PHÖÔNG TRÌNH LÖÔÏNG GIAÙC

1

PH??NG TR?NH L???NG GIA?C

GVBM : ?OA?N NGO?C DU?NG

BA?I 1 : Gia?i ca?c ph??ng tr?nh sau : (Ph??ng tr?nh l???ng gia?c c? ba?n)

1) sin 2x 7 3 12 2

2) 2 cos x 1 0 12

?S: x k x k (k Z)

8

24

?S: x k2 x k2 (k Z)

3

6

3) tan5x 3 10 3

4) 3cot 2x 3 0 7

5) sinx + cosx = 0

6) tanx ? cotx = 0

7) sin x 1 2

8) sin2x = 1 2

9) sin3x sin x 5 10

10) cos 2x cos5x 2 3 5

11) tan7x tan x 5 10

12) cot3x cot 2 x 5 3 10

13) cos2(x ?30o) ? sin2(x ?30o) = sin(x + 60o) 14) tan5x.tan3x = 1

15) 4cos2x 1 6 2 0

?S: x k (k Z) 75 5

?S: x 2 k (k Z) 21 2

S : x 3 k , k Z 4

S : x k , k Z 42

S : x k2 x 5 k2 x 7 k2

6

6

6

S : x k , k Z 42

?S: x 3 k x 11 k (k Z)

20

40 2

?S: x k 2 x 11 k 2 (k Z)

45 3

105 7

?S: x 3 k (k Z) 80 8

?S: x 9 k 3 (k Z) 70 7

S : x = 30o + k.120o, k Z S : x k , k Z

16 8 S : x 1 5 k , k Z

2 24

16) 8sin x 3 1 cos x sin x

S : x k x k , k Z

6

12 2

17) 8cos x.cos 2x.cos 4x sin 6x sin x

S : x k , k Z 14 7

18) cos x 3 sin x 2sin x 2 2 3

?S: x k2 x k2 (k Z)

3

6

19) sinx + cosxsin2x + 3 cos3x = 2(cos4x + sin3x) ?S: x k2 x k 2

6

42 7

20) 2

2 sin x

cos

x

1

12

?S: x = k ; x = k (k Z)

4

3

?H B 2009 DB?H 2007

21) cos3xcos3x ? sin3xsin3x = 2 3 2 8

22) sin2x ? 2 2 (sinx + cosx) ? 5 = 0

?S: x = k (k Z) 16 2

?S: x = ?3/4 + k2 (k Z)

DB?H 2006 DB?H 2004

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII TA?I LIE?U LUYE?N THI THPT QG ? ?A?I HO?C ........................................................................................................... GVBM : ?OA?N NGO?C DU?NG

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23) (1 2sin x)cos x 3 (1 2sin x)(1 sin x)

?S: x k2 , k Z 18 3

?H A 2009

24) 2 2 cos 3 x 3cos x sin x 0 4

25) tan4x + 1 = (2 sin 2 2x) sin 3x cos 4 x

26) cotx + sinx 1 tan x.tan x = 4

2

?S: x = k x = + k (k Z)

2

4

?S: x = + k2 x = 5 + k2

18 3

18 3

?S: x = + k x = 5 + k (k Z)

12

12

DB?H 2005 B?H 2002 ?H B 2006

(2 3) cos x 2sin 2 x

27)

2 4 1

2 cos x 1

28)

sin

x 2

cos

x 2

2

3 cos x 2

?S: x = 4 k2 (k Z) 3

?S: x k2 ; x k2 (k Z)

2

6

DB?H 2003 ?H D 2007

29) 3 cos5x 2sin 3x cos 2x sin x 0

?S: x k x k

18 3

62

?H D 2009

30)

tan 2

x 3tan2 x

cos 2x 1 cos2 x

?S: x = ? + k (k Z) 4

DB?H 2005

31) (2sin2x ? 1)tan22x + 3(2cos2x ? 1 ) = 0

?S: x = k (k Z) 62

DB?H 2006

32) sin 5x 2cos2 x 1

?S: x k 2 x k 2

63

14 7

?H B 2013

33) T?m nghie?m tre?n khoa?ng (0 ; ) cu?a ph??ng tr?nh 4sin2 x 2

3

cos

2x

1

2

cos

2

x

3 4

DB?H

2005

?S: x = 5 x = 17 x = 5

18

18

6

34) T?m x thuo?c ?oa?n [0 ; 14] nghie?m ?u?ng ph??ng tr?nh : cos3x ? 4cos2x + 3cosx ? 4 = 0 ?H D 2002

?S: x = , x = 3 , x = 5 , x = 7

2

2

2

2

BA?I 2 : Gia?i ca?c ph??ng tr?nh sau : (Gia?i ph??ng tr?nh l???ng gia?c ba?ng ca?ch bie?n ?o?i t?ch tha?nh to?ng)

