lượng giác toán học 12 nguyễn văn bốn thư viện tài nguyên dạy học tỉnh thanh hóa

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Ph
 ầ n I

A) Ph

ơng trình l

ợng giác cơ bản

I) sinx=a (1)

1) |a|>1 : (1) VN
2) |a|≤1 :

*) Nếu a là số đặc biệt: Thì a=sin α :

Thì:

(1)sinx=sin

x=+k2

x= +k2

/kZ.

hoặc x= 1k+k/kZ

*) Nếu a không là số đặc biệt: Thì đặt a=sin α với: −Π

2 <α<

Π

2 Ta viÕt: α=arcsina

Th×:

(1)⇔sinx=sinα⇔

x=arcsina+k2Π

x=Π −arcsina+k2Π

/k∈Z.

¿
¿
¿

L

u ý

: NÕu: *) a=0 (1) ⇔sinx=0⇔x=kΠ/k∈Z

*) a=-1: (1)⇔sinx=−1⇔x=−Π

2 +k2Π/k∈Z Hc x=
3Π

2 +k2Π/k∈Z .

*) a=1: (1)⇔sinx=1⇔x=Π

2 +k2Π/k∈Z .

(1)⇔sinx=sinβ0⇔

x=β0+k3600

x=1800− β0+k3600

/k∈Z.

¿
¿
¿

Tỉng qu¸t

(1)⇔sinf(x)=sing(x)⇔

f(x)=g(x)+k2Π

f(x)=Π − g(x)+k2Π

/k∈Z.

¿
¿
¿

II) cosx=a : (1)
1) |a|>1 : (1) VN
2) |a|≤1 :

*) Nếu a là số đặc biệt: Thì a=cos α :

Th×: (1)⇔cosx=cosα⇔x=± α+k2Π/k∈Z.

*) Nếu a không là số đặc biệt: Thì đặt a=cos α với: 0<α<Π Ta viết: α=arccosa
Thì: (1)⇔cosx=cosα⇔x=±arccosa+k2Π/k∈Z.

L

u ý

: NÕu: *) a=0 (1) ⇔cosx=0⇔x=Π

2 +kΠ/k∈Z

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*) a=1: (1)⇔cosx=1⇔x=k2Π/k∈Z .
*) (1)⇔cosx=cosβ0⇔x=± β0+k3600/k∈Z.

Tỉng qu¸t

: (1)⇔cosf(x)=cosg(x)⇔f(x)=± g(x)+k2Π/k∈Z.
III) tanx=a §K: x ≠Π

2 +kΠ/k∈Z

*) Nếu a là số đặc biệt: Thì a=tan α :
Thì: (1)⇔tanx=tanα⇔x=α+kΠ/k∈Z.

*) Nếu a khơng là số đặc biệt: Thì đặt a=tan α với −Π

2 <α<

Π

2 Ta viÕt: α=arctana

Th×:

(1)⇔tanx=tanα⇔
x=arctana+kΠ

x ≠Π
2+k1Π

¿{

k , k1∈Z .

L

u ý

: NÕu: *) a=0 (1) ⇔tanx=0⇔x=kΠ /k∈Z
*) a=-1: (1)⇔tanx=−1⇔x=−Π

4 +kΠ/k∈Z .

*) a=1: (1)⇔tanx=1⇔x=Π

4 +kΠ/k∈Z .

*) (1)⇔tanx=tanβ0⇔x=β0+k1800/k∈Z.

Tæng qu¸t

(1)⇔tanf(x)=tang(x)⇔

f(x)=g(x)+kΠ

f(x)≠Π

2+k1Π

g(x)≠Π

2 +k2Π

/k , k1, k2∈Z.

¿{ {

IV) cotx=a §K: x ≠ kΠ/k∈Z

*) Nếu a là số đặc biệt: Thì a=cot α :
Thì: (1)⇔cotx=cotα⇔x=α+kΠ/k∈Z.

*) Nếu a không là số đặc biệt: Thì đặt a=cot α với 0<α<Π Ta viết: arccota

Th×:

(1)⇔cotx=cotα⇔

x=arc cota+kΠ

x ≠ k1Π

/k , k1∈Z.

¿{

L

u ý

: NÕu: *) a=0 (1) ⇔cotx=0⇔x=Π

2 +kΠ/k∈Z

*) a=-1: (1)⇔cotx=−1⇔x=−Π

4 +kΠ/k∈Z .

*) a=1: (1)⇔cotx=1⇔x=Π

4 +kΠ/k∈Z .

*) (1)⇔cotx=cotβ0⇔x=β0+k1800/k∈Z.

Tæng qu¸t

(1)⇔cotf(x)=cotg(x)⇔

f(x)=g(x)+kΠ

f(x)≠ k1Π

g(x)≠ k2Π
/k , k1, k2∈Z.

¿{ {

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D¹ng: at+b=0

víi: a , b∈R , a ≠0 .t∈{sinf(x);cosf(x);tanf(x);cotf(x)}

PP giải: Tìm t đa về phơng trình cơ bản giải tìm x.
B) Ph ơng Trình bậc hai đối với một hàm số l ợng giác.

D¹ng: at

2

+bt+c =0

víi: a , b , c∈R , a≠0 .t∈{sinf(x);cosf(x);tanf(x);cotf(x)}

PP giải: Tìm t đa về phơng trình cơ bản giải tìm x.
C) Ph ơng trình bậc nhất đối với sinf(x) và cosf(x).

D¹ng: asinf(x)+bcosf(x) =c

(1)

PP gi¶i:

*) Khi a=0 hoặc b=0 bài toán trở thành dạng A) giải đợc

PP1 *) khi a2+b2≠0 : Chia 2 vế (1) cho

√

a2+b2 ta đa về dạng:

sin[f(x)± α]= c

√

a2+b2 hc cos[f

(x)± α]= c

√

a2+b2 Giải đợc.

PP2

(1)

asinf(x)+bcosf(x)=c

sin2f(x)+cos2f (x)=1

{

Đặt X=sinf(x),Y=cosf(x) (*)k: X,Y [1;1]

gi¶i

aX+bY=c

X2+Y2=1

¿{

Tìm đợc X,Y thay vào (*) tìm đợc f(x) từ đú giải x.
 PP3 *) Khi a=0 (hoặc b=0) bài toán trở thành dạng A) giải đợc

*) khi a ≠0 (hc b ≠0 ):Chia 2 vÕ (1) cho: a (hoặc b) đa phơng trình 
về

sin[f(x)± α]= c

√

a2

+b2 hc

cos[f(x)± α]= c

√

a2

+b2 Giải đợc.

PP4 +) kiÓm tra trùc tiÕp f(x)= Π+k2Π/k∈Z. (f(x)

2 ≠

Π

2 +kΠ)

+) khi f(x) +k2 Đặt t=tanf(x)

2 sinf(x)=
2t

1+t2;cosf(x)=

1t2

1+t2 Đa

(1) về dạng: At2+Bt+C=0 Giải đợc t thay vào phép đặt: tanf(x)

2 =t gii c.

Đặc biệt

: *)Khi c=0

(1) ⇔tanf(x)=−b

a víi: a 0 hc (1) ⇔cotf(x)=−
a

b víi: b 0 .

*)Khi a2+b2=c2 ¸p dơng B§T Bu-nhi-a-cèp-xki khi dÊu b»ng xÈy ra:
 (1) ⇔tanf(x)=a

b víi: b 0 hc (1) ⇔cotf(x)=
b

a víi: a 0 .

L

u ý : Phơng trình: asinf(x)+bcosf(x)=c cã nghiƯm khi vµ chØ khi: a2

+b2≥ c2
D) Ph ơng trình thuần nhất bậc 2 đối với sinx và cosx.

Dạng

: asin2x+bsinxcosx+ccos2x=0 Nếu vế phải b»ng d th× thay: d=d(sin2x+cos2x)

a,b,c R và a,b,c không đồng thời bằng 0.
PP1 giải:

*) KiÓm tra trùc tiÕp cosx=0

*) Chia hai vế cho cos2x đặt t=tanx (*) ta đợc: at2+bt+c=0 giải đợc t

Thay vào (*) giải đợc x.
 PP2 giải: Thay sin

2x

=1−cos 2x

2 ;cos

2x

=1+cos 2x

2 ;sinxcosx=
1

2sin 2x

đa phơng trình đã cho về dạng: Asin2x+Bcos2x=C giải đợc

E) Ph ơng Trình đối xứng đối với sinx và cosx.

Dạng

: a(sinx ± cosx)+bsinxcosx+c=0 (1)

PP1 gi¶i: Đặt: sinx+cosx=t (*) sinxcosx=t

1

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Từ (*) giải đợc x.( Nếu: sinx-cosx=t thì ⇒sinxcosx=1−t

2 )

PP2 gi¶i: sinx+cosx= √2sin

(

x+Π

)

Do đó đặt t=x+

Π

4 (*)thì (1) có dạng:

Asin2t+Bsint+C=0 giải đợc t . Thay vào (*) tìm đợc x.

C) Bài tập về ph

ơng trình l

ợng gi¸c

Theo s¸ch cơ bản-sách nâng cao & các sách tham khảo.

