Truy
Truy cp ti t?i liu hc tp, b?i ging min ph?
Dng: sin 2x cos 2x sin x cos x 0
Bin i a v dng: m cos x a sin x b n a sin x bc sin x d 0 Hoc m sin xa cos x b n a cos x bc cos x d 0
C?u :Gii c?c phng tr?nh sau:
1). 8 sin6 x cos6 x 3 3 cos 2x 11 3 3 sin 4x 9 sin 2x
2). T?m nghim x 0; ca phng tr?nh: 5 cos x sin x 3
2
sin
2x
4
3). 9 sin x 6 cos x 3 sin 2x cos 2x 8 [DB A11]
3 sin 2x cos 2x 5 sin x 2 3 cos x 3 3
4).
1
2 cos x 3
5). sin 2x 2 cos 2x 1 sin x 4 cos x
6). 2 2 sin 2x cos 2x 7 sin x 2 2 cos x 4 0
7).
2
sin
2x
4
sin
x
3
cos
x
2
8). cos 4x 3 sin 4x 9 cos 2x 3 sin 2x 5 0
LI GII
1). 8 sin6 x cos6 x 3 3 cos 2x 11 3 3 sin 4x 9 sin 2x
LI GII
8
1
3 4
sin2
2x
3
3 cos 2x 11 6
3 sin 2x cos 2x 9 sin 2x
Ph?n phi, chuyn v phi sang v tr?i sau ? r?t gn ta c:
6 sin2 2x 9 sin 2x 3 3 3 cos 2x 6 3 sin 2x cos 2x 0
2 sin2 2x 3 sin 2x 1 3 cos 2x 2 3 sin 2x cos 2x 0
Ch? ?: ax2 bx c a x x1 x x2 vi x1 , x2 l? nghim ca ax2 bx c 0 ?p dng: 2 sin2 2x 3 sin 2x 1 sin 2x 12 sin x 1 sin 2x 12 sin x 1 3 cos 2x sin 2x 1 0
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
2 sin 2x 1 sin 2x 1 3 cos 2x 0 2 sin 2x 1 0 sin 2x 3 cos 2x 1 0
Vi 2 sin 2x 1 0 sin 2x 1 x k hoc x 5 k,k Z
2
12
12
Vi sin 2x
3 cos 2x 1 1 sin 2x 3 cos 2x 1
2
2
2
sin
2x
3
sin
6
.
x k hoc x 7 k,k ?
4
12
Nghim phng tr?nh: x k , x 5 k , x k , x 7 k k ?
12
12
4
12
2). T?m nghim x 0; ca phng tr?nh: 5 cos x sin x 3
2
sin
2x
4
LI GII
? tng: Bin i v phi th?nh sin2x v? cos2x , sau ? bin i th?nh t?ch...
5 cos x sin x 3 sin 2x cos 2x
5 cos x sin x 3 sin 2x 2 cos2 x 1 2 cos2 x 5 cos x 2 2 sin x cos x sin x 0
Ch? ?: 2 cos2 x 5 cos x 2 2 cos x 1cos x 1 2 cos x 1cos x 1 sin x2 cos x 1 0 2 cos x 1cos x sin x 2 0 2 cos x 1 0 cos x sin x 2 0
Vi 2 cos x 1 0 cos x 1
cos x
cos
x
3
k2
m,k Z.
2
3
x
3
m2
Vi cos x sin x 2 0
2
cos
x
4
2
cos
x
4
2 (v? nghim)
V?
x 0;
0 0
k2 3 m2
3
3
k2
2 3
3
m2
4 3
1 6
1 k 1
6
3
m 2 3
k 0 m 0
Kt lun nghim ca phng tr?nh: x . 3
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
3). 9 sin x 6 cos x 3 sin 2x cos 2x 8 (1)
6 cos x 6 sin x cos x 1 2 sin2 x 9 sin x 8 0
6 cos x sin x 1 2 sin2 x 9 sin x 7 0
6 cos xsin x 1 sin x 12 sin x 7 0
sin x 16 cos x 2 sin x 7 0
sin x 1 0 6 cos x 2 sin x 7 0
Vi sin x 1 0 sin x 1 x k2, k ? 2
Vi 6 cos x 2 sin x 7 0 phng tr?nh v? nghim (v? 62 22 72 )
Nghim ca phng tr?nh l?: x k2, k ? 2
3 sin 2x cos 2x 5 sin x 2 3 cos x 3 3
4).
