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Truy cp ti t?i liu hc tp, b?i ging min ph?
C?u : Gii c?c phng tr?nh sau:
1). 1 2 sin x 2 sin 2x 2 cos x cos 2x 3 1 cos x
2 sin x 1 2). cos 2x tan2 x cos2 x cos3 x 1
cos2 x
3).
4 cos2
x 2
2
cos2
7 4
x
3 cos 2x 3 3
0
1 2 sin x
4). sin x.sin 2x 2 sin x.cos2 x sin x cos x 6 cos 2x
sin
x
4
5). 1 cot 2x 4 sin2 x 1 cos 4x
6). cos 2x 3 1 sin x 2 cos x 2 sin 2x 2 sin x 1
2 cos x 1
2 3 sin 2x 1 cos 2x 4 cos 2x.sin2 x 3
7).
0
2 sin 2x 1
8). sin2 x (1 cos 2x)2 2 cos 2x 2 sin 2x
LI GII
1). 1 2 sin x 2 sin 2x 2 cos x cos 2x 3 1 cos x
2 sin x 1
LI GII
iu
kin:
2 sin
x
1
0
sin
x
1 2
x x
k2 6 5 k2 6
,k
?
? tng: C?c bn ?: t ca v tr?i, t s ph?n t?ch c th?nh nh?n t chung, v? r?t gn c mu...ta l?m nh sau:
1 2 sin x 4 sin x cos x 2 cos x cos 2x 3 1 cos x
2 sin x 1
2 sin x 1 2 cos x2 sin x 1
cos 2x
3 1 cos x
2 sin 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?
2 sin x 11 2 cos x
cos 2x
3 1 cos x
2 sin x 1
1 2 cos x 2 cos2 x 1 3 3 cos x
2 cos2 x 2 3 cos x 3 0 cos x 3 cos x 1 2
Vi cos x 3 cos x cos x k2,k Z
2
6
6
Vi cos x 1 x k2 k Z.
Kt hp vi iu kin nghim ca phng tr?nh: x k2; x k2,k Z.
6
2).
cos 2x tan2 x
cos2 x cos3 x 1 cos2 x
LI GII
iu kin: cos x 0.
? tng: Bin i VP ta r?t gn c tan2 x ....
cos
2x
tan2
x
cos2 cos2
x x
cos2 cos2
x x
1 cos2
x
cos
2x
tan2
x
1
cos
x
1 cos2
x
cos 2x tan2 x 1 cos x 1 tan2 x
2 cos2 x 1 cos x 0 2 cos2 x cos x 1 0 cos x 1 hoc cos x 1 . 2
So vi iu kin cos x 0 hai gi? tr n?y u nhn.
Vi cos x 1 x k2 k Z.
Vi cos x 1 cos x cos 2 x 2 k2 k Z.
2
3
3
Nghim ca phng tr?nh: x k2; x 2 k2; x 2 k2,k Z.
3
3
3).
4 cos2
x 2
2
cos2
7 4
x
3 cos 2x 3 3
0
1 2 sin x
LI GII
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?
iu kin: 1 2 sin x 0 sin x 1 sin x sin
2
6
x x
k2
6
,k
5 k2
6
Z.
? tng: H bc, sau ? r?t gn...
Ta c?: g 4 cos2 x 4. 1 cos x 2 1 cos x.
2
2
g
2
cos2
7 4
x
1
cos
7 2
2x
1
cos
3
2
2x
1
cos
2
2x
1
sin
2x.
2 1 cos x 1 sin 2x 3 cos 2x 3 0
3 cos 2x sin 2x 2 cos x 0 3 cos 2x sin 2x 2 cos x
3 cos 2x 1 sin 2x cos x cos 2x.cos sin 2x.sin cos x
2
2
6
6
cos
2x
6
cos
x
x
5 k2 18 3
hoc
x 7 k2,k Z
6
Biu din nghim
-
7 5
6 17
6
18
5 18
6
29 18
Ta thy hai u m?t 7 v? 5 tr?ng nhau, vy nghim x 7 k2 loi.