$1)

Dạng cơ bản

a) Cơ bản sinx=a

1) sinx=1

2 ⇔x=

Π

6+k2Π ; x=
5Π

6 +k2Π

2) sinx=1

5 ⇔x=arcsin

5+k2Π ;x=Π −arcsin
1

5+k2Π

3) sinx=1

3 ⇔x=arcsin

3+k2Π ;x=Π −arcsin
1

3+k2Π

4) sin(x+450)=−√2

5) sin(x+2)=1

3 ⇔x=arcsin

3−2+k2Π ;x=Π −arcsin
1

3−2+k2Π

6) sin 3x=1 ⇔x=Π

6 +k
2Π

7) sin

(

2x

3 −

Π

)

=0 ⇔x=

Π

2 +k
3Π

8) sin(2x+200)=−√3

2 ⇔x=−40

+k1800; x=1100+k1800

9) sin3x-cos5x=0 ⇔x= Π

16 +k

Π

4 ; x=−

Π

4 +kΠ

10) sin(x+1)=2

3 ⇔x=−1+arcsin

3+k2Π ; x=Π −1−arcsin
2
3+k2Π

11) sin22x

2 ⇔x=±

Π

8 +kΠ ; x=±
3Π

8 +kΠ

12) sinx=2

3 ⇔x=arcsin

3+k2Π ;x=Π −arcsin
2
3+k2Π

13) sinx=−√3

2 ⇔x=−

Π

3 +k2Π ; x=
4Π

3 +k2Π

14) sinx=√2

2 ⇔x=

Π

4+k2Π ; x=
3Π

4 +k2Π

15) sin

(

2x −Π

)

=sin

(

Π

5 +x

)

⇔x=

Π

3+k
2Π

3 ; x=
2Π

5 +k2Π

16) sin3x=sinx ⇔x=kΠ ;x=Π

4 +k

Π

17) sin(x+200)=√3

2 ⇔x=40

+k3600; x=1000+k3600

18) sin 4x=sinΠ

5 ⇔x=

Π

20+k

Π

2 ; x=

Π

5 +k

Π

19) sinx+Π

5 =−

2 ⇔x=−

11Π

6 +k10Π ; x=
29Π

6 +k10Π

20) sin 2x=−1

2/x∈(0; Π) ⇔x=

7Π

12 ; x=
11Π

21) sin

(

x −2Π

)

=cos 2x ⇔x=
7Π

18 +k
2Π

3 ; x=−
7Π

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22) 8sin2xcos2xcos4x= √2 ⇔x=Π

32+k

Π

4 ; x=
3Π

32 +k

Π

23) 2sin2x=1

24) sin2x

4cos

2x

4=1

25) cos2x-sin2x=1

26) sinx=√5

27) sin(2x+1)=−√2

28) sin2

(

x −Π

)

=cos

(

Π2 +3x

)

29) sin(2x-1)=sin(x+3) ⇔x=4+k2Π ; x=−2

3+(2k+1)

Π

30) tan(2x+1)cot(x+1)=1 ⇔x=kΠ

31) sin2

(

5x+2Π

)

=cos

(

4x+Π

)

⇔x=−

6Π

35 +k
4Π

21 ; x=
22Π

95 +k

4Π

32) sin(x2-4x)=0 ⇔x

=2±√4+kΠ/k∈Z , k ≥−1

33)

|

sinx+1

|

=
1

2 ⇔x=kΠ ;x=−

Π

2 +k2Π

34) sin(8cosx)=1

⇔x=±arccos−3Π

16 +k2Π ;x=±arccos

Π

16+k2Π ; x=±arccos
5Π

16 +k2Π

35) 2√2 sin

(

x+Π

)

=
1
sinx +

cosx ⇔x=±

Π

4 +kΠ

36) |cosx|+sin 3x=0

⇔x∈

{

−Π

8 +k2Π ;−

Π

4 +k2Π ;
3Π

8 +k2Π ;
5Π

8 +k2Π ; x=
9Π

8 +k2Π ;
5Π

4 +k2Π

}

37) 3 sinx

1+cosx=0 ⇔x=k2Π

37) sinxsin2x=-1 VN

38) 8sinxcosxcos2x=-1 ⇔x=− Π

24+k

Π

2 ; x=
7Π

24 +k

Π

39) 4sinxcosxcos2x=-1 ⇔x=−Π

8 +k

40) Tìm nghiệm dơng bé nhất của: sin(x2)=sin(x2+2x) đ/s: x=√3−1

41)

√

Π2

9 − x

2.sin 2x

=0 x

{

0; 3

}

b) Cơ bản cosx=a

1) cosx=1

3 ⇔x=±arccos

3+k2Π

2) cos(x+600)=√2

2 ⇔x=−15

+k3600; x=−1050+k3600

3) cosx=cosΠ

6 ⇔x=±

Π

6 +k2Π

4) cos 3x=−√2

2 ⇔x=±

Π

4 +k
2Π

5) cosx=−1

5) cos(x+300)=√3

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6) cosx=2

3 ⇔x=±arccos

3+k2Π

7) cos 3x=cos 120 ⇔x=±40+k1200

8) cos(x −1)=2

3 ⇔x=1±arccos

3+k2Π

9) cos

(

3x

2 −

Π

)

=−
1

2 ⇔x=

11Π

18 +k

4Π

3 ; x=−
5Π

18 +k
4Π

10) cos22x

4 ⇔x=±

Π

6 +kΠ ; x=±

Π

3 +kΠ

11) 2 cos2x

1−sin 2x=0 ⇔x=−
Π

4 +kΠ

12) tanx.tan3x=1 ⇔x=Π

8 +k

Π

13) cosx=−√2

2 ⇔x=±

3Π

4 +k2Π

14) cos(3x −150)=−√2

2 ⇔x=50

+k1200; x=−400+k1200

15) cos x

2=cos√2 ⇔x=±2√2+k4Π

16) cos

(

x+ Π

)

=
2

5 ⇔x=−

Π

18 ±arccos
2

5+k2Π

17) cos(x −5)=√3

2 /x∈(− Π ; Π) ⇔x=5−
11Π

6 ; x=5−

13Π

18) cos 3x=sin 2x

19) cos 2x −sin(x −1200)=0

20) cos(x+300)+2cos2150=1 ⇔

x=1200+k3600; x=−1800+k3600

21) 4cos2x+3=0 x∈

(

0;Π

)

22) tan(2x+450)tan

(

1800−x

)

=1 ⇔x=300+k1200

23) 4cos2x=3

24) tan5x.tan3x=-1

25) cos(3x −150)=√3

26) tan

(

2x+Π

)

tan

(

Π −

x

)

27) cos(sinx)=1 ⇔x=kΠ

28) tan2xtan23x=1 ⇔x=Π

8 +k

Π

4 ; x=

Π

4 +k

Π

29) tan

(

x+ Π

)

cot

(

Π

4 − x

)

=2−√3 ⇔x=kΠ ;x=−

Π

3 +kΠ

30) tan(2x+1).tan(3x-1)=1 ⇔x=Π

10+k

Π

31) tan5x.tanx=1 ⇔x=Π

12+k

Π

32) cos(2x+1)=cos(2x-1) ⇔x=k Π

33) cos2x

=2+√3

4 ⇔x=±

Π

12+kΠ

34) sin4x=2cos2x-1 ⇔x=Π

4+kΠ ; x=

Π

12+k

Π

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35) 8cos4x-cos4x=1 ⇔x=± Π

3 +kΠ

36) 2cos2x-1=sin3x x=

10+k
2

37)Tìm nghiệm dơng bÐ nhÊt cđa: cos(Πx2)=cos

(

Π(x2+2x+1)

)

®/s: x=√3−1

2 <

1
2

38) coxcos2x=1+sinxsin2x ⇔x=k2Π

39) |sinx|

sinx =cosx −

2 víi:0<x<2 Π x=

4

40)Tìm nghiệm dơng bé nhất của: sin(x2

)=cos

(

x2+2x 1

)

đ/s: x=√
3−1

2 <1

41) cos(Πsinx)=cos(3Πsinx) ⇔x=± Π

6 +kΠ ; x=k

Π

42) cosΠx=x2−4x+5 ⇔x=2

43) cos5x+x2=0 VN

c) Cơ bản tanx=a

1) tanx=tanΠ

5 ⇔x=

Π

5 +kΠ

2) tan 2x=−1

3 ⇔x=

1
2arctan

−1
3 +k

Π

3) tan(3x+150)=√3 ⇔x=150+k600

4) tan(x-300)cos(2x-1500 )=0 ⇔

x=300+k1800

5) tan(2x+600)cos(x+750 )=0 ⇔x

=−300+k900

6) tan2x-2tanx=0 ⇔x=kΠ

7) cos2x.tanx=0 ⇔x=kΠ ;x=Π

4 +k

Π

8) tan(x −150)=√3

3 ⇔x=45

+k1800

9) tan

(

Π

12+12x

)

=−√3 x=

5

144+k

10) Với giá trị nào của x thì giá trị của hàm số y=tan(

4 x) vµ y=tan2x b»ng

nhau?

