Tải bản đầy đủ

Phương trình lượng giác thường gặp – lê văn đoàn

SỞ GD & ĐT TP. HỒ CHÍ MINH

ĐỀ CƯƠNG HỌC TẬP LỚP 11

TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

§ 3. PHƯƠNG TRÌNH LƯNG GIÁC THƯỜNG GẶP
Dạng toán 1. Phương trình bậc hai và bậc cao theo một hàm lượng giác
Quan sát và dùng các cơng thức biến đổi để đưa phương trình về cùng một hàm lượng giác
(cùng sin hoặc cùng cos hoặc cùng tan hoặc cùng cot) với cung góc giống nhau, chẳng hạn:
Dạng

Đặt ẩn phụ

Điều kiện

a sin2 X  b sin X  c  0

t  sin X


1  t  1

a cos2 X  b cos X  c  0

t  cos X

1  t  1

a tan2 X  b tan X  c  0

t  tan X

a cot2 X  b cot X  c  0

t  cot X

X 


 k
2

X  k

Nếu đặt t  sin2 X , cos2 X hoặc t  sin X , cos X thì điều kiện là 0  t  1.

Nhóm 1. Phương trình bậc hai cơ bản
1. Giải: 2 sin2 x  sin x  1  0.

2. Giải: 4 sin2 x  12 sin x  7  0.

Đặt sin x  t thì t  [1;1].

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Phương trình trở thành 2t 2  t  1  0

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 t  1 (nhận) hoặc t  

1
(nhận).
2

 Với t  1  sin x  1  x 


 k 2.
2

1
1
 sin x  
2
2

 x     k 2
  

6
 sin x  sin    
.

7

6


x 
 k 2

6

 


7
S    k 2;   k 2;
 k 2, k   .
 2

6
6
 Với t  

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 Nhận xét. Có thể trình bày nhanh như sau:

..............................................................................

sin x  1

2 sin x  sin x  1  0  
...
sin x   1

2

 

5
Đáp số: S  
 k 2, k   .
  k 2;
 6

6

2

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Trang - 69 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

3. Gii: 2 cos2 x 5 cos x 2 0.

4. Gii: cos2 x 3 cos x 2 0.

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ỏp s: S {/3 k 2, k }. ...............

ỏp s: S {k 2, k }. .............................

5. Gii: 2 tan2 x 2 3 tan x 3 0.

6. Gii: tan2 x (1 3) tan x 3 0.

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S: S
k ; x k , k . ...
6

3




ỏp s: S
k ; k , k . ..
3

4

7. Gii:

3 cot2 x (1 3) cot x 1 0.

8. Gii:

3 cot2 x (1 3) cot x 1 0.

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ỏp s: S
k ; x k , k .
4

3




ỏp s: S
k ; k , k . ..
4

3

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S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

sin2 1 cos2
Nhúm 2. S dng cụng thc sin2 cos2 1 2
.
2
cos 1 sin
9. Gii: 6 cos2 x 5 sin x 2 0.

10. Gii: 2 sin2 x 3 cos x 3 0.

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7
ỏp s: S
k 2 . .........
k 2;
6

6




ỏp s: S
k 2; k 2, k . ......


3

11. Gii: 2 cos2 x 5 sin x 4 0.

12. Gii 2 sin 2 x ( 3 4) cos x 2 3 2.

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5
ỏp s: S
k 2, k .
k 2;
6

6



ỏp s: S
k 2, k . .................
6


13. Gii: 4 sin 4 x 12 cos2 x 7.

14. Gii: 3 sin2 x 2 cos4 x 2 0.

PT 4(sin2 x )2 12(1 sin2 x ) 7 0

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t sin2 x t thỡ t [0;1] v phng trỡnh

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tr thnh 4t 2 12t 5 0

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k

ỏp s: S
, k . .....................

4

2



k
ỏp s: S

, k . .............
k ;


4
2

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Trang - 71 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 3. S dng cụng thc cos 2x

2 cos2 x 1 (1)
1 2 sin2 x (2)

khi cung gúc gp ụi nhau.

