Chuyên đề 8. Lượng giác

7/31/2019 Chuyn 8. Lng gic

1/13

Chuyen e 8: LNG GIACTOM TATGIAO KHOA

A. KIEN THC C BAN:I. n v o goc va cung:

1. o:

.y

o180

O betgoc01Goc 1801

=

33

2. Radian: (rad)

rad0180 =

3. Bang oi o sang rad va ngc lai cua mot so goc (cung ) thong dung:

o 00 300 450 600 900 1200 1350 1500 1800 3600

Radian 06

4

3

2

3

2

4

3

6

5 2

II. Goc lng giac & cung lng giac:1. nh ngha:

x

y

B

M

(im gc)

+

t

O A

(im ngn)

2kAB +=

x

y

+

2. ng tron lng giac:

So o cua mot so cung lng giac ac biet:

k

CA

k

C

k

A

+

+

++

2DB,

k,

22

-D

2k

22B

2k

(tia gc)Z)(k2),( += kOyOx

t

(tia ngn)

O

+

x

y

OC A

B

D


2/13

34

III. nh ngha ham so lng giac:

1. ng tron lng giac: A: iem goc x'Ox : truc cosin ( truc hoanh )

+

x

y

OC A

B

D

1

1

1=R1

1

'x

'uu

t

't'y

y

'

Oy : truc sin ( truc tung ) t'At : truc tang u'Bu : truc cotang

2. nh ngha cac ham so lng giac:a. nh ngha: Tren ng tron lng giac cho AM= .

Goi P, Q lan lt la hnh chieu vuong goc cua M tren x'Ox va y'OyT, U lan lt la giao iem cua tia OM vi t'At va u'Bu

Ta nh ngha:

cos

sin

tg

cot

OP

OQ

AT

g BU

=

=

=

=

t

't

ty

b. Cac tnh chat :

Vi moi ta co :1 sin 1 hay sin 1

1 cos 1 hay cos 1

tg xac nh2

k

+

cotg xac nh k c. Tnh tuan hoan

sin( 2 ) sin

cos( 2 ) cos

( )

cot ( ) cot

k

k

tg k tg

g k g

+ =

+ =

+ =

+ =

)( Zk

'u

x

u

'y

t

1

Q

'x O

B

T

M

AP

U

Trc cosin

Trc sinTrc cotang

+

Trc tang


3/13

IV. Gia tr cac ham so lng giac cua cac cung (goc ) ac biet:Ta nen s dung ng tron lng giac e ghi nh cac gia tr ac biet

- 3

-1

- 3 /3

(iem goc)

t

t'

y

y'

xx'

uu'

- 3 -1 - 3 /3

1

1

-1

-1

- /2

5/6

3/4

2/3

-/6

-/4

-/3

-1/2

- 2 /2

- 3 /2

-1/2- 2 /2- 3 /23 /22 /21/2

3 /2

2 /2

1/2

A

/3

/4

/6

3 /3

3

B /2 3 /3 1 3

O

+

35

00 300 450 600 900 1200 1350 1500 1800 3600Goc

Hslg0

6

4

3

2

3

2

4

3

6

5 2

sin 02

1

2

2

2

3 1

2

3

2

2

2

1 0 0

cos 1

2

3

2

2

2

1 0

2

1

2

2

2

3

-1 1

tg 0

3

3

1 3 kx 3 -1

3

3

0 0

cotg kx 3 1

3

3 0

3

3

-1 3 kx kx


4/13

V. Ham so lng giac cua cac cung (goc) co lien quan ac biet:o la cac cung :

1. Cung oi nhau : va - (tong bang 0) (Vd:6

&6

,)

2. Cung bu nhau : va - ( tong bang ) (Vd:6

5&

6

,)

3. Cung phu nhau : va2

( tong bang

2

) (Vd:

3&

6

,)

4. Cung hn kem2

: va

2

+ (Vd:

3

2&

6

,)

5. Cung hn kem : va + (Vd:6

7&

6

,)