11) tan x

3=3

12) tan2x=tanx

13) tan5x=tan250 ⇔

x=50+k360

14) tan 3x=tan3Π

5 ⇔x=

Π

5 +k

Π

15) tan(x −150)=5 ⇔x=arctan 5+12Π +kΠ

16) tan(2x −1)=√3 ⇔x=Π

6+
1
2+k

Π

17) tan(2x −150)=1/x∈(−1800;900) ⇔x=−1500;x=−600; x=300

18) tan3x=tanx ⇔x=kΠ

19) tan x

2=tanx ⇔x=k2Π

20) tan(2x+100)+cotx=0 ⇔

x=800+k1800

21) 1+tanx

1−tanx=tan 3x ⇔x=
Π

8 +k

Π

22) tan

(

x+Π

)

+cot

(

Π

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23) tan(Πcosx)=tan(2Πcosx) x=k

d) Cơ bản cotx=a

1) cot 4x=cot2Π

7 ⇔x=

Π

14 +k

Π

2) cot3x2 ⇔x=1

3arccot(−2)+k

Π

3) cot(2x −100)= 1

√3 ⇔x=35

+k900

4) sin2xcotx=0

⇔x=Π

2+kΠ

5) cos2xcot(x- Π

4 ) =0 ⇔x=

3Π

4 +kΠ

6) (cotx+1)sin3x=0 ⇔x=−Π

4 +kΠ ; x=

Π

3 +kΠ ;x=
2Π

3 +kΠ

7) cot(3x −1)=−√3 ⇔x=1

3+
5Π

18 +k

Π

8) sin3xcotx=0 ⇔x=Π

2 +kΠ ; x=k

Π

3 /k ≠3m , m∈Z

9) cot2 x

2=
1

3 ⇔x=±

2Π

3 +k2Π

10) cotx=−1

3 ⇔x=arccot

−1

3 +kΠ

11) cot3x=1 ⇔x= Π

12+k

Π

12) cot2x+1

6 =tan
1

3 ⇔x=

3Π −3

2 +k3Π

13) cot 2x=cot(−1

3) ⇔x=−

1
6+k

Π

14) cot(x

4+20

)=−√3 ⇔x=−2000+k7200

15) cot 3x=tan2Π

5 ⇔x=

Π

30+k

Π

16) cot 3x=− 1

√3 /x∈

(

−

Π

2 ;0

)

⇔x=−

4Π

9 ; x=−

Π

17) cot 2x=cot(x+Π

2 ) VN

19) cot(x-2)=5

20) cot(x2+4x+3)=cot6 ⇔x

=−2±√7+kΠ/k∈Z , k ≥ −2

21) tan(x-150)cot(x+150)=1/3 ⇔x

=450+k1800

$2)

D¹ng th

êng gặp

1.Dạng: at+b=0

1) 2sinx-3=0

2) √3 tanx+1=0

3) 3cosx+5=0 VN
4) √3 cotx −3=0 ⇔x=Π6 +kΠ

5) √3 tan 2x+3=0 ⇔x=−Π

6 +k

Π

6) 2 cosx −√3=0 ⇔x=±Π6 +k2Π

7) √3 tan 3x −3=0 ⇔x=Π

9 +k

Π

</div>
(9)<div class='page_container' data-page=9>

8) 5−3 tan 3=0/x∈

(

−Π

6 ;

Π

)

9) 3sinx+2sin2x=0 ⇔x=kΠ ;x=±arccos(−3

4)+k2Π

10) 2 cos

(

3x+Π

)

11) tan

(

x+Π

)

√3

3 =0

12) sin

(

3cosΠx

)

=
1

2.Đ a về dạng: at+b=0

1) 5cosx-2sin2x=0 ⇔x=Π

2+kΠ

sin2x-2cosx=0

⇔x=Π

2+kΠ

3)

2 sin2x+√2sin 4x=0

⇔x=±38Π+kΠ ; x=k Π2

4)

(sinx+1)(2 cos 2x −√2)=0

⇔x=−Π

2 +k2Π ; x=±

Π

8+kΠ

5)

sin2x-sinx=0 ⇔x=kΠ ;x=Π

2 +k2Π

6) (1+2cosx)(3-cosx)=0

⇔x=±2Π

3 +k2Π

(

cot x

3−1

)(

cot

x

2+1

)

⇔x=
3Π

4 +k3Π ; x=−

Π

2+k2Π

8) (3 tanx+√3)(2 sinx −1)=0

⇔x=5Π

6 +kΠ ;x=

Π

6 +k2Π

9) (2+cosx)(3cos2x-1)=0 ⇔x=±1

2arccos
1

3+kΠ

10) Cos3x-cos4x+cos5x=0 ⇔x=Π

8 +k

Π

4 ; x=±

Π

3 +k2Π

11) sin7x-sin3x-cos5x=0 ⇔x=Π

10+k

Π

5 ;x=

Π

12+kΠ ; x=
5Π

12 +kΠ

12) cos2x-sin2x=sin3x+cos4x ⇔x=k Π

3 ; x=

Π

6+k2Π ; x=
5Π

6 +k2Π

13) sinxsin 2xsin 3x=1

4sin 4x ⇔x=

Π

8 +k

Π

4 ; x=k

Π

14) 1+sinxcos2x=sinx+cos2x ⇔x=Π

2+k2Π ; x=kΠ

15) 9sinx+6cosx-3sin2x+cos2x=8 ⇔x=Π

2 +k2Π

16) tanx

(

2 sin x

3−1

)

=0 ⇔x=kΠ

3.D¹ng: at2 +bt+c=0

1) 2sin2x+3sinx-2=0

2) 3cot2x-5cotx-7=0

3) 3cos2x-5cosx+2=0

4) 3tan2x-2

√3 tanx+3=0

5) 2 sin2x

2+√2 sin

x

2−2=0 ⇔x=

Π

2 +k4Π ; x=
3Π

2 +k4Π

6) 6cos2x+5sinx-2=0 ⇔x=−Π

6 +k2Π ; x=
7Π

</div>
(10)<div class='page_container' data-page=10>

7) √3 tanx −6 cotx+2√3−3=0 ⇔x=Π

3 +kΠ ; x=arctan(−2)+kΠ

8) 3cos26x+8sin3xcos3x-4=0

9) tanx+tan

(

x+Π

)

=1 ⇔x=kΠ ;x=arctan 3+kΠ

10) 2cos2x-3cosx+1=0 ⇔x=k2Π ; x=±Π

3 +k2Π

11) sin2x

2+−2 cos

x

2+2=0 ⇔x=k4Π

12) 8cos2x+2sinx-7=0 ⇔x=Π

6 +k2Π ; x=
5Π

6 +k2Π

13) 2tan2x+3tanx+1=0 ⇔x=−Π

4 +kΠ ; x=arctan

−1

2 +kΠ

14) tanx-2cotx+1=0 ⇔x=Π

4+kΠ ; x=arctan(−2)+kΠ

15) 2cos2x-3cosx+1=0 ⇔x=k2Π ; x=∓Π

3 +k2Π

16)

sinx+1,5 cotx=0

⇔x=±2Π

3 +k2Π

17)

√1−cosx=sinx(x∈[Π ;3Π])

⇔x=2Π ; x=5Π

18

) 2sin2x+5sinx-3=0 −1¿

kΠ

6 +kΠ

⇔x=¿

19) cot23x-cot3x-2=0 ⇔x=Π

4+k

Π

3 ; x=
1

3arctan 2+k

Π

20) 4 cos2x −2

(1+√2)cosx+√2=0

⇔x=± Π3 +k2Π ;x=±Π4 +k2Π

21)

2 cos 2x+2 cosx −√2=0

⇔x=± Π

4 +k2Π

22) 5tanx-2cotx-3=0 ⇔x=Π

4 +kΠ ; x=arctan

−2
5 +kΠ

23) cos2x+sinx+1=0 ⇔x=−Π

2 +k2Π

24) √3 tan2x −(1+√3)tanx+1=0 ⇔x=Π

4 +kΠ ; x=

Π

6 +kΠ

25) 3cos2x+10sinx+1=0 x∈

(

−Π

2 ;

Π

)

26) cot2x-3cotx-10=0 x∈(0;Π)

27) (tanx+cotx)2-(tanx+cotx)=2 ⇔x=Π

4 +kΠ

28) 2sin2x-3cosx=2

x∈

[

00;3600

]

29) tanx+2cotx=3 x∈

[

1800;3600

]