1
()
2 cos x 3
iu kin 2 cos x 3 0 x 5 k2 6
() 3 sin 2x cos 2x 5sin x 2 3 cos x 3 3 2 cos x 3
3 sin 2x cos 2x 5 sin x 3 cos x 3 0
3 sin 2x 3 cos x cos 2x 5sin x 3 0
3 cos x 2 sinx 1 2 sin2 x 5 sin x 2 0
3 cos x 2 sinx 1 2 sinx 1sin x 2 0
2 sinx 1 3 cos x sin x 2 0 2 sinx 1 0 hoc 3 cos x sin x 2 0
x k2 hoc x 5 k2, k ?
6
6
So vi iu kin nghim ca phng tr?nh x k2 6
5). sin 2x 2 cos 2x 1 sin x 4 cos x
()
() sin 2x sin x 2 cos 2x 4 cos x 1 0
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
sin 2x sin x 4 cos2 x 4 cos x 3 0
sin x2 cos x 1 2 cosx 32 cos x 1 0 2 cos x 1sin x 2 cos x 3 0
2 cos x 1 0 hoc sin x 2 cos x 3 0 (v? nghim, v? 12 22 32 ) x k2,k ?
3 Kt lun: C?c tp nghim cn t?m x k2, k ?
3 6). 2 2 sin 2x cos 2x 7 sin x 2 2 cos x 4 0 ()
() 2 2 sin 2x 2 2 cos x cos 2x 7 sin x 4 0
2 2 cos x2 sin x 1 2 sin2 x 7 sin x 3 0
2 2 cos x2 sin x 1 sin x 32 sin x 1 0
2 sin x 1 2 2 cos x sin x 3 0 2 sin x 1 hoc 2 2 cos x sin x 3 .
Vi 2 sin x 1 x k2 hoc x 5 k2, k ?
6
6
Vi 2 2 cos x sin x 3 cos x 1 x k2, k ?
(vi 2 2 cos v? 1 sin ).
3
3
Kt lun: C?c tp nghim cn t?m x k2 , x 5 k2, x k2, k ?
6
6
7).
2
sin
2x
4
sin
x
3
cos
x
2
()
() sin 2x cos 2x sin x 3cos x 2 sin 2x sin x cos 2x 3cos x 2 0
sin x2 cos x 1 2 cos2 x 3 cos x 1 0
sin x2 cos x 1 cos x 12 cos x 1 0
2 cos x 1sin x cos x 1 0 2 cos x 1 hoc sin x cos x 1 .
2 cos x 1 hoc sin x cos x 1
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
x k2 hoc x k2 hoc x k2, k ? .
3
2
Kt lun: C?c tp nghim cn t?m x k2 , x k2 , x k2, k ?
3
2
8). cos 4x 3 sin 4x 9 cos 2x 3 sin 2x 5 0
2 cos2 2x 1 6 sin 2x.cos 2x 9 cos 2x 3 sin 2x 5 0
6 sin 2x.cos 2x 3 sin 2x 2 cos2 2x 9 cos 2x 4 0
3 sin 2x2 cos 2x 1 2 cos 2x 1cos 2x 4 0 2 cos 2x 13 sin 2x 2 cos 2x 4 0
2 cos 2x 1 0 hoc 3 sin 2x 2 cos 2x 4 0
Vi 2 cos 2x 1 0 cos 2x 1 x k, k ?
2
3
Vi 3 sin 2x 2 cos 2x 4 0 . Phng tr?nh v? nghim (v? (3)2 22 42 ).
1.30: Gii c?c phng tr?nh :
1). 1 sin x sin x cos x cos x
2). 1 sin x sin 2x cos x cos 2x 0 3). sin x sin 2x sin 3x cos x cos 2x cos x3x
4). sin 2x 2 cos x 3 sin x 3 1
1). 1 sin x sin x cos x cos x 1 1 1 sin x sin x cos x cos2 x
LI GII
1 cos2 x sin x sin x cos x 0 sin2 x sin x sin x cos x 0
sin x sin x 1 cos x 0 sin x 0 sin x cos x 1 Vi sin x 0 x k k ?
Vi sin x cos x 1
2
sin
x
4
1
x
2
k2
hoc
x
k2,k ?
.
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
Nghim phng tr?nh x k2 , x k2,k ? .
2 2). 1 sin x sin 2x cos x cos 2x 0
sin x sin 2x cos x 1 cos 2x 0 sin x 2 sin x cos x cos x 2 cos2 x 0
sin x 2 sin x cos x cos x 2 cos2 x 0
sin x 1 2 cos x cos x 1 2 cos x 0
1 2 cos xsin x cos x 0 1 2 cos x 0 sin x cos x 0
Vi 1 2 cos x 0 cos x 1 cos 2 x 2 k2.