1. Cung oi nhau: 2. Cung bu nhau :

cos( ) cos

sin( ) sin

( )

cot ( ) cot

tg tg

g g

=

=

=

=

cos( ) cos

sin( ) sin

( )

cot ( ) cot

tg tg

g g

=

=

=

=

36

3. Cung phu nhau : 4. Cung hn kem2

cos( ) sin2

sin( ) cos2

( )2

cot ( ) t2

tg cotg

g g

=

=

=

=

cos( ) sin2

sin( ) cos2

( )2

cot ( ) t2

tg cotg

g g

+ =

+ =

+ =

+ =

5. Cung hn kem :

cos( ) cos

sin( ) sin

( )

cot ( ) cot

tg tg

g g

+ =

+ =

+ =

+ =

tang , cotangHn kem

Hn kem2

sin bang coscos bang tr sin

Phu cheo

Bu sinoi cos


5/13

V du 1: Tnh )4

11cos(

,

4

21tg

V du 2: Rut gon bieu thc: )3cos()2cos()2

cos( xxxA ++++=

VI. Cong thc lng giac:1. Cac he thc c ban:

22

22

11 tg = cos

11 cotg =

sintg . cotg = 1

+

+

2 2cos sin 1

sintg =

coscos

cotg =sin

+ =

37

V du: Chng minh rang:

1. xxxx 2244 cossin1sincos =+2. xxxx 2266 cossin31sincos =+

2. Cong thc cong :

cos( ) cos .cos sin .sin

cos( ) cos .cos sin .sin

sin( ) sin .cos sin .cos

sin( ) sin .cos sin .cos

tg +tgtg( + ) =

1 .

tg tgtg( ) =

1 .

tg tg

tg tg

+ =

= +

+ = +

=

+

V du: Chng minh rang:

+ =

= +

1.cos sin 2 cos( )4

2.cos sin 2 cos( )

4

3. Cong thc nhan oi:

=

=

=

=

=

=

2 2

2

2

4 4

2

cos2 cos sin2 cos 1

1 2sin

cos sin

sin2 2sin .cos

22

1

tgtg

tg

2

2cos1cos

2 +

=

2

2cos1sin

2

=

2sin2

1cossin =


6/13

4 Cong thc nhan ba:

38

3

3

cos3 4cos 3cos

sin3 3sin 4sin

=

=

5. Cong thc ha bac:

2cos1

2cos1;

2

2cos1sin;

2

2cos1cos

222

+

=

=

+= tg

6.Cong thc tnh sin ,cos ,tg theo2

t tg

=

22

2

21

2;

1

1cos;

1

2sin

t

ttg

t

t

t

t

+=

+

=

+=

7. Cong thc bien oi tch thanh tong :

[ ]

[ ]

[ ]

1cos .cos cos( ) cos( )

21

sin .sin cos( ) cos( )21

sin .cos sin( ) sin( )2

= + +

= +

= + +

V du:1. Bien oi thanh tong bieu thc: xxA 3cos.5cos=

2. Tnh gia tr cua bieu thc:12

7sin

12

5cos

=B

8. Cong thc bien oi tong thanh tch :

cos cos 2 cos .cos2 2

cos cos 2sin .sin2 2

sin sin 2sin .cos2 2

sin sin 2 cos .sin2 2

sin( )

cos cos

sin( )

cos cos

tg tg

tg tg

+ + =

+ =

+ + =

+ =

++ =

=

4

3sinsin3sin

3

=

4

cos33coscos

3 +

=


7/13

V du: Bien oi thanh tch bieu thc: 3xsin2xsinsin ++= xA 9. Cac cong thc thng dung khac:

cos sin 2 cos( ) 2 sin( )4 4

cos sin 2 cos( ) 2 sin( )

4 4

+ = = +

= + =

8

4cos35sincos

4

4cos3sincos

66

44

+=+

+=+

B. PHNG TRNH LNG GIACCac bc giai mot phng trnh lng giac

Bc 1: Tm ieu kien (neu co) cua an so e hai ve cua pt co nghaBc 2: S dung cac phep bien oi tng ng e bien oi pt en mot pt a biet cach giaiBc 3: Giai pt va chon nghiem phu hp ( neu co)Bc 4: Ket luan