1) sin4x.sin7x = cos3x.cos6x

S : x k x k , k Z

20 10

2

DB?H 2004

2) cosx.cos7x = cos3x.cos5x

S : x k x k , k Z

2

4

3) cos11x.cos3x = cos17x.cos9x

S : x k x k , k Z

20

6

4) sin18x.cos13x = sin9x.cos4x

S : x k x k , k Z

9

44 22

5) 4cos x.sin x sin x cos 2x 6 6

6) cos x cos x cos 3x sin x sin x sin 3x 1

22

2 22

S : x k2 , k Z 5

S : x k ; x k2 ; x k2 ; x 5 k2

4

2

6

6

BA?I 3 : Gia?i ca?c ph??ng tr?nh sau : (Gia?i ph??ng tr?nh l???ng gia?c ba?ng ca?ch bie?n ?o?i to?ng tha?nh t?ch)

1) cosx + cos2x + cos3x + cos4x = 0

S : x k x k2 x k2 , k Z

2

55

2) 1 + cosx + cos2x + cos3x = 0

S : x k x k2 , k Z

2

33

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII TA?I LIE?U LUYE?N THI THPT QG ? ?A?I HO?C ........................................................................................................... GVBM : ?OA?N NGO?C DU?NG

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3) sin2x + sin4x = sin6x 4) cosx + cos2x + cos3x + cos4x = 0 5) sin3x ? sinx + sin2x = 0 6) 1 + sinx + cos3x = cosx + sin2x + cos2x

S : x k x k x k , k Z

2

3

S : x k x k2 , k Z

2

55

S : x k2 x k2 x k , k Z 3

S : x k2 x k2 x k x 7 k2

3

6

6

BA?I 4 : Gia?i ca?c ph??ng tr?nh sau : (Gia?i ph??ng tr?nh l???ng gia?c ba?ng co?ng th??c ha? ba?c)

1) sin23x ? cos24x = sin25x ? cos26x 2) cos2 x cos2 2x cos2 3x cos2 4x 3

2 3) sin24x + sin23x = sin22x + sin2x

4) sin2 x sin2 3x 2cos2 2x 0 5) cos2 x cos2 2x cos2 3x 3

2 6) cos2 x cos2 2x cos2 3x cos2 4x 2

S : x k x k , k Z

2

9

S : x k2 x k

5

84

S : x = k x = k x = k , k Z

2

5

2

S : x k x k , k Z

2

82

S : x k x k , k Z

3

84

S : x k x k , k Z

42

10 5

BA?I 5 : Gia?i ca?c ph??ng tr?nh sau : (Ph??ng tr?nh l???ng gia?c co? ch??a to?ng cu?a sinx va? cosx)

1) sinx + cosx = 2

2) sin2x ? cos2x + sinx = cosx

S : x k2 , k Z 4

S : x = k2 x = + k2 , k Z 63

3) cos4 x sin4 x 1 4 4

S : x k x k , k Z

2

4

4) (2cosx ? 1)(sinx + cosx) = 1

S : x = k2 x = k2 , k Z 63

5) 1 tan x 2 2 sin x

S : x = /4 + k2, k Z

BA?I 6 : Gia?i ca?c ph??ng tr?nh sau : (Ph??ng tr?nh ba?c nha?t ?o?i v??i sinx, cosx)

1) 5 sinx + 2cosx = 4

S : x

2) 3 cos x sin x 2

S : x = 11 k2 x = 7 k2 , k Z

12

12

3) 3sinx + 4cosx = 5

4) sin2x + 6 cosx = 3cos2x + 2 sinx

S : x = k2 , k Z

S : x k x 5 k2 x k2 , k Z

3

12

12

5) sin4x + 3 cos4x = 3 6) 3 cos3x sin 3x 2 7) 3sin 3x 3 cos9x 1 4sin3 3x 8) cos 7x.cos5x 3 sin 2x 1 sin 7x.sin5x 9) 4sin3 x.cos 3x 4cos3 xsin3x 3 3 cos 4x 3

S : x k x k , k Z

2

12 2

S : x k2 x 5 k2 , k Z

36 3

36 3

S : x k2 x 7 k2 , k Z

18 9

54 9

S : x k x = k , k Z 3

S : x k x k , k Z

24 2

82

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII TA?I LIE?U LUYE?N THI THPT QG ? ?A?I HO?C ........................................................................................................... GVBM : ?OA?N NGO?C DU?NG

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10) Cho ph??ng tr?nh : 2sin x cos x 1 a sin x 2cos x 3

(a la? tham so?)

a) Gia?i ph??ng tr?nh khi a 1 3

b) T?m a ?e? ph??ng tr?nh co? nghie?m.

