z x y x y

x y xy x y y z xy zx yz

z zy y y y z y x x x

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PHN II - KT LUN V KIN XUT

1.Kt lun:

Qua thi gian nghin cu sng kin v vn dng sng kin vo ging dy ti rt

ra c mt s kt qu sau:

hnh thnh phng php t duy,suy lun ton hc cho hc sinh THPH.Bn

cnh sng kin ny cng gip cho gio vin, hc sinh luyn tp k nng gii cc

bi ton i s v cc php bin i lng gic, thc y qu trnh ging dy v hc

tp mn Ton c tt hn.

Gio vin:

To ra tm th hng th, sn sng lnh hi tri thc mn hc thc y tnh tch

cc t duy ca hc sinh, khc phc tm th ngi, s khi tip cn ni dung mn hc.

Nu c nhiu hnh thc t chc dy hc kt hp mn hc s tr ln hp dn v

ngi hc thy c ngha ca mn hc.

V phng php dy hc, cn ch hn n phng php lnh hi tri ca HS,

gip cc em c kh nng tip thu sng to v vn dng linh hot tri thc trong tnh

hung a dng

Rn luyn cho hc sinh thi quen, tnh k lut trong vic thc hin cc k nng

gii ton thng qua vic luyn tp; nhm khc phc tnh ch quan, hnh thnh tnh

c lp, tnh t gic ngi hc, thng qua hnh thnh v pht trin nhn cch

ca cc em.

Phi thng xuyn hc hi trau ri chuyn mn tm ra phng php dy hc

ph hp.

Phi nhit tnh, gng mu quan tm ti hc sinh, gip cc em cc em

khng cm thy p lc trong hc tp.

Lun to ra tnh hung c vn , kch thch hng th tm ti hc tp hc sinh.

Cho hc sinh thy ng dng ca l thuyt vo thc hnh.

t ra cu hi gi m ph hp vi i tng hc sinh.

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Hc sinh:

Kh nng tip thu kin thc mi tt hn khi bit phn tch mt bi ton .Cc

em c th vn dng cc qui trnh hay cc phng php gii cc v d vo cc bi

tp c th.Cc em bit huy ng cc kin thc c bn, cc tri thc lin quan

gii cc bi tp ton,bit la chn hng gii bi tp ph hp.Trnh by li

gii hp l cht ch, ngn gn v r rng hn

2. Khuyn ngh:

a)Nn c cc chuyn t chn gio vin v hc sinh c th trao i thng

thn vi nhau v cc vn , t c th rt ra cc phng php ph hp vi tng

i tng hc sinh.

b) Trong lp gio vin nn phn nhm hc theo trnh nhn thc ca cc em

Do kinh nghim cn thiu, thi gian nghin cu v ng dng cha di nn

ti ca ti khng trnh khi cn nhiu hn ch..

Rt mong c s ng gp ca cc ng nghip ti c th hon thin hn

ti ca mnh.

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TI LIU THAM KHO

[1]. Sch gio khoa,sch bi tp 11(c bn v nng cao),

NXB Gio Dc Nm 2007

[2].Tuyn chn theo chuyn chun b thi tt nghip THPT v thi vo H- C mn ton,Nh xut bn Gio dc Nm 2010 .

[3]. thi tuyn sinh. Mn Ton,Nh xut bn Gio dc Nm 1994.

[4]. Phan Huy Khi .Tuyn tp cc chuyn luyn thi i hc. Nh xut bn H Ni Nm 2002.

[5].

. nm 2012

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