15. Gii: 2 cos 2x 8 cos x 5 0.

16. Gii: cos 2x 5 sin x 2 0.

PT 2(2 cos2 x 1) 8 cos x 5 0

PT 1 2 sin2 x 5 sin x 2 0

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ỏp s: S
k 2, k . ..................
3





ỏp s: S
k 2, k .
k 2;
6

6

17. Gii: cos2x 9 cos x 5 0.

18. Gii: cos 4x 9 sin 2x 8 0.

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PT (1 2 sin2 2x ) 9 sin 2x 8 0

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2

ỏp s: S
k 2, k . ...............

3




ỏp s: S
k , k . .......................
4


19. Gii: 5 cos x 2 sin

x
7 0.
2

Phng trỡnh 5 cos 2

x
x
2 sin 7 0
2
2

20. Gii: sin2 x cos 2x cos x 2 0.
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ỏp s: S { k 4, k }. ......................

ỏp s: S {k 2, k }. .............................

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Trang - 72 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 4. Va h bc, va nhõn ụi khi tn ti cung gúc gp 4 ln nhau

2
sin x 1 cos 2x

2
H bc:
cos2 x 1 cos 2x

2


cos 2x 2 cos 2 x 1
Nhõn ụi:

cos 4x 2 cos2 2x 1 ......
cos 6x 2 cos2 3x 1

21. Gii: cos 4x 12 sin2 x 1 0

22. Gii: 1 cos 4x 2 sin2 x 0.

Ta cú: cos 4x 12 sin2 x 1 0

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(2 cos2 2x 1) 12.

1 cos 2x
1 0
2

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2 cos2 2x 1 6(1 cos 2x ) 1 0

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2 cos2 2x 6 cos 2x 8 0

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ỏp s: S k , k . ...............................

ỏp s: S /2 k ; /6 k . ........

23. Gii: cos 4x 2 cos2 x 1 0.

24. Gii: 8 cos2 x cos 4x 1.

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ỏp s: S
k ; k , k . .........


3



ỏp s: S
k , k . ...................
3


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Trang - 73 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

25. Gii: 6 cos2 3x cos12x 7.

26. Gii: 5(1 cos x ) 2 sin 4 x cos 4 x .

Phng trỡnh 6 cos2 3x cos 4.3x 7

PT 5(1 cos x ) 2 (sin 2 x )2 (cos2 x )2

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ỏp s: S
k , k . .......................
2


2

ỏp s: S
k 2, k . ...............

3


27. Gii: cos 2x 3 cos x 4 cos2

x

2

PT (2 cos 2 x 1) 3 cos x 4.

1 cos x
2

28. Gii: cos 2x 2 cos x 2 sin 2

x

2

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2

ỏp s: S
k 2, k . ...............

3




ỏp s: S
k 2 k . .................
3


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S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 5. S dng cụng thc liờn quan n tan, cot a v phng trỡnh bc hai

1
1
& cot x
cot x
tan x .
1
2
1 cot x
sin 2 x

tan x.cot x 1 tan x
1
1 tan 2 x &
2
cos x
29. Gii: tan x cot x 2.

30. Gii: 2 tan x 3 cot x 3 0.

sin x 0
k
K:
sin 2x 0 x


cos x 0
2


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1
2
tan x

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ỏp s: S
k , k . ......................
4


1


ỏp s: S
k ; arctan k . .....
4

2



31. Gii: 5 tan x 2 cot x 3 0.

32. Gii:

Ta cú tan x cot x 2 tan x

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3 tan x 6 cot x 2 3 3 0.

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2


ỏp s: S
k ; arctan k . .
4

5





ỏp s: S
k ; arctan(2) k . ...
3


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Trang - 75 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

33. Gii phng trỡnh:

3
4
2

3

2
tan
x
.
34.
Gii
phng
trỡnh:
tan x 7.
cos2 x
cos2 x

k , k .
2

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Phng trỡnh 3(1 tan2 x ) 3 2 tan 2 x

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ỏp s: S k , k . ................................