I. nh ly c ban: ( Quan trong )

u = v+k2sinu=sinv u = -v+k2

u = v+k2cosu=cosv

u = -v+k2

tgu=tgv u = v+k (u;v )2

cotgu=cotgv u = v+k (u;v k )

k

+

39

( u; v la cac bieu thc cha an va Zk )

V du : Giai phng trnh:

1. sin3 sin( 2 )4

x x

= 2.4

3cos)

4cos(

=x

3. 4.xx 2sin3cos = 4 41

sin cos (3 cos6 )4

x x+ = x

II. Cac phng trnh lng giac c ban:1. Dang 1: sinx = m ; cosx = m ; tgx = m ; cotgx = m ( Rm )

* Gpt : sinx = m (1)

Neu 1m > th pt(1) vo nghiem Neu 1m th ta at m = sin va ta co

x = +k2(1) sinx=sin

x = ( - )+k2

* Gpt : cosx = m (2)


8/13

Neu 1m > th pt(2) vo nghiem Neu 1m th ta at m = cos va ta co

x = +k2(2) cosx=cos

x = +k2

* Gpt: tgx = m (3) ( pt luon co nghiem Rm )

at m = tg th(3) tgx = tg x = +k

* Gpt: cotgx = m (4) ( pt luon co nghiem Rm )

at m = cotg th(4) cotgx = cotg x = +k

Cac trng hp ac biet:

sin 1 x = 22

sinx = 0 x = k

sin 1 x = 22

cos 1 x = 2

cosx = 0 x = + k2

cos 1 x = 2

x k

x k

x k

x k

= +

= +

= +

=

V du:

1) Giai cac phng trnh :

a) =1

sin22

x b)2

cos( )4 2

x

=

c) 03)6

2sin(2 =+

x d) 03)3

cos(2 =+

x

e) f)12cos2sin =+ xx xxx 2cossincos44

=+

2) Giai cac phng trnh:

a) 4 41 cos sin 2cos2x x+ = x c) 024sin)cos(sin4 44 =++ xxx

b) 6 6sin cos cos4x x+ = x d) 3 31

sin .cos cos .sin4

x x x x =

e) 4)2

.1(sincot =++x

tgtgxxgx

40


9/13

2. Dang 2:2

2

2

2

sin sin 0

cos cos 0

0

cot cot 0

a x b x c

a x b x c

atg x btgx c

a g x b gx c

+ + =

+ + =

+ + =

+ + =

( 0a )

Cach giai

41

:

at an phu : t = sinx ( t = cosx; t = tgx; t = cotgx)

Ta c phng trnh : 2 0at bt c+ + = (1)Giai phng trnh (1) tm t, roi suy ra x

Chu y : Phai at ieu kien thch hp cho an phu (neu co)

V du :

a) b)22cos 5sin 4 0x x+ =5

cos2 4 cos 02

x x + =

c) 22sin 4 5cosx x= + d) 2cos cos2 1 cos2 cos3x x x= + + x

e) 4 41

sin cos sin 22

x x x+ = f) 0)22

cos()cos(sin244 =+ xxx

g) 4 4sin cos 1 2sin2 2

x xx+ = h) 0cos.sincossin 44 =++ xxxx

k) 0sin22

cos.sin)sin(cos266

=

+

x

xxxxl) 32cos)

2sin21

3sin3cos(sin5 +=

+

++ x

x

xxx

3.Dang 3:cos sin (1) ( a;b 0)a x b x c+ =

Cach giai:

Chia hai ve cua phng trnh cho 2 2a b+ th pt2 2 2 2 2 2

(1) cos sina b

x xa b a b a b

+ =+ +

c

+(2)

at 2 2 2 2b

cos va sina

a

a b b = =+ +

vi [ )0;2 th :

2 2

2 2

c(2) cosx.cos + sinx.sin =

a

ccos(x- ) = (3)

a

b

b

+

+

Pt (3) co dang 1. Giai pt (3) tm x.