30) 2 tan2x+3= 3

cosx ⇔x=k2Π

31) tan2x+2tanx-1=0

32) √2sinx −1=2−3 sinx −1¿

karcsin5

9+k2Π

⇔x=¿

33) 2sin2x+cosx-1=0

34) 8sin2x-cosx=5 ⇔x=± Π

3 +k2Π ;x=±arccos(−
3

</div>
(11)<div class='page_container' data-page=11>

35) 3 tan2x

+5= 7

cosx

36) cos2x-3sinx=2 ⇔x=−Π

2 +k2Π ; x=−

Π

6 +k2Π ; x=
7Π

6 +k2Π

37) tanx+√3 cotx=1+√3 ⇔x=Π

3 +kΠ ; x=

Π

4 +kΠ

38) 3 cot2x+2√2sin2x=(2+3√2)cosx ⇔x=± Π4 +kΠ ; x=±Π3+k2Π

39) sin2x+ 1

sin2x=sinx+

sinx ⇔x=

Π

2+k2Π

40) 2(cos2x+ 4

cos2x )=9(cosx −

cosx)+1 ⇔x=(2k+1)Π ; x=±

2Π

3 +k2Π

41) cot2x+ 1

cos2x +

2(tanx+cotx)+2=0 ⇔x=−

Π

4 +kΠ

42) (4sinx-5cosx)2-13(4sinx-5cosx)+42=0

⇔x=arccot4

5+arcsin
6

√41+k2Π ; x=Π+arccot
4

5−arcsin
6

√41+k2Π

43) 3 cosx+4 sinx+ 6

3 cosx+4 sinx+1=6 ⇔x=arctan

3+k2Π

44) sin22x+sin24x=3/2 ⇔x=Π

8 +k

Π

4 ; x=±

Π

6 +k

Π

45) sin4x=tanx ⇔x=kΠ ;x=±1

2arccos

√3−1

2 +kΠ

46) cos2x+sin2x+2cosx+1=0 ⇔x=(2k+1)Π

47) 1−tanx

1+tanx=1+sin 2x ⇔x=kΠ

48) sin 5x

5 sinx=1 VN

49) |cotx|=tanx+ 1

sinx ⇔x=−

2Π

3 +k2Π

50) cos4x

3 =cos

x ⇔x=k3Π ; x=±Π

4 +k3Π ; x=±
5Π

4 +k3Π

51) 3 cosx+2|sinx|=2 ⇔x=Π

2 +kΠ

52) 3 cosx+2|sinx|=3 ⇔x=k2Π ; x=±arccos 5

13+k2Π

53) 6tanx+5cot3x=tan2x ⇔x=±1

2arccos
1

3+kΠ ; x=±
1
2arctan

−1
4 +kΠ

54) 1

2tan

x − 2

cosx+

2=0 ⇔x=±

Π

3+k2Π

55) cos22x+sin2x=1/2 ⇔x=Π

4 +k

Π

2 ; x=±

Π

6 +kΠ

56) 2cos22x+3sin2x=2 ⇔x=kΠ ;x=±1

2arccos

−1

4 +kΠ

57) cos 2x+2cosx=2sin2x

2 ⇔x=±

Π

3+k2Π

58) 2-cos2x=sin4x VN

59) 3 tanx+√3 cotx −3−√3=0 ⇔x=Π6+kΠ ; x=Π4 +kΠ

60) 3sin22x+7cos2x-3=0 ⇔x=Π

4 +k

Π

</div>
(12)<div class='page_container' data-page=12>

62) 4sin4x+12cos2x-7=0 ⇔x=Π
4 +k

Π

63) cot2x −

(1−√3)cotx −√3=0

⇔x=−Π6 +kΠ ; x=Π4 +kΠ

64) 12 cosx+5 sinx+ 5

12 cosx+5 sinx+14+8=0

⇔x=Π+arcsin 5

13+k2Π ;x=arcsin
5

13 ±arccos

−9

13 +k2Π

65) 5cos4x+3sin4x-3=0 ⇔x=Π

2 +kΠ ; x=±

Π

6 +kΠ 66)

3 cos 2x+2(1+√2+sinx)sinx −3−√2=0 ⇔

{

Π6 +k2Π ;56Π+k2Π ;Π4 +k2Π ;34Π+k2Π

}

67) cos2x+sin2x+2cosx+1=0 ⇔x=(2k+1)Π

68) 2cos2x-sin2x-4cosx+2=0 ⇔x=k2Π ; x=±arccos1

3+k2Π

69) 3

sin2x+3 tan

2x

+4(tanx+cotx)−1=0 ⇔x=−Π

4 +kΠ

4.D¹ng : asin2 x+bsinxcosx+ccos2 x =0

1) 2sin2x-5sinxcosx-cos2x=-2 ⇔x=Π

4 +kΠ ; x=arctan
1

4+kΠ

2) 2sin2x+sinxcosx-3cos2x=0 ⇔x=Π

4 +kΠ ; x=arctan

−3

2 +kΠ

3) 3sin2x-4sinxcosx+5cos2x=2 ⇔x=Π

4 +kΠ ; x=arctan 3+kΠ

4) sin2x+sin2x-2cos2x=1/2 ⇔x=Π

4 +kΠ ; x=arctan(−5)+kΠ

5) 4sin2x+3

√3 sin2x-2cos2x=4 ⇔x=Π

2+kΠ ; x=

Π

6 +kΠ

6) 25sin2x+15sin2x+9cos2x=25 ⇔x=Π

2 +kΠ ; x=arctan
8

15+kΠ

7) 4sin2x-5sinxcosx-6cos2x=0 ⇔x=arctan 2+kΠ ; x=arctan(−3

4)+kΠ

8) sin2

x-√3 sinxcosx+2cos2x=1 ⇔x=Π

2 +kΠ ; x=

Π

6 +kΠ

9) 2sin2x+3

√3 sinxcosx-cos2x=4 VN

10) 3sin2x+4sin2x+ (8

√3−9) cos2x=0 ⇔x=−Π

3 +kΠ ; x=arctan

(

√3−
8
3

)

+kΠ

11) sin2x+sin2x-2cos2x=1/2 ⇔x=Π

4 +kΠ ; x=arctan(−5)+kΠ

12) cos2x-3sin2x=0 ⇔x=±Π

6 +kΠ

13) 3sin2x-sin2x-cos2x=0 ⇔x=Π

4 +kΠ ; x=arctan

−1

3 +kΠ

14)3sin22x-sin2xcos2x-4cos22x=2 ⇔x=1

2arctan(−2)+k

Π

2 ; x=
1

2arctan 3+k

Π

15) 2sin2x+3sinxcosx+cos2x=0 ⇔x=Π

4 +kΠ ; x=arctan

−1
2 +kΠ

16) 2 sin2x+(3+√3)sinxcos sx+(√3−1)cos2x=−1 ⇔x=−Π4 +kΠ ; x=−Π6 +kΠ

17) sin2x

2+sinx −2 cos

2x

2=1/2 ⇔x=

Π

</div>
(13)<div class='page_container' data-page=13>

18) sin2x-3sinxcosx+2cos2x=0 ⇔x=Π

4 +kΠ ; x=arctan 2+kΠ

19) sin2x-2

√3 sinxcosx+3cos2x=0 ⇔x=Π

3 +kΠ

20) sin2x+√3 sinxcosx+√3 cos2x=−1

2sin 2x ⇔x=
3Π

4 +kΠ ;x=
2Π

3 +kΠ

21) √3 sin2x −(1−√3)sinxcosx −cos 2x=sin2x ⇔x=34Π+kΠ ;x=Π6 +kΠ

22) 2sin2x-5sinxcosx-8cos2x=-2 ⇔x=arctan 2+kΠ ; x=arctan(−3

4)+kΠ

23) sin2x+sin2x=1/2 ⇔x=α

2+k

Π

2 víi:

sinα= 1

√5
cosα= 2

√5
0<α<Π

¿{

25) 4sin2x+2sin2x+2cos2x=1 ⇔x=−Π

4 +kΠ ; x=arctan

−1

3 +kΠ

26) 4sin2x+3

√3 sin2x-2cos2x=4

27) 4cos2x+3sinxcosx-sin2x=3 ⇔x=Π

4+kΠ ; x=arctan

−1

4 +kΠ

28) 2sin2x-sinxcosx-cos2x=2 ⇔x=Π

2 +kΠ ; x=arctan(−3)+kΠ

29) 4sin2x-2sin2x+3cos2x=1 VN

30) 5sin2x+2sinxcosx+cos2x=2 ⇔x=−Π

4 +kΠ ; x=arctan
1

3+kΠ

31) sin2x-2sin2x+3cos2x=1 ⇔x=Π

2 +kΠ ; x=arctan
1

2+kΠ

32) 3sin2x-3sinxcosx+4cos2x=1 VN

33) sin2x-2sin2x=2cos2x ⇔x=Π

2+kΠ ; x=

Π

4 +kΠ

34) 2sin2 2x-3sin2xcos2x+cos22x=2 ⇔x=Π

4 +k

Π

2 ; x=
1

2arccot(−3)+k

Π

35) 4 sinxcos

(

x −Π

)

+4 sin(Π+x)cosx+2 sin

(

3Π

2 − x

)

cos(Π+x)=1

⇔x=Π

4 +kΠ ; x=arctan
1

3+kΠ

36)sin2x+sinxcos4x+cos24x=3/4 ⇔

{

Π

18+k
2Π

3 ;

7Π

30 +k
2Π

5 ;−

Π

30+k
2Π

5 ;
5Π

18 +k
2Π

}

37) 2√2(sinx+cosx)cosx=3+cos 2x VN

38) 4√3 sinxcosx+4 cos2x −2 sin2x=5

2 ⇔x=

Π

3+kΠ ; x=arc cot(−3√3)+kΠ

D¹ng: asinx+bcosx =c

1) sinx+√3 cosx=1 ⇔x=Π

2+k2Π ; x=−

Π

6+k2Π

2) √3 sin3x −cos 3x=√2

3) cosx −√3 sinx=√2 ⇔x=− Π

12+k2Π ; x=−
7Π

</div>
(14)<div class='page_container' data-page=14>

4) 3 sin 3x −4 cos 3x=5 ⇔x=α

Π

6 +k
2Π

3 víi:

sinα=4

5
cosα=3

5
0<α<Π

¿{

5) 2 sinx+2 cosx −√2=0 ⇔x=−12Π +k2Π ; x=127Π+k2Π

6) 12sin 2x+5 cos 2x −13=0 ⇔x=Π

4 −

α

2+kΠ víi:

sinα= 5

13
cosα=12

13
0<α<Π

¿{

7) 9sinx+6cosx-3sin2x+cos2x=8 ⇔x=Π

2+k2Π

8) tanx −3 cotx=4(sinx+√3 cosx) ⇔x=−Π

3 +kΠ ; x=
4Π

9 +k
2Π

9) 2 sinx+cosx=1 ⇔x=k2Π ; x=Π −2α+k2Π víi:

sinα= 1

√5
cosα= 2

√5
0<α<Π

¿{

10) 4 sinx+3 cosx=5 ⇔x=Π

2 − α+k2Π víi:

sinα=3

5
cosα=4

5
0<α<Π

¿{

11) √3 sinx −cosx=1 ⇔x=Π+k2Π ; x=Π

3 +k2Π

12) 2 sin3x+√5 cos 3x=−3 ⇔x=Π3+α+k23Π víi:

sinα=2

3
cosα=√5

3
0<α<Π

¿{

</div>
(15)<div class='page_container' data-page=15>

13) 4 sinx+3 cosx=−5 ⇔x=Π+α+k2Π víi:

sinα=4

5
cosα=3

5
0<α<Π

¿{

14) 2 sin2x −2 cos 2x=√2 ⇔x=5Π

24 +kΠ ;x=
13Π

24 +kΠ

15) 5sin2x-6cos2x=13 VN

16) sinx+sin2x

2=0,5 ⇔x=arctan

2+kΠ

17) sinx −2cosx=3 VN

18) sinx=√2sin 5x −cosx ⇔x=Π

8+k

Π

3 ; x=

Π

16+k

Π

19) cos2x-sin2x=0 ⇔x=±1

2arccos
1

3+kΠ

20) 2 cosx −|sinx|=1 ⇔x=±arccos4

5+k2Π

21) 2 sinx −2 cosx=1−√3 ⇔x=Π

6 +k2Π ; x=
4Π

3 +k2Π

22) 2 sin(x+100)−√12 cos(x+100)=3

23) sin 5x+√3 cos 5x=2 cos 3x ⇔x=12Π+kΠ ; x=48Π+kΠ4

24) √3 sin3x −cos 3x=1 ⇔x=−Π3 +k23Π

25) √13sin 4x+3 cos 2x=4 sinxcosx ⇔x=−α2+kΠ ; x=Π6+α+kΠ3 víi:

sinα= 3

√13
cosα= 2

√13

¿{

26) 3 sinx+4 cosx=5 ⇔x=α+k2Π víi:

sinα=3

5
cosα=4

5
0<α<Π

¿{

27) sin 4x+√3 cos 4x=√2 ⇔x=5Π

48 +k

Π

2 ; x=−

Π

48+k

Π

28) 3 cosx+2|sinx|=2 ⇔x=Π

2 +kΠ

29) 3 cosx+2|sinx|=3
⇔x=Π

2+kΠ ; x=k2Π ; x=±arccos
5

13+k2Π

30) 2cos3x+cos2x+sinx=0 ⇔x=Π

2 +k2Π ; x=

Π

4 +kΠ

31) sinx+√3 cosx

sinx −cos 450 =0 ⇔x=−

Π

</div>
(16)<div class='page_container' data-page=16>

32) 2 sin2x+3 cos 2x=√13 sin 14x ⇔x= α

12+k

Π

6 ; x=

Π −α

16 +k

Π

8 víi:

sinα= 2

√13
cosα= 3

√13

¿{

33) 2√3 sinx+3 cosx=9

2 ⇔x=arcsin 2

√

17±arccos

2√21+k2Π

34) sin8x-cos6x= √3 (sin6x+cos8x) ⇔x=Π

4+kΠ ; x=

Π

12+k

Π

35) Tìm x sao cho: y=1+sinx

2+cosx là mét sè nguyªn. ⇔x=

Π

2+kΠ ; x=Π+k2Π

6. D¹ng: a(sinx ± cosx)+bsinxcosx+c=0

1) sinx+cosx=1+cosxsinx ⇔x=k2Π ; x=Π

2 +k2Π

2) 5sin2x +sinx+cosx+6=0 VN

3) (√2−1)(sinx+cosx)+2 sinxcosx=√2−1

⇔x=k2Π ; x=Π

2 +k2Π ; x=
3Π

4 +k2Π

4) (1−√2)(sinx+cosx)+2 sinxcosx=√2−1

⇔x=Π+k2Π ; x=−Π

2 +k2Π ; x=

Π

4 +k2Π

5) (2+√2)(sinx+cosx)+4 sinxcosx=−2−√2

⇔x=Π+k2Π ; x=−Π

2 +k2Π ; x=−
5Π

12 +k2Π ;x=
11Π

12 +k2Π

6) 4cosxsinx-2(sinx+cosx)=-1 ⇔x=Π

4 ±arccos

√2±√6

4 +k2Π

L u ý : √2−√6

4 =cos

7Π

12 ;

√2+√6

4 =cos

Π

7) 2(sinx+cosx)+sin2x+1=0

8) sinx+cosx=1-sin2x ⇔x=k2Π ; x=Π

2 +k2Π

9) 12(sinx-cosx)=12+sin2x ⇔x=Π+k2Π ; x=Π

2 +k2Π

10)3(sinx+cosx)+2sin2x+3=0

⇔x=−Π

4 +arcsin

−√2

4 +k2Π ; x=
3Π

4 −arcsin

−√2

4 +k2Π

11) sinx-cosx+4sinxcosx+1=0 ⇔x=k2Π ; x=3Π

2 +k2Π

12) sin3x+cos3x=1 ⇔x=k2Π ; x=Π

2 +k2Π

13) sin3x+cos3x=cosx ⇔x=kΠ ;x=Π

4 +kΠ

14) 1+sin2x=sinx+cosx ⇔x=k2Π ; x=Π

2 +k2Π ; x=−

Π

4 +kΠ

15) 2sin2x+3sinx=-3cosx ⇔x=Π

4 ±arccos
1

2√2+k2Π

16) sin2x(sinx+cosx)= √2 ⇔x=Π

4 +kΠ

17) sin2x-4(sinx-cosx)=4 ⇔x=k2Π ; x=−Π

</div>
(17)<div class='page_container' data-page=17>

18) cotx-tanx=sinx+cosx ⇔x=−Π

4 +kΠ ; x=−

Π

4 ±arccos

√2−1

√2 +k2Π

19) cos3x=sin3x+1 ⇔x=(2k −1)Π ; x=Π

2 +k2Π

20) |sinx −cosx|+4 sin 2x=1 ⇔x=kΠ

21) 1+sin32x+cos32x=3

2sin 4x ⇔x=

Π

2 +kΠ ; x=−

Π

4+kΠ

22) 1+cos2x+sinx=2cos2x

2 ⇔x=k2Π ; x=

Π

2 +k2Π ; x=

Π

4+kΠ

23) sin3x+cos3x=cos2x ⇔x=k2Π ; x=3Π

2 +k2Π ; x=
3Π

4 +kΠ

24)3(cotx-cosx)-5(tanx-sinx)=2 ⇔x=arctan3

5+kΠ ; x=

Π

4 ±arccos
1−√2

√2 +k2Π

25) tanx-2 √2 sinx=1 ⇔x=−Π

4 +k2Π ; x=
5Π

12 +k
2Π

26) cos3x-cos2x=sin3x

⇔x∈

{

−Π

2 +k2Π ;−

Π

4 +kΠ ;k2Π ;
5Π

4 −arcsin

√2

4 +k2Π ;

Π

4 +arcsin

√2

4 +k2

}

7. Một số dạng khác:

1)

(

cos x

4−3 sinx

)

sinx+

(

1+sin

x

4−3 cosx

)

cosx=0

VN

2) sin2x+sin23x=2sin22x ⇔x=kΠ ;x=Π
8 +k

Π

3) 2 sin x

2cos

x −2 sin x
2sin

x=cos2x −sin2x

4) sin2xsin5x=sin3xsin4x ⇔x=k Π

5) cosxcos5x=cos2xcos4x ⇔x=kΠ

6) cos4xcos5x=cos2xcos3x

7) sinx+sin2x=cosx+cos2x ⇔x=Π+k2Π ; x=Π

6 +k
2Π

8) sin2x+sin4x=sin6x ? ⇔x=kΠ

2 ; x=k

Π

9) cos2x+cos22x+cos23x+cos24x=2 ⇔x= Π

10+k

Π

5 ;x=

Π

4 +k

Π

2 ; x=

Π

2 +kΠ

10) sin24x+sin23x=sin22x+sin2x ⇔x=Π

2 +kΠ ; x=k

Π

11) (1-tanx)(1+sin2x)=1+tanx ⇔x=kΠ ;x=−Π

4 +kΠ

12) tanx+cot2x=2cot4x ⇔x=± Π

3 +kΠ hay x=k

Π

3 /k∈Z , k ≠3m, m∈Z

13) sinx+sin2x+sin3x=cosx+cos2x+cos3x ⇔x=±2Π

3 +k2Π ; x=

Π

8 +k

Π

14) tanx+tan2x=sin3xcosx ⇔x=k Π

15) 1

sin 2x+

1
cos 2x=

sin 4x VN

16) sinx+cosx=cos 2x

1−sin2x ⇔x=k2Π ; x=−

Π

2 +k2Π ; x=−

Π

4 +kΠ

17) 1+cos 2x

cosx =

sin 2x

1−cos 2x ⇔x=
Π

6+k2Π ; x=
5Π

</div>
(18)<div class='page_container' data-page=18>

18) tan2x-sin2x+cos2x-1=0 ⇔x=kΠ ;x=Π

8 +k

Π

19) sin4x+cos4x=3/4 ⇔x=Π

8+k

Π

20) sin22x −sin2x=sin2Π

4 ⇔x=±

Π

4 +kΠ ; x=±

Π

6 +kΠ

21) sin2x+tanx=2 ⇔x=Π

4+kΠ

22) cosxcos2x=cos3x ⇔x=kΠ

23) tan2x

=1+cosx

1+sinx ⇔x=(2k+1)Π ; x=

Π

4 +kΠ

24) tanx+tan 2x=sin 3x

cosx ⇔x=k

Π

3 /k∈Z , k ≠3m , m∈Z

25) sin5x+sin3x+sinx=0 ⇔x=kΠ

26) cosx+cos2x+cos3x=0 ⇔x=Π

4 +k

Π

2 ; x=∓
2Π

3 +k2Π

27) sinx+sin3x+sin 5x

cosx+cos 3x+cos 5x=0 ⇔x=kΠ ;x=∓

Π

3 +k2Π

28) cos2

(

x+Π

)