ỏp s: S /4 k ;arctan 3/4 k .

iu kin: cos x 0 x

35. Gii phng trỡnh:

3
3 cot x 3.
sin2 x

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36. Gii phng trỡnh:

1
cot x 3.
sin2 x

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ỏp s: S
k , k . .......
k ;
2

6



ỏp s: S
k ; arccot(2) k . ...
4


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Trang - 76 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

37. Gii: 9 13 cos x

4
0.
1 tan 2 x

iu kin: cos x 0. p dng cụng thc:

1
1

cos2 x
2
2
cos x
1 tan x
thỡ phng trỡnh tr thnh:
1 tan2 x

38. Gii phng trỡnh 2 tan 2 x 3

3

cos x

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9 13 cos x 4 cos2 x 0
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ỏp s: S k 2, k . .............................

ỏp s: S k 2, k . .............................

1
2
5
39. Gii: tan 2 x
0.
2
cos x 2

40. Gii:

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3 sin x cos x

1

cos x

Hng dn: Chia hai v cho cos x 0.
iu kin: cos x 0 x


k , k .
2

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ỏp s: S
k 2, k . .................
3





ỏp s: S
k , k . ..............
k ;


3

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Trang - 77 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 6. Phng trỡnh quy v phng trỡnh bc hai (dng nõng cao)
41. 4 cos2 (6x 2) 16 cos2 (1 3x ) 13.

42. cos(2x 150) 3 sin(15 x ) 1 0.

Ta cú: 4 cos2 (6x 2) 16 cos2 (1 3x ) 13

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2

4 cos 2.(3x 1) 16 cos(3x 1) 13




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43. Gii: cos2 x 4 cos x 4.
3

6

Nhn xột:






6
3
2
6
2 3





PT cos2 x 4 cos x 4
3



2 3


5


9.
44. Gii: 5 cos 2x 4 sin

x

3
6

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cos2 x 4 sin x 4 0
3

3


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Trang - 78 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

45. cos2x 3 sin 2x 3 sin x 4 cos x .

46.

Cú cos 2x 3 sin 2x 3 sin x 4 cos x

..............................................................................

3 sin 2x cos 2x 3 sin x cos x 4


t t 3 sin x cos x 2 sin x

6

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Vỡ sin x [1;1] t [2;2].

6
2

2

2

t 3 sin x cos x 3 sin 2x
2

t

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2

Khi ú t ( 3 sin x cos x )
2

3 sin 2x 3 sin x cos2x cos x 2.

2

3 sin 2x (2 sin x 1) 2

t 2 2 3 sin 2x cos 2x .
2

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Phng trỡnh tr thnh t t 6 0

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47. 2sin2 x 3 sin2x 4 4.( 3sinx cos x).

48. 2sin2 x 3 sin2x 2 3 sinx 2cos x 2.

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Trang - 79 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020


2

4


1.
49. 2 cos2 x

9

cos
x


cos2 x
cos x

iu kin: cos x 0 x


k , k .
2
2

t

2

2
cos x t
cos x t 2
cos x
cos x


4
4


cos x cos2 x t 2
2
cos x cos x




1
sin x 1 7.

50. 4 sin2 x

4


sin x
sin2 x

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4
cos x
t 2 4. Khi ú phng
2
cos x
trỡnh tr thnh: 2(t 2 4) 9t 1 0

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S: S { k 2; 2/3 k 2, k }. .

S: S {/6 k 2; 7/6 k 2, k }.

2

51. cos2 x

1
2
2 2 cos x

2
cos x
cos x

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2


4
2 cos2 x
1.
52. 9

cos
x
cos x
cos2 x



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ỏp s: S {k 2, k }. ..............................

ỏp s: S { 2/3 k 2, k }. ...........