6
6
6
C?n nghim x 5 k2 c biu din ba u m?t 5 , 17 , 29 u kh?c hai u m?t v? 5 .
18 3
18 18 18
66
Vy nghim x 5 k2 nhn. 18 3
Kt lun nghim phng tr?nh x 5 k2 . 18 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?
bit nghim
x k2 ,k, n ? , c? bao nhi?u u m?t,
n
ta ly
k2
:
k2 n
n
vy nghim n?y
c? n u m?t, sau ? chn k = 0,1, 2, 3,..., n ? 1.
4). sin x.sin 2x 2 sin x.cos2 x sin x cos x 6 cos 2x
sin
x
4
LI GII
iu
kin:
sin
x
4
0
x
4
k
x
4
k,
k
?
2 sin2 x.cos x 2 sin x.cos2 x sin x cos x 6 cos 2x
sin
x
4
2 sin x.cos x sin x cos x sin x cos x
sin
x
4
6 cos 2x
sin x cos x1 2 sin x cos x 3 cos 2x sin x cos x
1 2 sin x cos x 3 cos2 x sin2 x
sin x cos x2 3 sin x cos xsin x cos x
sin x cos x 3 sin x cos x
1 3 sin x 3 1 cos x
3 1
tan x
tan x tan 5 x 5 k,k Z.
1 3
12
12
So vi iu kin nghim ca phng tr?nh x 5 k,k Z
12
5). 1 cot 2x 4 sin2 x
1 cos 4x
LI GII
iu
kin:
sin 2x 0 1 cos 4x
0
sin 2x cos 4x
0 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?
1
cos 2x sin 2x
4 sin2 x 2 sin2 2x
sin 2x cos 2x 2 sin2 x
sin 2x
sin2 2x
sin 2x cos 2x sin 2x 1 cos 2x
sin2 2x 1 sin 2x cos 2x cos 2x 0
sin 2x 1sin 2x 1 cos 2xsin 2x 1 0
sin 2x 1sin 2x 1 cos 2x 0
sin 2x 1 0 sin 2x cos 2x 1 0
Vi sin 2x 1 0 sin 2x 1 (nhn) 2x k2 x k.
2
4
Vi
sin 2x
cos 2x
1
cos
2x
4
cos
4
x k hoc x k . 4
Ta thy
x k th? 4
sin
2x
sin
2
k2
1
0
v?
cos 4x cos k4 1 0 . Vy
nghim
x k nhn. 4
Tng t thay 2 nghim x k , x k th? nghim x k l?m sin 2x = 0, n?n nghim n?y b loi. 4
So vi iu kin nghim ca phng tr?nh: x k; x kk Z
4
4
6). cos 2x 3 1 sin x 2 cos x 2 sin 2x 2 sin x 1
2 cos x 1
LI GII
iu kin: 2 cos x 1 0 cos x 1 x k2,k Z.
2
3
Ph?n t?ch t s ca VP th?nh nh?n t, sau ? r?t gn...
cos 2x 3 1 sin x 2 cos x 1 2 sin x2 cos x 1
2 cos x 1
cos 2x 3 1 sin x 2 cos x 11 2 sin x
2 cos x 1
1 2 sin2 x 3 3 sin x 1 2 sin x
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 sin2 x 2 3 sin x 3 0 sin x 3 hoc sin x 1 . 2
Vi sin x 3 sin x sin x k2 hoc x 2 k2,k Z
2
3
3
3
Vi sin x 1 x k2,k Z
2
So vi iu kin nghim ca phng tr?nh: x 2 k2; x k2,k Z.
3
2
7). 2 3 sin 2x 1 cos 2x 4 cos 2x.sin2 x 3 0
2 sin 2x 1
LI GII
iu kin: 2 sin 2x 1 0 sin 2x 1 2
2x 2x
k2 6 5 k2 6
x x
12 5 12
k
,k
k
Z.