+cos

(

x2+

Π

)

=1 ⇔x=Π+k2Π ; x=k

2Π

29) cos2

(

x+Π

)

−cos

(

x2+

Π

)

=0 ⇔x=k2Π ; x=−

Π

3 +k
2Π

30)

|

tan

(

2x −Π

)

cot

(

x+

Π

)

|

=1 ⇔x=kΠ ;x=

Π

3 +kΠ

31) cos7xcos6x=cos5xcos8x
32)

sinx
2
1+sin x

1−sinx
2
cosx

⇔x=Π

2+kΠ

33)

2 tan x
2
1+tan2x

=cosx

2 ⇔x=

Π

3+k

4Π

34) cos6xcos2x=1 ⇔x=k Π

35) sin6xsin2x=1 VN

36) 2(sin22x+sin2x)=3 ⇔x=Π

4 +k

Π

2 ; x=∓

Π

3 +kΠ

37) 6cos2x-cosx=-cos3x ⇔x=Π

2 +kΠ ; x=∓

Π

3 +k2Π

38) 2tanx+tan2x=tan4x ⇔x=kΠ

39) 2 sin3x

4 sin
5x

4 =sin

x

2sin
3x

2 −cos

x

2cos
3x

2 ⇔x=Π+k2Π

40) √2cos 5x=sin(2x+Π)+sin(2x+Π

2 )cot 3x

⇔x=Π

10+k

Π

5 ;x=

Π

12+k
2Π

3 ; x=

Π

4 +k
2Π

41) √2cos(2x+Π

4 )=cos(x+

Π

4)−sin(x+

Π

4 ) ⇔x=

Π

4 +k2Π ; x=
5Π

12 +k
2Π

</div>
(19)<div class='page_container' data-page=19>

42) (tanx+√3)2−√3=(1+√3)tanx+3 ⇔x=Π

4+kΠ ; x=

− Π

13 +kΠ

43) sin 2x+sinx

cosx=2 ⇔x=

Π

4 +kΠ

44) cos2xsin2x+1=0 ⇔x=Π

2 +kΠ

45) tan2x-2sin2x=sin2x ⇔x=kΠ ;x=Π

8 +k

Π

46) 4 sin

(

x+Π

)

+6 cos

(

x −

Π

)

=5√2 ⇔x=

Π

12+k2Π ; x=
7Π

12 +k2Π

47) 1

sinx+

cosx=tanx −

tanx ⇔x=−

Π

4 +kΠ

48) sinx+sin2x+sin3x+sin4x=0 ⇔x=Π+k2Π ; x=k2Π

49)8sin3xcosx-3sin2x+2sin2xcos2x+cos4x=1 ⇔x=kΠ ;x=arctan1

2+kΠ ; x=arctan
3
2+kΠ

50) 2sinxcos2x-1+2cos2x-sinx=0 ⇔x=± Π

6 +kΠ

51) sinx −sin3x+sin 5x

cosx −cos 3x+cos 5x=0 ⇔x=k

Π

52) 2cot2x-3cot3x=tan2x

53) cosxcosx

2cos
3x

2 −sinxsin

x

2sin
3x

2 =

1
2

⇔x=−Π

2 +k2Π ; x=

Π

6 +k
2Π

3 ; x=−

Π

4 +kΠ

54) cos 3x

sin 2x =

cos 5x

sin 2x

55) tan2x=3tanx

56) tan2x+cotx=8cos2x ⇔x=Π

2 +kΠ ; x=

Π

24+k

Π

2 ;x=
5Π

24 +k

Π

57) cos4

3 x+sin

2x+2 sin

6 x=cos

2x

58) cos2x+4sin4x=8cos6x ⇔x=Π

4 +k

Π

59) sin4x+cos4x=3−cos 6x

4 ⇔x=

Π

2 +kΠ ; x=

Π

10+k

Π

60) 2cos24x+sin10x=1 ⇔x=−Π

4 +kΠ ; x=−

Π

36+k

Π

61) tan(3x+1500)−tan(1300− x)=2sin(1000+2x)

62) 2cos2x-sin2x=2(sinx+cosx)

63) tan

(

2x −Π

)

tan

(

2x+

Π

)

4 cos22x

tanx −cotx

64) sin

x+cos10x

2 =

sin4x+cos4x

2 cos22x+sin22x

65) sin3x+cos3x

2 cosx −sinx=cos 2x ⇔x=
Π

2+kΠ ; x=−

Π

4 +kΠ ;x=arctan
1
2+kΠ

66) 1−cos 2x

2sinx =

sin 2x

1+cos 2x

67) cos 2x −cosx=2 sin23x

2 ⇔x=k

2Π

3 ; x=kΠ

68) sin4xcos5x=sin2xcos3x ⇔x=k Π

2 ; x=(2k+1)

Π

</div>
(20)<div class='page_container' data-page=20>

69) sin3x+sin5x+sin7x=0 ⇔x=kΠ

5 ; x=±

Π

3 +kΠ

70) tanx+tan2x=tan3x ⇔x=kΠ

71) 3+2sinxsin3x=3cos2x ⇔x=kΠ

72) 2sinxcos2x-1+2cos2x-sinx=0 ⇔x=−Π

2 +k2Π ; x=±

Π

6 +kΠ

73) sin2x+sin22x+sin23x+sin24x=2 ⇔x=Π

4 +k

Π

2 ; x=

Π

10+k

Π

74) 3tanx+2cot3x=tan2x ⇔x=± Π

6 +kΠ ; x=±

arccos(−1 3)

2 +kΠ

75) (2sinx-cosx)(1+cosx)=sin2x ⇔x=(2k+1)Π ; x=Π

6 +k2Π ; x=
5Π

6 +k2Π

76) tan2x-2sin2x=sin2x ⇔x=kΠ ;x=Π

8 +k

Π

77) sin4x

+cos4x=3

4 ⇔x=±

Π

8+k

Π

78) sin6x+cos6x=1 ⇔x=k Π

79) 1-sinxcosx(2sin2x-cos22x)=0 ⇔x=Π

4 +k

Π

80) tanx-3cot3x=2tan2x ⇔x=±arccos(−1

√2)

2 +kΠ

81) 6tan2x-2cos2x=cos2x ⇔x=± Π

6 +kΠ

82) 4 sinx+3 cosx=4(1+tanx)− 1

cosx ⇔x=k2Π ; x=arccos

5±arccos
1
5+k2Π

83) sin2xsin5xsin7x=1 VN

84) sin22x+cos23x=1 ⇔x=kΠ ;x=k Π

85) tan2x+cot2x=2sin5

(

x+Π

)

⇔x=

Π

4 +k2Π

86) (cos4x-cos2x)2=4+cos23x ⇔x=Π

2+kΠ

87) 2sin5x+3cos8x=5 VN

88) cos2xcos25x=1 ⇔x=kΠ

89) sinxcos4xcos8x=1 ⇔x=Π

2 +k2Π

90) 2(cos6x+cosx)=4+cos2 3x

2 VN

91) sin3xsin3x+cos3xcos3x=1 ⇔x=kΠ

92) sinx+2sin2x=3+sin3x VN
93)

sin6x+cos6x

tan

(

x −Π

)

tan

(

x+

Π

)

=−1

4 VN

94) sin

x+cos6x

cos2x −sin2x =

4tan 2x VN

95) sin2x+cos2x+tanx=2 ⇔x=Π

4+kΠ

96) sin3

(

x − Π

)

=√2 sinx ⇔x=
3Π

</div>
(21)<div class='page_container' data-page=21>

97) 2 sin3

(

34Π+

x

)

=sin

(

Π

4 +
3x

)

⇔x=

Π

2+k2Π ; x=(k −1)Π

98) sin2x+sin22x=1 ⇔x=(2k+1)Π

99) sin2x+sin22x+sin23x=3/2 ⇔x=± Π

3+kΠ ; x=

Π

8+k

Π

100) cos2x+cos22x=1 ⇔x=(2k+1)Π

4 ; x=±

Π

3 +kΠ

101) cos2x+cos22x+cos23x=1 ⇔x=Π

6 +k

Π

3 ; x=

Π

4 +k

Π

102) cos2x+cos22x+cos23x+cos24x=3/2 ⇔x=Π

8+k

Π

4 ; x=±
2Π

5 +kΠ ; x=±

Π

5 +kΠ

103) sinxcos2x=sin2xcos3x- 1

2 sin5x ⇔x=kΠ ;x=k

Π

104) sinx(1+cosx)=1+cosx+cos2x ⇔x=Π

2 +k2Π

105) sinxsin2xcos5x=1 VN

106) tan2x+cot2x=2sin2y

⇔

x=Π

4 +k

Π

y=(2l+1)Π

¿{

107) 1−sinx¿

8
sin4x+¿

⇔x=Π

6 +k2Π ; x=
5Π

6 +k2Π

108) sin4x+cos4x −cos2x+1

4sin

2x −2=0 ⇔x=(2k+1)Π

109) sin3x+cos3x= 1

sin4x+cos4x ⇔x=k2Π ; x=

Π

2 +k2Π

110) sinxsin2xsin3x= 1

4 sin4x ⇔x=

Π

8 +k

Π

4 ; x=k

Π

111) sin4x+cos4x=1

2sin 2x ⇔x=

Π

4 +kΠ

112) 2sinx+cotx=2sin2x+1 −1¿

kΠ

6 +kΠ ; x=−
Π

4 ±arccos√
2−2

2 +k2Π

⇔x=¿

113) sinxcos 4x −sin22x=4 sin2

(

Π

4 −

x

)