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Trang - 80 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

53. Gii phng trỡnh cos

4x
cos2 x .
3

3 x 1

3x
sin .
54. Gii: sin

10 2 2
10
2

2x 1 1
Phng trỡnh cos 2. cos 2x
3 2 2


3 x 1
3 x
sin 3
PT sin

10 2 2
10 2



2x 1 1
2x
cos 2. cos 3.
3 2 2
3

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k
S: S
, k . .............
k 3;


4
2

3

2
S: S
k 2, k . ......
k 2;
5

5

55. Gii: 2 cos2

3x
4x
1 3 cos

5
5

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56. Gii phng trỡnh: cos

8x
2x
cos2

3
3

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k 4
k 4
3 1 21
S
;
, cos

. ..
3
3
2
4



3 k 3
4
3
S
; k 2, cos
. .....
4
2
3
4

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Trang - 81 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

57. 3 cos 4x 8 cos6 x 2 cos2 x 3 0.

58. 4 3 sin x sin 3 x 3 cos2 x cos6 x .

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x
59. 2 cos2 x 1 tan x tan cos 2x 3.

2
x
x
sin x
2
Ta cú: 1 tan x tan 1

2
cos x
x
cos
2
x
x
x
cos x cos sin x sin
cos
2
2
2 1

x
x cos x
cos x.cos
cos x cos
2
2
sin

60.



3
1 tan x tan x

tan
x

2
3

sin
x

2
cos2 x

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Trang - 82 -


SỞ GD & ĐT TP. HỒ CHÍ MINH

ĐỀ CƯƠNG HỌC TẬP LỚP 11

TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

Đề rèn luyện về nhà số 01
Giải các phương trình lượng giác sau:
1)

2 cos2 x  ( 2  2)cos x  2  0.

2)

2 sin 3 x  sin2 x  2 sin x  1  0

3)

6  4 cos2 x  9 sin x  0.

4)

4 sin 4 x  12 cos2 x  7

5)

cos 2x  3 cos x  2  0.

6)

4 cos 2x  5 sin x  1.

7)

5 tan x  2 cot x  3  0.

8)

1  cos 4x  2 sin2 x  0.

9)

sin 2 8x  3 sin 4x cos 4x 



3
ĐS: S  
 k 2, k   .
k 2; 


4
 

ĐS: S  
  k , k   .
 2



1
1
ĐS: S  
arcsin  k 2;   arcsin  k 2 .


4
4
  k 

ĐS: S  
, k   .
 
 4

2
 2

ĐS: S  
 k 2;   k 2, k   .

 3

 

3
3
ĐS: 
  k2;arcsin  k2;   arcsin  k2.
 2

8
8

5
 0.
2

 

 2 
ĐS: S  
  k ; arctan    k , k   .
 4

 5 


 


ĐS: S  
  k ;   k , k   .
 2

6
 

k
ĐS: S  
, k   .
 
16

4

Đề rèn luyện về nhà số 02
Giải các phương trình lượng giác sau:
1)

2 sin2 x  3 sin x  1  0.

2)

3 cot2 x  2 3 cot x  1  0.

3)

4 sin x  3  2(1  sin x ) tan 2 x .

4)
5)
6)
7)
8)

5
 0.
2
1
2
5
 tan 2 x 
  0.
2
cos x 2
1
3 sin x  cos x 

cos x
x
cos 2x  3 cos x  4 cos2 
2
1
1
sin x 
 sin2 x 

sin x
sin2 x
cos 2x  4 cos x 

 


5
ĐS: S  
 k 2;
 k 2  .
  k 2;
 2

6
6

ĐS: x    k , (k  ).
3

7
ĐS: x    k 2  x 
 k 2, (k  ).
6
6

ĐS: x    k 2, (k  ).
3

ĐS: x    k 2, (k  ).
3

ĐS: x  k   x   k , (k  ).
3
2
ĐS: x  
 k 2, (k  ).
3

ĐS: x   k 2, (k  ).
2

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Trang - 83 -


SỞ GD & ĐT TP. HỒ CHÍ MINH

ĐỀ CƯƠNG HỌC TẬP LỚP 11

TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

Dạng toán 2. Phương trình lượng giác bậc nhất đối với sin và cos (pt cổ điển)
Dạng tổng qt: a sin x  b cos x  c () , a, b   \ {0}.
Điều kiện có nghiệm của phương trình: a 2  b 2  c 2 , (kiểm tra trước khi giải)
Phương pháp giải:
a
b
c
 Chia 2 vế a 2  b 2  0, thì () 
sin x 
cos x 
()
2
2
2
2
2
a b
a b
a  b2