2 3 sin 2x 2 3 sin 2x cos 2x 2 cos 2x 1 cos 2x 3 0
2 3 sin 2x 2 cos 2x 3 sin 4x 2 cos2 2x 3 0 2 3 sin 2x 2 cos 2x 3 sin 4x 1 cos 4x 3 0
2 3 sin 2x cos 2x 3 sin 4x cos 4x 2 0
2
3 2
sin
2x
1 2
cos
2x
3 sin 4x 1 cos 4x 1 0
2
2
2
sin
2x.
cos
6
cos
2x.
sin
6
sin
4x. sin
3
cos
4x.
cos
3
1
0
2
sin
2x
6
cos
4x
3
1
0
Ta c?
4x 3
2
2x
6
?p dng c?ng thc nh?n ?i, ta c:
2
sin
2x
6
1
2
sin2
2x
6
1
0
2
sin
2x
6
1
sin
2x
6
0
sin
2x
6
0
sin
2x
6
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?
Vi
sin
2x
6
0
x k hoc x 7 k ,k Z
12
12
Vi
sin
2x
6
1
2x
6
2
k2
x
3
k
kZ.
So vi iu kin nghim ca phng tr?nh: x 7 k; x k,k Z.
12
3
8). sin2 x (1 cos 2x)2 2 cos 2x 2 sin 2x
iu kin sin 2x 0
1 cos2 x 4 cos4 x 2(2 cos2 x 1) cot x.cos2 x 5 cos2 x 3 4 sin x.cos x
Chia hai v cho cos2 x ( da v?o iu kin cos x 0 )
1 tan
x
5
3 cos2
x
1 5 3(1 tan2 x) 3 tan3 x 2 tan x 1 0 tan x
tan x 1 x k (k ? ) 4
So vi iu kin nghim ca phng tr?nh x k (k ? ) 4
C?u : Gii c?c phng tr?nh sau:
1). 1 cos x sin x tan x
2
sin
2x
7 4
2). (4 sin2 x 1) cos x 2 sin x(2 cos x 3) 3 2 3 sin2 x cos x 2 sin x 1
3).
2 sin x 1 tan x.tan x
2
sin
x
7 4
1
2
4). tan x.cos 3x 2 cos 2x 1 3(sin 2x cos x) 1 2 sin x
5).
2(1
cot 2x.cot sin2 x
x)
1 cos4
x
48
6). sin 3x cos x cos 2x
2
tan
2 x 2 tan x 1 tan2 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?
7). sin 2x 3 cos 2x 3 sin x cos x 3 1 2 sin x 1
8). sin x 1 cot x 2 1 cos x 1 cos x
sin 2x cos 2x 4 9).
2
sin
x
4
3 cos
x
1
cos x 1
10). 1 2 sin x 1 sin x 1 2 sin x 3 cos x
LI GII
1). 1 cos x sin x tan x
2
sin
2x
7 4
(1)
iu
kin
cos x tan x
0 0
x x
k 2 k
Ta
c?
:
sin
2x
7
4
sin
2x
2
4
sin
2x
4
(1)
(1 cos x).cos x sin x sin x
2
sin
2x
4
C?
2
sin
2x
4
sin
2x
cos
2x
,
v?
quy
ng
mu
c:
(1 cos x).cos x sin2 x (sin 2x cos 2x).sin x
cos x cos2 x sin2 x sin 2x.sin x cos 2x.sin x Bin i cos2 x sin2 x cos 2x v? sin 2x.sin x 2 sin2 x.cos x c: cos x cos 2x 2 sin2 x.cos x cos 2x.sin x (cos x 2 sin2 x.cos x) cos 2x cos 2x.sin x 0
cos x(1 2 sin2 x) cos 2x cos 2x.sin x 0
cos x.cos 2x cos 2x cos 2x.sin x 0
cos
2x(cos
x
sin
x
1)
0
cos cos
2x 0 x sin
x
1
0
Truy cp ti t?i liu hc tp, b?i ging min ph?
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