−
7

2 thoả m·n: |x −1|<3 ⇔x=−

Π

6 ;x=
7Π

114) 6sinx-2cos3x=5sin2xcosx ⇔x=Π

4+kΠ

115) 3(cos 2x+cot2x)

cot 2x −cos 2x −2 sin 2x=2 ⇔x=

7Π

12 +kΠ

116) √1−sinx+√1+sinx=2 cosx ⇔x=k2Π

117) 4 cos2

x+3 tan2x −4√3 cosx+2√3 tanx+4=0 ⇔x=−Π6 +kΠ

upload.123doc.net) sin4x+cos4x

sin 2x =

2(tanx+cotx) VN

119) sin

2x −2

sin22x −4 cos2x=tan

2x ⇔x

=Π

4 +k

Π

120) √sinx+sinx −sin2x+cosx=1 ⇔x=k2Π ; x=Π ±arcsin√5−1

</div>
(22)<div class='page_container' data-page=22>

121) cos 2x+cos3x

4 −2=0 ⇔x=k8Π

122) 1-tan2x=2tanxtan2x ⇔x=Π

8+k

Π

123)

(

sin

3x

2+
1
sin3x

)

(

cos3 x

2+
1
cos3x

)

=81

4 cos

24x

⇔x=Π

2+kΠ

124) sinx-2sin2x-sin3x=2 √2 VN

125) sin4x-4sinx-cos4x+4cosx=1 ⇔x=Π

4 +kΠ

126)

sin42x+cos42x

tan

(

Π

4 − x

)

tan

(

x+

Π

)

=cos44x

⇔x=kΠ

127) tan2x=1+cosx

1−sinx ⇔x=(2k+1)Π ; x=−

Π

4 +kΠ

128) (cosx −2 sin3x)sin 3x+(1+sinx −2cos 3x)cos 3x=0

129) 2cos3 x=sin3x ⇔x=Π

4 +kΠ ; x=arctan(−2)+kΠ

130) sin6x+cos6x= 5

6 (sin4x+cos4x) ⇔x=

Π

8 +k

Π

131) (cos2x-cos4x)2=6+2sin3x ⇔x=Π

2 +k2Π

132) sin8x

+cos8x=2(sin10x+cos10x)+5

4cos 2x ⇔x=

Π

4+k

Π

133) √3 sin2x −2cos2x=2√2+2 cos 2x ⇔x=Π2 +kΠ

134) tan22xtan23xtan5x=tan22x-tan23x+tan5x ⇔x=k Π

135) 3(cos 2x+cot2x)

cot 2x −cos 2x −2 sin 2x=2 ⇔x=−
Π

12+kΠ ; x=

Π

12+kΠ

136) sin22x −cos28x=sin

(

17Π

2 +10x

)

⇔x=

Π

20 +k

Π

10 ; x=

Π

6 +k

Π

137) sin

4 x

2+cos

4x

2
1−sinx −tan

xsinx=1+sinx

2 +tan

x ⇔x=

Π

4 +k

Π

139) sin8x

+cos8x=17

16 cos

22x

⇔x=Π

8 +k

Π

140) 4cosx-2cos2x-cos4x=1 ⇔x=k2Π ; x=Π

2 +kΠ

141) 3tan3x+cot2x=2tanx+ 2

sin 4x ⇔x=±

2arccos

−1
4 +kΠ

142) cos10x+2cos24x+6cos3xcosx=cosx+8cosxcos33x ⇔x=k2Π

143) sin4x+cos4

(

x+Π

)

=
1

4 ⇔x=kΠ ;x=

Π

4 +kΠ

144) 1+sinx

2sinx −cos

x

2sin

2x

=2cos2

(

Π

4 −

x

)

⇔x=kΠ

145) cos2x-cos6x+4(3sinx-4sin3x+1)=0 ⇔x=Π

2 +kΠ

146) tan2xtan3xtan5x=tan2x-tan3x-tan5x ⇔x=k Π

</div>
(23)<div class='page_container' data-page=23>

147) |cosx+2 sin 2x −cos 3x|=1+2 sinx −cos 2x
⇔x∈

{

kΠ ;Π

2 +k2Π ;
Π

3 +k2Π ;
2Π

3 +k2Π

}

148) sinx+sin3x+sin 3x

cosx+cos 3x+cosx =√3 ⇔x=

Π

6+k

Π

149) 1+2 sin2x −3√2 sinx+sin 2x

2 sinxcosx −1 =1 ⇔x=
3Π

4 +k2Π

150) tan2x=1−cos

x

1−sin3x

⇔x∈

{

k2Π ;Π

4 +kΠ ;−

Π

4 +arcsin

(

1−

√2

)

+k2Π ;
3Π

4 −arcsin

(

1−

√2

)

+k2Π

}

151) cot 2x+cot 3x+ 1

sinxsin 2xsin 3x=0 VN

152) 3sinx+2cosx=2+3tanx ⇔x=k2Π ; x=arctan−2

3 +kΠ

153) sin3xsin3x+cos3xcos3x=cos34x ⇔x=kΠ ;x=kΠ

154) sin3xsin3x+cos3xcos3x= √2

4 ⇔x=±

Π

8 +kΠ

155) 1

sinx+ √

cosx=8 sinx ⇔x=
Π

6 +kΠ ; x=−

Π

12+k

Π

156) sin3x

(1+cotx)+cos3x(1+tanx)=2√sinxcosx ⇔x=Π4 +k2Π

157) tan(1200+3x)-tan(1400-x)=2sin(800+2x)  x=-400+k600

158) sinx+ 1

sinx +cosx+

1
cosx=

10
3

⇔x=−Π

4 +arcsin

2−√19

3√2 +k2Π ; x=
3Π

4 −arcsin

2−√19

3√2 +k2Π

159) 3 cos4x

3 =2 cos

2x

+1 ⇔x=k3Π ; x=±3

2arccos

1−√21

4 +k3Π

160) √2(2 sinx −1)=4(sinx −1)−cos

(

2x+Π

)

−sin

(

2x+

Π

)

⇔x=

Π

2+k2Π

161) 6 sinx −2cos3x

=5 sin 4xcosx

2 cos 2x VN

162) 1−cos 4x

2 sin 2x =

sin 4x

1+cos 4x VN

163) cos3x+sin3x=sinx-cosx ⇔x=Π

2 +kΠ

164)

√

sin2x −2 sinx

+2=2 sinx −1 ⇔x=Π2 +kΠ

165) cosx(2 sinx+3√2)−2 cos

2x −1

1+sin 2x =1 ⇔x=

Π

4 +k2Π

166) tanx+tan2x+tan3x+cotx+cot2x+cot3x=6 ⇔x=Π

4 +kΠ

167) sinx+

√

2−sin2x+sinx

√

2−sin2x=3 ⇔x=Π2+k2Π

168) 3tan2x+4sin2x-2

√3 tanx-4sinx+2=0 ⇔x=Π

6+k2Π

169) x2-2xcosx-2sinx+2=0 VN

170)

(

sin2x − 1

sin2x

)

(

cos2x − 1

cos2x

)

2+siny −sin

2y ⇔

(x , y)∈

(

Π

4 +k

Π

2 ;

Π

</div>
(24)<div class='page_container' data-page=24>

171) cos2x.sin(sinx)+sinx.cos(sinx)=0

172) 7cos2x+8sin100x=8 ⇔x=Π

2 +kΠ

173) cos 3x+

√

2−cos23x=2(1+sin22x) ⇔x=k2Π

174) sinx+cosx= √2 (2-sin3x) VN

175) x2-2xsinxy+1=0 ⇔

{

(

1; Π

2 +k2Π

)

,

(

−1;

Π

2 +k2Π

)

}

176) (cos2x-cos4x)2=5+sin3x ⇔x=Π

2+k2Π

177) cos4x −sin4x=|sinx|+|cosx| ⇔x=kΠ

178) cos2x-cos6x+4(3sinx-4sin3x+1)=0 ⇔x=Π

2 +k2Π

179) sin2x+sin2y+sin2(x+y)=9/4

⇔

{

(

−Π

3 +mΠ ;−

Π

3 −kΠ+mΠ

)

,

(

Π

3 +kΠ+mΠ ;

Π

3 +mΠ

)

}

180) sin

10x

+cos10x

4 =

sin6x

+cos6x

4 cos22x+sin22x ⇔x=k

Π

181) cosx

√

cosx −1+cos 3x

√

cos 3x−1=1 VN

182) sin2x+ 1

4 sin23x=sinxsin23x ⇔x=kΠ ;x=

Π

6 +k2Π ; x=
5Π

6 +k2Π

183)

(

sin2x+ 1

sin2x

)

(

cos2x+ 1

cos2x

)

=12+1

2siny ⇔(x , y)∈

(

Π

4 +k

Π

2 ;

Π

2 +l2Π

)