Giả sử: cos  

a
2

a b

2

, sin  

b
2

2

,

  0;2   thì:

a b
c
c
()  sin x cos   cos x sin  
 sin(x  ) 
: dạng cơ bản.
2
2
2
a b
a  b2
sin a  cos b  cos a  sin b  sin(a  b)
Lưu ý. Hai cơng thức sử dụng nhiều nhất là: 


cos a  cos b  sin a  sin b  cos(a  b)

Các dạng có cách giải tương tự:

 2
2

 a  b cos nx
, (a 2  b2  0)
 a.sin mx  b.cos mx  
PP
2
2


 Chia : a 2  b2 .
 a  b sin nx



 a.sin mx  b.cos mx  c.sin nx  d.cos nx, (a 2  b 2  c2  d 2 )

Nhóm 1. Dạng cơ bản a sin X  b cos X  c.

61. Giải: sin x  3 cos x   3.

62. Giải: sin x  3 cos x  1.

Điều kiện có nghiệm:
2

2

2

2

..............................................................................
2

a  b  1  ( 3)  4  ( 3) : đúng.

..............................................................................

Chia 2 vế cho a 2  b 2  2 thì

..............................................................................

phương trình 

1
3
3
sin x 
cos x  
2
2
2



3
 cos x sin  
3
3
2
 

   sin  
3 
 3 

 sin x cos

 sin x



 x  k 2
 x       k 2


3
3


 x  5  k 2
 x    4   k 2


3
3
3




5
Kết luận: S  
 k 2, k   . ......
k 2;


3

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 


Đáp số: S  
 k 2, k  
  k 2;
 6

2

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Trang - 84 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

63. Gii:

3 sin x cos x 1.

64. Gii:

3 cos x sin x 2.

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S: S
k 2; k 2, k . .......
3




5
S: S
k 2, k . ......
k 2;
12

12

65. Gii:

3 sin 2x cos 2x 2.

66. Gii:

3 sin 3x cos 3x 2.

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7


S: S
k , k . .........
k ;
24

24

5 k 2 11 k 2

S: S
;

, k . .

36

3
36
3

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S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

67. Gii:



3 sin x sin x 2.
2




68. Gii: sin 2x 3 sin( 2x ) 1.
2




p dng cụng thc sin cos thỡ
2




p dng cụng thc sin cos v
2


phng trỡnh ó cho tr thnh:

sin( ) sin thỡ phng trỡnh tr thnh

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S: S
k 2, k . .........................
6




S: S
k ; k , k . ......................
3


69. Gii:





3 sin x sin x 2.
4

4


Nhn xột:






4
4
2
4
2 4





PT 3 sin x sin x 2

4
4
2

70. Gii:





3 sin x sin x 2.
3

6


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S: S
k 2; k 2, k . ......
6

3



S: S
k 2, k . ............................
3


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Trang - 86 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

71.



3 cos2x sin 2x 2 sin 2x 2 2
6




72. 2 3 sin x cos x 3 sin x 2 3

3

Chia 2 v cho 2 thỡ phng trỡnh tr thnh

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5

S: S
k , k . ...........................
24




2
S: S
k 2, k . .......
k 2;
3

3

73. sin 3x cos x 3 cos2x 3 cos 3x sin x

74. cos7x cos5x 3 sin2x 1 sin7x sin5x.

(sin 3x cos x cos 3x sin x ) 3 cos2x 3

..............................................................................

sin 2x 3 cos 2x 3

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S: S
k , k . .............
k ;
3

2



S: S
k ; k , k . ...................
3


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Trang - 87 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 2. Dng

75. Gii:

a sin x b cos x a 2 b 2 sin( x )
a sin x b cos x a b cos( x )

3 sin x cos x 2 sin

2



12

Chia 2 v cho a 2 b 2 2 phng trỡnh tr
thnh

3
1

sin x cos x sin
2
2
12

2

, ( a2 b2 0 )



76. Gii: sin x 3 cos x 2 sin x

6
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3
S: S
k 2, k . ...
k 2;
12

4



S: S
k , k . ..............................
4


77. Gii: sin 3x 3 cos 3x 2 sin 2x .

78. Gii: cos x 3 sin x 2 cos 3x .

..............................................................................