2
3
3
Vi sin x cos x 0
2
sin
x
4
0
x
4
k
k z
3). sin x sin 2x sin 3x cos x cos 2x cos x3x
sin 3x sin x sin 2x cos 3x cos x cos 2x
2 sin 2x.cos x sin 2x 2 cos 2x.cos x cos 2x
sin 2x 2 cos x 1 cos 2x 2 cos x 1
2 cos x 1sin 2x cos 2x 0 2 cos x 1 0 sin 2x cos 2x 0
Vi 2 cos x 1 0 cos x 1 x 2 k2
2
3
Vi sin 2x cos 2x 0
2
sin
2x
4
0
x
8
k 2
,
k
?
Vy nghim ca phng tr?nh l?: x 2 k2, x k ,k ?
3
82
4). sin 2x 2 cos x 3 sin x 3 1
1 2 sin x cos x 2 cos x 3 sin x 3 0
2 cos xsin x 1 3 sin x 1 0 sin x 1 2 cos x 3 0
sin x 1 hoc cos x 3 x k2 hoc x k2,k ?
2
2
6
Nghim phng tr?nh x k2 , x k2,k ?
2
6
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
1.31: Gii c?c phng tr?nh:
1). cos 2x cos x 2 sin2 3x 2
2). sin x cos x cos 2x 1 sin 2x
3). 1 cos 2x sin 2x cos x 1 cos 2x
4). sin2 4x sin2 3x sin2 2x sin2 x
5). sin2 x sin2 3x cos2 2x cos2 4x
LI GII
1). cos 2x cos x 2 sin2 3x cos 2x cos x 2. 1 cos 3x
2
2
cos 2x cos x 1 cos 3x
cos 3x cos x cos 2x 1 0
2 sin 2x sin x 1 cos 2x 0 2 sin 2x sin x 2 sin2 x 0
2 sin x sin 2x sin x 0 2 sin2 x 2 cos x 1 0 Vi sin x 0 x k k ?
2 sin x 2 sin x cos x sin x 0
sin x 0 2cosx + 1 = 0
Vi 2 cos x 1 0 cos x 1 x 2 k2 k ? .
2
3
Vy nghim ca phng tr?nh x k , x 2 k2k ? .
3
2). sin x cos x cos 2x 1
1 sin 2x
iu kin: 1 sin 2x 0 sin 2x 1 2x k2 x k k ?
2
4
1
sin x cos x
cos2 x sin2 x
sin x cos x2
sin x cos x
sin x cos xcos x sin x cos x sin x2
sin x cos x sin x cos x sin x cos xcos x sin x sin x cos x
cos x sin x
sin x cos xcos x sin x 1 0 sin x cos x 0 cos x sin x 1 0
Vi sin x cos x 0
2
sin
x
4
0
x
4
k
x
4
k
Truy cp ti t?i liu hc tp, b?i ging min ph?
Truy cp ti t?i liu hc tp, b?i ging min ph?
Vi cos x sin x 1 0
2
cos
x
4
1
cos
x
4
1 cos
2
4
x x
4
4
4
4
k2 k2
x x
k2
2
k2
k?
So vi iu kin nghim phng tr?nh: x k, x k2, x k2,k ?
4
2
3). 1 cos 2x sin 2x 1
cos x 1 cos 2x
iu
kin:
cos x 0 1 cos 2x
0
cos cos
x0 2x 1
x k 2 2x k2
x x
2 k
k
k?
S dng c?ng thc nh?n ?i sin 2x 2 sin x cos x, 1 cos 2x 2 sin2 x, 1 cos 2x 2 cos2 x
1
2 cos2 x cos x
2 sin x cos x 2 sin2 x
2 cos x cos x sin x
2 cos x sin x cos x
(v?
cos x 0 )
2 sin x 1 sin x 1 sin x sin x k2 x= 5 k2 k z
2
6
6
6
4). sin2 4x sin2 3x sin2 2x sin2 x 1
? tng: C? b?nh phng ta h bc, sau ? bin i tng th?nh t?ch, v? t nh?n t chung...
1 1 cos 8x 1 cos 6x 1 cos 4x 1 cos 2x
2
2
2
2
cos 8x cos 6x cos 4x cos 2x 2 cos 7x.cos x 2 cos 3x.cos x
cos x cos 7x cos 3x 0 cos x 0 cos 7x cos 3x 0
Vi cos x 0 x k k ?
2
Vi cos 7x cos 3x 0 cos 7x cos 3x x k hoc x k ,k ?
2
5
Kt lun nghim ca phng tr?nh: x k , x k , x k , k ?
2
2
5
5). sin2 x sin2 3x cos2 2x cos2 4x (1)
1 1 cos 2x 1 cos 6x 1 cos 4x 1 cos 8x
2
2
2
2
Truy cp ti t?i liu hc tp, b?i ging min ph?
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