10/13

Chu y :2 2 2Pt acosx + bsinx = c co nghiem a b c +

42

V du : Giai cac phng trnh :a) + = 3 sin 1x xcos b) 2sin3cos =+ xx

c) 4 44( d)sin cos ) 3 sin 4 2x x x+ + =x

tgx cos13 =

e) 31sincos2

2sincos2

=

xx

xx

d. Dang 4:(1)2 2sin sin .cos cos 0 (a;c 0)a x b x x c x+ + =

Cach giai 1:

2 21 cos2 1 cos2sin va cos2 2

x xx x += = Ap dung cong thc ha bac :

va cong thc nhan oi :1

sin .cos sin 22

x x= thay vao (1) ta se bien oi pt (1) ve dang 3x

Cach giai 2: ( Quy ve pt theo tang hoac cotang )

Chia hai ve cua pt (1) cho ta c pt:2cos x

2 0atg x btgx c+ + =

ay la pt dang 2 a biet cach giai

Chu y: Trc khi chia phai kiem tra xem x2

k

= + co phai la nghiem cua (1) khong?

V du : Giai phng trnh:031coscos.sin)31(sin3

22 =++ xxxx d. Dang 5:

(1)(cos sin ) sin .cos 0a x x b x x c+ + + =

Cach giai :

at cos sin 2 cos( ) vi - 2 24

t x x x t

= + =

Do2

2 t 1(cos sin ) 1 2sin .cos sinx.cosx=2

x x x x

+ = +

Thay vao (1) ta c phng trnh :2 1

02

tat b c

+ + = (2)


11/13

Giai (2) tm t . Chon t thoa ieu kien roi giai pt: 2 cos( )4

x t

= tm x.

V du : Giai phng trnh :sin 2 2 2(sin cos ) 5 0x x x + =

Chu y : Ta giai tng t cho pt co dang : (cos sin ) sin .cos 0a x x b x x c + + =

V du : Giai phng trnh :

sin 2 4(cos sin ) 4x x x+ =

43

4. Cac phng phap giai phng trnh lng giac thng s dung :a. Phng phap 1: Bien oi pt a cho ve mot trong cac dang pt lng giac c ban a biet

V du: Giai phng trnh:

02

3

2sincossin44

=++ xxx

b. Phng phap 2: Bien oi pt a cho ve dang tch soC s cua phng phap la da vao cac nh ly sau ay:

hoacA=0

. 0B=0

A B

=

A=0

. . 0 B=0

C=0

A B C

=

V du : Giai cac phng trnh :a. si b.2 2 2n sin 2 sin 3 2x x x+ + = 2 2 2 2sin 3 cos 4 sin 5 cos 6x x x = x

in cos2 cos 0x x x+ =

c. 2s d.3 03)4

sin(2cos222sin =++++

xxx

c. Phng phap 3: Bien oi pt ve dang co the at an so phuMot so dau hieu nhan biet :

* Phng trnh cha cung mot mot ham so lng giac ( cung cung khac luy tha)

V du : Giai cac phng trnh :a. cos 01cos2cos3 =+ xxx

3

=+ xxx

b. 4 01cos42coscosc.

12 cos2 8cos 7

cosx x

x + =

d. 22cossin 24 =+ xx

* Phng trnh co cha (cos sin ) va sinx.cosxx x

V du : Giai phng trnh : a. + + =3 3 31 sin cos sin 2x2

x x

b. 1)cos(sin2cossin 33 +=+ xxxx


12/13

BAI TAP REN LUYENDANG 1: Giai phng trnh lng giac

S dung 1 trong 3 phng phap sau Bien oi phng trnh ve dang phng trnh lng giac c ban Bien oi phng trnh ve dang phng trnh tch so Bien oi phng trnh ve dang co the at an so phu chuyen ve phng trnh ai so

Bai 1: Giai cac phng trnh lng giac sau1) 03)