Phng trỡnh 2 cos 3x cos x 3 sin x

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4 k 2
S: S

, k . .......
k 2;
3

15
5




k
S: S

, k . ...........
k ;
6

12
2

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Trang - 88 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 3. Dng a sin( mx) b cos( mx) c sin(nx) d cos( nx)

a 2 b 2 c2 d 2 0
79. cos 2x 3 sin 2x

3 sin x cos x .

80.

3(cos 2x sin 3x ) sin 2x cos 3x .

1
3
3
1
cos 2x
sin 2x
sin x cos x
2
2
2
2

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S: x

2
k 2
k 2 x
, k . ...
3
3

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S: x



k 2
k 2 x
.....
6
10
5

81. cos 3x sin x 3(cos x sin 3x ).

82. sin 8x cos 6x 3(sin 6x cos 8x ).

PT cos 3x 3 sin 3x 3 cos x sin x

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S: x


k
k x
, (k ).
12
8
2

S: x



k
k x

, (k ).
4
12
7

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Trang - 89 -


SỞ GD & ĐT TP. HỒ CHÍ MINH

ĐỀ CƯƠNG HỌC TẬP LỚP 11

TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

Đề rèn luyện về nhà số 03
Câu 1. Giải các phương trình lượng giác sau:
a)

sin x  3 cos x  2.

b)

 2 6 
cos 7x  3 sin 7x   2, x   ;  
 5 7 

5
 k 2, k  .
6
53
5
59
ĐS: x 
 x
 x

84
12
84

c)

2 cos 3x  3 sin x  cos x  0.

ĐS: x 

d)

2 sin 17x  3 cos 5x  sin 5x  0.

ĐS: x

e)

( 3  1)sin x  ( 3  1)cos x  2 2 sin 2x .

ĐS: x

f)

sin 8x  cos 6x  3(sin 6x  cos 8x ).

ĐS: x

g)

ĐS: x 



3 cos 2x  sin 2x  2 sin 2x    2 2.
6 


ĐS: x

 k

, k  .
3
2

k
 k
 
 x 

66 11
9
6
5
7  k 2

 k 2  x 


12
36
3


k
  k  x 


4
12
7
5

 k , k  .
24

Câu 2. Tìm tham số m để phương trình m cos 2x  (m  1)sin 2x  m  2 có nghiệm.
Câu 3. Tìm giá trị lớn nhất và nhỏ nhất của hàm số y 

2 cos x  sin x  1

2  sin x  cos x

Đề rèn luyện về nhà số 04
Câu 1. Giải các phương trình lượng giác sau:
a)
b)
c)



sin   2x   3 sin(  2x )  1.
2



cos x  3 sin x  2 cos   x  .
3

sin x  cos x  2 2 sin x cos x .

ĐS: x  


7
 k 2  x 
 k 2.
6
6

ĐS: x  k , (k  ).
ĐS: x 


 k 2
 k 2  x  

4
4
3

 9
 1 3
 5

2

5
cos   x  
sin   x  
 ĐS: x   k 2  x 
 k 2.
 2
 2
2
3
6


2 2
2 2
1  cos 2x
 k
e) 1  cot 2x 

ĐS: x  
, k  .
2
3
2
sin 2x

f)
3  2 cos2 x (sin 2x  cos 2x tan x )  3 cos 2x . ĐS: x  k   x   k , k  .
6

k
 k
g) 4 sin 3 x cos 3x  4 cos3 x sin 3x  3 3 cos 4x  3. ĐS: x  

 x 

24
2
8
2
d)

1 3

Câu 2. Tìm tham số m để phương trình sin x  5 cos x  1  m(2  sin x ) có nghiệm.
Câu 3. Tìm giá trị lớn nhất và nhỏ nhất của hàm số y 

3 cos x  2 sin x  1

2  sin x

Biªn so¹n vµ gi¶ng d¹y: Ths. Lª V¨n §oµn – 0933.755.607 – 0929.031.789

Trang - 90 -


SỞ GD & ĐT TP. HỒ CHÍ MINH

ĐỀ CƯƠNG HỌC TẬP LỚP 11

TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ

Mơn: Tốn, Năm học: 2019 – 2020

Dạng toán 3. Phương trình lượng giác đẳng cấp
Dạng tổng qt: a. sin2 X  b.sin X cos X  c. cos2 X  d

(1) a, b, c, d  .