184) 2 tanx+cotx=2sin 2x+ 1

sin 2x ⇔x=±
Π

6 +kΠ ; x=

Π

4 +k

Π

185) sin3x+cos3x=2-sin4x ⇔x=Π

2+k2Π

186) cos16xsin4x=1 ⇔x=Π

8 +k

Π

187) cos13x+sin14x=1 ⇔x=k2Π ; x=Π

2 +kΠ

188) 3tan2x+4cos2x+2

√3 tanx-4 √3 cosx+4=0 ⇔x=−Π

6 +k2Π

189) cos2x-4cosx-2xsinx+x2+3=0  x=0

190) 3cot2x+4cos2x-2

√3 cotx-4cosx+2=0 ⇔x=Π

3 +k2Π

191) cos2007x+sin2008x=1 ⇔x=k2Π ; x=Π

2 +kΠ

192) cos2008x+sin2008x=1 ⇔x=kΠ

193) cos2008x+sin2009x=1 ⇔x=kΠ ;x=Π

2 +k2Π

194) cos2009x+sin2009x=1 ⇔x=k2Π ; x=Π

2 +k2Π

195) sin2(x- Π )-sin(3x- Π )-=sinx ⇔x=kΠ ;x=Π

3 +k
2Π

196) 2√3 sinx= 3 tanx

2√sinx −1−√3 ⇔x=30

+k3600; x=500+k3600; x=1700+k3600

197) sinx(sinx+cosx)−1

cos2x

+sinx −1 =0 ⇔x=−

Π

2 +k2Π ; x=

Π

</div>
(25)<div class='page_container' data-page=25>

198) √1−cosx −√1+cosx

cosx =4 sinx ⇔x=arcsin

1−√5

4 +k2Π ; x=Π −arcsin
1−√5

4 +k2Π

hoặc ⇔x=− Π

10+k2Π ; x=
11Π

10 +k2Π

199) sin3x-7sin2xcosx+11sinxcos2x-6cos3x=0

⇔x=Π

4 +kΠ ; x=arctan 2+kΠ ; x=arctan 3+kΠ

200) 9sin3x-5sinx+2cos3x=0 ⇔x=arccot 2+kΠ ; x=1

2arc cot 4+kΠ

201) sin2x+sinx+cos3x=0 ⇔x=−Π

2 +k2Π ; x=

Π

4 ±arccos√
2−2

2 +k2Π

202) tan2x=1+cos

x

1+sin3x ⇔x∈

{

(2k+1)Π ;

Π

4 +kΠ ;

Π

4 ±arccos

(

√2

2 −1

)

+k2Π

}

203) 4sin3xsin3x-4cos3xcos3x= −5

2 ⇔x=±

Π

12+k

Π

204) tan2xtan23xtan4x=tan2x-tan23x+tan4x ⇔x=kΠ ;x=Π

4 +k

Π

205) sin6x+sin8x+sin16x+sin18x+16sin3x=0 ⇔x=kΠ

206) 2sin3x(1-4sin2x)=1

⇔x=Π

14+k
2Π

7 ∧k ≠
7m+3

2 ,m∈Z ; x=

Π

10+k
2Π

5 ∧k ≠

5n+2

2 , n∈Z

207) (cosx −2 sin 4x)sin 4x+(1+sinx −2 cos 4x)cos 4x=0

⇔x=Π

2 +k2Π

208) cos4x+(cos2x-sinx)2=5 ⇔x=Π

2 +k2Π

209) tan2x+tan22x+cot23x =1 VN

210) 3(tan2x+tan22x+tan23x)=tan2xtan22xtan23x ⇔x=kΠ

211) cosx-3 √3 sinx=cos7x ⇔x=kΠ

212) sin4x+cos4x=−1

2cos

2x VN

213) sin6x+cos6x+ 1

2 sin4x=0 ⇔x=

3Π

8 −

4arccos
4
5+k

Π

214) 8cos4x-4cos2x+sin4x-4=0 ⇔x=kΠ

2 ; x=

Π

8+k

Π

215) 1+sinx-cosx-sin2x+2cotx+2=0 ⇔x=Π

4 +kΠ ; x=arccos
3

√10±arccos
1

√10+k2Π

216) sinx −sin1x =sin2x − 1

sin2x ⇔x=

Π

2+kΠ

217) 2tan2x+3tanx+2cot2x+3cotx+2=0 ⇔x=−Π

4 +kΠ

218) 2sin3x+4cos3x=3sinx ⇔x=Π

4 +kΠ

219) 3 sin2x

2cos

(

3Π

2 +

x

)

+3 sin

2x

2cos

x

2=sin

x

2cos

2x

2+sin

(

Π2+

x

)

cos

x

⇔x=−Π

2 +k2Π ; x=±

Π

3+k2Π

220) cosxcos3x-sin2xsin6x-sin4xsin6x=0 ⇔x=Π

2 +kΠ ; x=

Π

18+k

Π

221) sin4xsin5x+sin3xsin4x-sin2xsinx=0 ⇔x=kΠ

5 ; x=k

Π

2 ; x=±

Π

3 +kΠ

222) sin5x+sin3x=sin4x ⇔x=kΠ

4 ; x=±

Π

</div>
(26)<div class='page_container' data-page=26>

223) cosx+cos3x+2cos5x=0

⇔x=Π

2+kΠ ; x=±
1
2arccos

1−√17

8 +kΠ ; x=±
1
2arccos

1+√17

8 +kΠ

224) cos22x+3cos18x+3cos14x+cos10x=0 ⇔x=Π

4 +k

Π

2 ; x=

Π

32+k

Π

225) sin23x+sin24x=sin25x+sin26x ⇔x=k Π

2 ; x=k

Π

226) sin2x+sin22x+sin23x=3/2 ⇔x=± Π

3 +kΠ ; x=

Π

8 +k

Π

227) sin22x+sin24x=sin26x ⇔x=kΠ

4 ; x=±

Π

12+k

Π

228) cos23x+cos24x+cos25x=3/2 ⇔x=± Π

3 +kΠ ; x=

Π

16 +k

Π

229) 8cos4x=1+cos4x ⇔x=± Π

3 +kΠ

230) cos4x+sin4x=cos4x ⇔x=kΠ

231) sin2x+2cos2x=1+sinx-4cosx ⇔x=±Π

3+k2Π

232) (2sinx-cosx)(1+cosx)=sin2x ⇔x=Π+k2Π ; x=Π

6 +k2Π ; x=
5Π

6 +k2Π

233) sin2xtanx+cos2xcotx-sin2x=1+tanx+cotx ⇔x=− Π

12+kΠ ; x=
7Π

12 +kΠ

234) sin6x+3sin2xcosx+cos6x=1 ⇔x=kΠ

235) cosxsin3x-sinxcos3x= √2

8 ⇔x=−

Π

16+k

Π

2 ; x=
5Π

16 +k

Π

236) (2sinx-1)(2sin2x+1)=3-4cos2x

⇔

{

kΠ ;Π

4 ±arccos
1−√3

2√2 +k2Π ;

Π

4 ±arccos
1+√3

2√2 +k2Π

}

237) sin

(

Π

2+2x

)

cot 3x+sin(Π+2x)−√2 cos 5x=0 ⇔

{

Π

10+k

Π

5 ;

Π

12+k
2Π

3 ;

Π

4 +k
2Π

}

238) tan2x+cos4x=0 ⇔x=Π

4+k

Π

2 ; x=±
1
2arccos

√5−1

2 +kΠ

239) sin4

(

x+Π

)

=
1
4+cos

2x −cos4x

⇔x=k Π

240) (2sinx+1)(3cos4x+2sinx-4)+4cos2x=3 ⇔x=k Π

2 ; x=−

Π

6 +k2Π ; x=
7Π

6 +k2Π

241) √2sin3

(

x+Π

)

=2sinx ⇔x=

Π

4+kΠ

242)2sinx+cotx=2sin2x+1 ⇔x=Π

6+k2Π ; x=
5Π

6 +k2Π ; x=−

Π

4 ±arccos
1−√5

2√2 +k2Π

243) tan2x(1-sin3x)+cos3x-1=0 ⇔x=Π

4 +kΠ ; x=k2Π ; x=

Π

4 ±arccos√
2−1

√2 +k2Π

244) 1+cot 2x=1−cos 2x

sin22x ⇔x=

Π

4+k

Π

245) 6 sinx −2cos3x

=5 sin 4xcosx

2 cos 2x VN

246) √1+cosx+√1−cosx

cosx =4 sinx víi:0<x<2 Π ⇔

{

Π

6 ;
3Π

10 ;
7Π

6 ;
13Π

}

</div>

lượng giác toán học 12 nguyễn văn bốn thư viện tài nguyên dạy học tỉnh thanh hóa

<b> A) Ph</b>

<b> ơng trình l</b>

<b> ợng giác cơ bản </b>

<i>L</i>

<i> u ý</i>

<i> </i>

Tỉng qu¸t

<i>L</i>

<i> u ý</i>

<i> </i>

Tỉng qu¸t

<i>L</i>

<i> u ý</i>

<i> </i>

Tæng qu¸t

<i>L</i>

<i> u ý</i>

<i> </i>

Tæng qu¸t

D¹ng: at+b=0

D¹ng: at

<b>+bt+c =0</b>

<b> D¹ng: asinf(x)+bcosf(x) =c</b>

PP gi¶i:

<i><b> *) Khi a=0 hoặc b=0 bài toán trở thành dạng A) giải đợc</b></i>

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√

√

√

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Đặc biệt

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<b> ơng trình l</b>

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<i><b> Theo s¸ch cơ bản-sách nâng cao & các sách tham khảo.</b></i>

<i> Dạng cơ bản</i>

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