4sin(2cos222sin =++++

xxx 2) 07cos2sin

2

5cos

2sin

2

3cos

2

7sin =++ xx

xxxx

3)6

cos.3)2

3(cos)2

2(cos)2

(cos 222

=++++ xxx

4))

4(sin2

2sin1

2sin

2sin

2cos

2

44

+

+=

x

x

x

xx

5) xxxx 2sin3cos8sin7cos =+

6) 12sincossin2 +=+ xxx

Bai 2 : Giai cac phng trnh lng giac sau

1. 8.32sin cos2 cos 0x x x+ + = 2 2 2sin ( ). cos 02 4 2

x xtg x

=

2. 2 27

sin .cos 4 sin 2 4sin ( )4 2 2

xx x x

= 9.

2cos (cos 1)2(1 sin )

sin cos

x xx

x x

= +

+

3. 9sin 6cos 3sin2 cos2 8x x x x+ + = 10.1

2 cos .s3

tg x tgx x x = in3

4.4 4sin cos 1 1

cot 25sin 2 2 8sin 2

x xg x

x x

+= 11.

12 cos2 8cos 7

cosx x

x + =

5.2

4

4

(2 sin 2 )sin31cos

x xtg x

x

+ = 12. 2cos2 1cot 1 sin sin21 2

xgx x x

tgx = +

+

6. 3 ( 13.2sin ) 6 costgx tgx x x + + 0=2

cot 4sin 2sin2

gx tgx xx

+ =

7. 14.2cos2 cos .(2 1) 2x x tg x+ = 2cos cos sin .(1 . )2

xtgx x x x tgx tg+ = +

DANG 2: Phng trnh lng giac co cha tham soS dung phng phap sau

Chon an phu thch hp va tm ieu kien ung cho an phu va chon (tuy thuoc vao x)

Chuyen phng trnh ve phng trnh ai so Lap luan e chuyen bai toan a cho theo an phu va chon S dung phng phap giai tch hoac ai so e tm tham so theo yeu cau cua e bai

Bai 1: Tm m e phng trnh sau co nghiem:

02sin4

12coscossin

244 =+++ mxxxx

Bai 2: nh m e phng trnh : mxx

gxtgxxx =++++++ )cos

1

sin

1cot(

2

11cossin

44


13/13

co nghiem

2;0

x

Bai 3: Cho ham so: 1)coscos

2()cos

cos

4(2

2

2=++ x

xmx

x

Tm m e phng trnh co nghiem thuoc ).2

;0(

Bai 4: Cho phng trnh : 01)cot(3sin3 22 =+++ gxtgxmxtgx

Tm tat ca cac gia tr cua m e phng trnh co nghiem.

Bai 5: Xac nh m e phng trnh :4 42(sin x cos x) cos4x 2sin2x m 0+ + + =

co t nhat mot nghiem thuoc oan [0; ]2

Bai 6: Cho phng trnh : mxxx = )sin(cos42sin (1)Tm tat ca cac gia tr cua m e phng trnh (1) co nghiem.

Bai 7: Tm m e phng trnh :4 4 6 6 2

4(sin x cos x) 4(sin x cos x) sin 4x m+ + =

co nghiem.Bai 8: Cho phng trnh cos4 6sin cos 0x x x m+ =

nh m e phng trnh co nghiem 0;4

x

.

Bai 9: Tm m e phng trnh : 0)cos)(sincos.(sin2cos2 =++ xxmxxx

co nghiem tren oan

2;0

Bai 10: Cho phng trnh: mtgxxx

xx=

+22

66

sincos

sincos

Vi gia tr nao cua m th phng trnh co nghiemBai 11: Cho phng trnh: mxx =+ 44 )1(sinsin

Vi gia tr nao cua m th phng trnh co nghiem

Bai 12: Tm m e phng trnh : co nghiem22 2sin 2x m(1 cosx)+ = + x [ ; ]2 2

--------------------------Het--------------------------

45

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Chuyên đề 8. Lượng giác