Dấu hiệu nhận dạng: Đồng bậc hoặc lệch nhau hai bậc của hàm sin hoặc cosin (tan
và cotan được xem là bậc 0).
Phương pháp giải:
cos X  0

 Bước 1. Kiểm tra X   k    2
có phải là nghiệm hay khơng ?
sin X  1
2


 Bước 2. Khi X 

cos X  0

 k , (k  )   2
. Chia hai vế (1) cho cos2 X :

sin
X

1
2


sin2 X
sin X cos X
cos2 X
d
(1)  a
b
c

2
2
2
cos X
cos X
cos X
cos2 X

 a tan2 X  b tan X  c  d (1  tan2 X )

 Bước 3. Đặt t  tan X để đưa về phương trình bậc hai theo ẩn t  x .
 Lưu ý. Giải tương tự đối với phương trình đẳng cấp bậc ba và bậc bốn.
Nhóm 1. Đẳng cấp bậc hai
83. Giải 2 cos2 x  4 sin x cos x  4 sin2 x  1.
Với x 

cos x  0

 k , k     2
.
sin x  1
2


Phương trình trở thành 4  1 : sai  loại.
Với x 


 k , chia 2 vế cho cos2 x  0
2

Phương trình trở thành:

2

cos2 x
sin x cos x
sin 2 x
1

4

4

2
2
2
cos x
cos x
cos x
cos2 x

84. 2 cos2 x  3 3 sin 2x  4 sin 2 x  4.
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 2  4 tan x  4 tan2 x  1  tan2 x

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 5 tan2 x  4 tan x  1  0

 tan x  1
x    k 


4


.
1
  
 tan x  




x  arctan    k 
5
 5


..............................................................................

 

 1
 S    k ; arctan    k , k   .
 4
 5 



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 


Đáp số: S  
 k , k   . ......
  k ;
 2

6

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Trang - 91 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

85. 4 cos2 x 3 sin x cos x sin2 x 3.

86. 2 sin2 x 3 3 sin x cos x cos2 x 2.

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1
S: S
k ; arctan k . .........
4

4






S: S
k , k . .............
k ;
2

6

87. Gii: sin2 x 2 cos2 x 3 sin x cos x .

88. Gii: sin2 x 3 sin x cos x 1.

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S: S
k ; arctan 2 k , k . ....
4




1
S: S
k ; arctan k , k .
4

2

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Trang - 92 -


S GD & T TP. H CH MINH

ẹE CệễNG HOẽC TAP LễP 11

TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH

Mụn: Toỏn, Nm hc: 2019 2020

Nhúm 2. ng cp bc ba, bc bn
89. Gii phng trỡnh: cos x 2 sin 3 x .

cos x 0

k 3
thỡ phng
sin x 1
2

trỡnh ó cho tr thnh 0 2.13 : sai.
Vi x


k , chia 2 v ca phng trỡnh
2
cho cos3 x 0 thỡ phng trỡnh tr thnh:
Vi x

cos x
1
sin 3 x


2
cos x cos2 x
cos 3 x

90. Gii phng trỡnh: sin x 2 cos 3 x .
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1 tan2 x 2 tan3 x

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S: x


k , k . ..................................
4

91. Gii: cos3 x sin 3 x sin x cos x .

S: x


k , k . ..................................
4

92. Gii: 7 cos x 4 cos3 x 4 sin 2x .

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S: S
k , k . ..............................
2





S: S
k ; k 2, k . .......
2

3

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Trang - 93 -


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