?S : x / 4 k ; 1/2 a 2

BA?I 7 : Gia?i ca?c ph??ng tr?nh sau : (Ph??ng tr?nh ba?c hai theo mo?t ha?m so? l???ng gia?c)

1) 2cos2x + cosx = 1 2) 2cos2(x 90) 5sin(x 90) 4

S : x = + k2 x = k2 , k Z S : x = 1200 + k.3600 x = 2400 + k.3600

3) 2cos2x + 2 cosx ? 2 = 0

4) 4cos2x ? 2 3 1 cosx + 3 = 0 5) tan2x + 1 3 tanx ? 3 = 0

6) cos2x + 9cosx + 5 = 0

7) cos2x + sinx + 1 = 0 8) sin22x ? 2cos2x + 3 = 0

4 9) cos2 3x.cos 2x cos2 x 0

10) sin2x cos 2x 3sin x cos x 2 0

11) cos x(2sin x 3 2) 2cos2 x 1 1 1 sin 2x

12) cos 2x 2 + 4sin x

3

3

=

5 2

13) tanx + 3 cotx ? 1 3 = 0

14) cos3x ? 2cos2x = 2

15) 2sin2x ? cos2x ? 4sinx + 2 = 0

16) 3cos 2x 4cos3 x cos3x 0

17) sin4 2x cos4 2x sin 2x.cos 2x 18) 3 2sin xsin3x 3cos 2x

S : x k2 , k Z 4

S : x k2 x k2 , k Z

3

6

S : x k x k , k Z

4

3

S : x 2 k2 , k Z 3

S : x = ?/2 + k2, k Z

S : x k x k , k Z

6

6

S : x k , k Z 2

S : x k2 ; x k2; x k2 ; x 5 k2

2

6

6

S : x k2 , k Z 4

S : x k2 x k2 , k Z

6

2

S : x k x k , k Z

4

3

S : x k x 2 k2 , k Z

2

3

S : x k2 x k2 x k2 2

S : x k2 x k2 , k Z 3

S : x k , k Z 82

S : x k , k Z

19) tan(7 x) 2 cot 9 x 3 0 2

20) sin 5x 1 5sin x

21) cot x tan x 2 cos 4x sin 2x

22) sin8 x cos8 x 17 cos2 2x 16

23) cot x tan x 4sin 2x 2 sin 2x

S : x k , k Z 4

S : x

S : x k , k Z 3

S : x k , k Z 84

S : x k , k Z 3

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII TA?I LIE?U LUYE?N THI THPT QG ? ?A?I HO?C ........................................................................................................... GVBM : ?OA?N NGO?C DU?NG

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24) sin4 x cos4 x 1 cot 2x 1

5sin 2x

2

8sin 2x

25) 2 cos6 x sin6 x sin x cos x 0 2 2sin x

26) 3 2sin xcos x 1 cos2 x 1 1 sin 2x

S : x k , k Z 6

S : x 5 2k , k Z 4

S : x k2 , k Z

27) 1 1 2 sin 2x cos x sin 4x

28) sin4 2x cos4 2x cos4 4x

tan

4

x

tan

4

x

S : x k2 x 5 k2 , k Z

6

6

S : x k , k Z 2

29) 2sin2x + 7sinx ? 4 = 0

?S: x k2 x 5 k2 , k Z

6

6

THPTQG 2016

30) cotx tanx 4sin 2x 2 sin2x

31) 2 cos6 x sin6 x sin x cos x 0 2 2sin x

?S: x = + k (k Z) 3

?S: x = 5 + h2 (h Z) 4

?H B 2003 ?H A 2006

32)

(1

sin

x

cos

2x) sin

x

4

1

cos x

1 tan x

2

?S: x k2 x 7 k2

6

6

?H A 2010

33) sin 2x cos 2x tan x cot x cos x sin x

?S: x = k2 (k Z) 3

DB?H 2007

34) cos23x.cos2x ? cos2x = 0

?S: x = k (k Z) 2

?H A 2005

35) 5sinx ? 2 = 3(1 ? sinx)tan2x

?S: x = + k2 ; x = 5 + k2

6

6

?H B 2004

36) cos 4 x sin 4 x cos x sin3x 3 0 ?S: x = + k (k Z)

4 4 2

4

?H D 2005

37) sin4 x cos4 x 1 cot 2x 1

5sin 2x

2

8sin 2x

?S: x = k , k Z 6

DB?H 2002

38) cos2x + cosx(2tan2x ? 1) = 2

?S: x = + k2 x = k2 (k Z) DB?H 2003 3

39) 4(sin3x + cos3x) = cosx + 3sinx

?S: x = + k x = + k (k Z) DB?H 2004

3

4

BA?I 8 : Gia?i ca?c ph??ng tr?nh sau : (Gia?i ph??ng tr?nh l???ng gia?c ba?ng ca?ch ??a ve? ph??ng tr?nh t?ch)

1) 2sin2x ? cos2x = 7sinx + 2cosx ? 4

S : x k2 x 5 k2 , k Z

6

6

2) cos3x 4cos 2x 3cos x 4 0

S : x / 2 k , k Z

3) sin4x ? cos4x = 1 + 2 2 sinxcosx

S : x k x k, k Z 2

4) 1 + tanx = sinx + cosx

S : x k x k2 4

5) sin 2x 2cos 2x 1 sin x 4cos x

S : x = /3 + k2 (k Z)

6) sin3 x cos 3x cos3 xsin3x sin34x

S : x k ; x k , k Z

12 4

12 4

7)

tan

x

cos

x

cos

2

x

sin

x1

tan

x

tan

x 2

S : x k2 , k Z

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII TA?I LIE?U LUYE?N THI THPT QG ? ?A?I HO?C ........................................................................................................... GVBM : ?OA?N NGO?C DU?NG

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