PHƯƠNG TRÌNH BẬC NHẤT VỚI SINX VÀ COSX
[Pages:19] - T?I LIU HC TP MIN PH?
PHNG TR?NH BC NHT VI SINX V? COSX
I. L? THUYT
PHNG TR?NH BC NHT:
a sin u b cos u c DNG: a sin u b cos u c a cos u b sin u c
iu kin phng tr?nh c? nghim l? : a2 b2 c2
Gi s gii phng tr?nh: a sin u b cos u c *
C?ch gii chia hai v ca (*) cho a2 b2
a
b
c
Ta c :
sin u
cos u
a2 b2
a2 b2
a2 b2
a
b
t
cos
sin .
a2 b2
a2 b2
c sin u.cos sin .cos u
sin u c
(**)
a2 b2
a2 b2
c
t
sin .
a2 b2
(**) sin u sin . Gii phng tr?nh c bn.
II. B?I TP MU
C?u 1: Gii c?c phng tr?nh sau:
1). cos x 3 sin x 2
2). 3 sin x 4 cos x 5
3). 3 sin x cos x 2
4). sin x cos x 1
6 5). sin x cos x
2
6). 5 sin 2x 12 cos 2x 13
Group:
- T?I LIU HC TP MIN PH?
7). sin 8x cos 6x 3 sin 6x cos 8x 8). sin x cos x 2 2 sin x.cos x
9). 2 sin2 x 3 sin 2x 3
2
10). 3 cos x 4 sin x
3
3 cos x 4 sin x 6
3 2
11).
2
sin
4
x sin x
4
2
LI GII
1). cos x 3 sin x 2 (1) Ta c? a 1, b 3 ,c 2 a2 b2 2 . Chia hai v ca (1) cho 2 c:
1 1 cos x
3 sin x
2
cos x.cos sin x.sin
2
2
2
2
3
32
7
cos x
3
cos
4
x x
3 3
k2 4
k2
4
x x
k2
12
,k
k2
12
Kt lun nghim ca phng tr?nh:
7 x k2,x
k2,k
12
12
2). 3 sin x 4 cos x 5 1 . Ta c? a 3, b 4,c 5 a2 b2 5 . Chia hai v ca (1) cho 5 c:
1
3
4
sin x cos x 1 .
t
3 cos
4
sin
5
5
5
5
sin x.cos cos x.sin 1
sin x 1
x k2
x k2 Vy nghim
2
2
ca phng tr?nh: x k2,k
2
3). 3 sin x cos x 2 1 .
Ta c? a 3 , b 1,c 2 a2 b2 2 . Chia hai v ca (1) cho 2 c:
1
3
1
sin x cos x
2
2
sin x.cos cos x.sin
2
2
2
6
62
Group:
- T?I LIU HC TP MIN PH?
5
sin x
6
sin
4
x x
6 6
k2 4
k2
4
x x
k2
12
,k
11
k2
12
Vy nghim ca phng tr?nh ? cho l? x 5 k2, x 11 k2,k
12
12
4). sin x cos x 1 (1)
Ta c? a 1, b 1,c 1 a2 b2 2 . Chia hai v ca (1) cho 2 c:
1 1 sin x 1 cos x 1
1
sin x.cos cos x.sin
2
2
2
4
42
sin x sin
4 4
x x
4
4
4
k2
k2
4
x
x
2
k2 k2
,
k
Vy nghim ca phng tr?nh: x k2, x k2,k
2
5). sin x cos x
6 (1) Ta c? a 1, b 1,c
6
a2 b2
2 . Chia hai v ca (1) cho
2
2
2
c: 1 1 sin x 1 cos x
3
sin x.cos cos x.sin
3
2
2
2
4
42
sin x
4
sin
3
x x
4 4
k2 3
k2
3
x x
k2
12
,k
5
k2
12
Vy nghim ca phng tr?nh: x k2, x 5 k2,k
12
12
6). 5 sin 2x 12 cos 2x 13 (1) .Ta c? a 5, b 12,c 13 a2 b2 13 . Chia hai v ca (1) cho 13
c: 1
5
12 sin 2x cos 2x 1 . t
cos
5
12 sin .
13
13
13
13
sin 2x cos sin cos 2x 1 sin 2x 1 2x k2 x k .
2
24
Vy nghim ca phng tr?nh: x k,k
24
Group:
- T?I LIU HC TP MIN PH?
7). sin 8x cos 6x 3 sin 6x cos 8x 1
1 sin 8x 3 cos 8x 3 sin 6x cos 6x
1
3
3
1
sin 8x cos 8x sin 6x cos 6x
2
2
2
2
sin 8x.cos cos 8x.sin sin 6x.cos cos 6x.sin
3
3
6
6
sin 8x
3
sin 6x
6
8x 8x
3 3
6x k2
6
6x k2
6
x x
k
4
k
12 7
8). sin x cos x 2 2 sin x.cos x (1)
1 sin x cos x
2 sin 2x
1
1
sin x cos x sin 2x
2
2
sin x.cos cos x.sin sin 2x
4
4
sin
x
4
sin
2x
2x 2x
x
k2
4
x
4
k2
x x
4 4
k2
k2 3
,
k
Vy nghim ca phng tr?nh: x k2, x k2 ,k
4
43
9). 2 sin2 x 3 sin 2x 3 1 cos 2x 3 sin 2x 3 3 sin 2x cos 2x 2
3
1
sin 2x cos 2x 1 sin 2x.cos cos 2x.sin 1
2
2
6
6
sin
2x
6
1
2x k2 x k
62
3
Vy nghim ca phng tr?nh: x k,k
3
10). 3 cos x 4 sin x
2
3 1
3 cos x 4 sin x 6
Group:
- T?I LIU HC TP MIN PH?
t t 3 cos x 4 sin x 6 3 cos x 4 sin x t 6
1
t
6
2
3
t2
3t
2
0
t
1
t
t 2
3
4
3
4
Vi t 1 3 cos x 4 sin x 5 cos x sin x 1 . t cos sin .
5
5
5
5
cos x.cos sin x.sin 1
cos x 1 x k2 x k2 .
2
2
3
4
4
3
4
Vi t 2 3 cos x 4 sin x 4 cos x sin x . t cos sin .
5
5
5
5
5
cos x.cos sin x.sin sin cos x sin
cosx
cos
2
x x
k2
2
k2
2
x x
2 k2
2
,k .
k2
2
Nghim
phng
tr?nh:
x
k2
,
x
2 k2,x
k2,k
2
2
2
3 2 11). 2 sin x sin x .
4 4 2
sin x cos x sin x cos x 3 2
2.
3 sin x cos x 3
2
2
2
3
1
3
3
1
sin x cos x . t cos sin
10
10
10
10
10
sin
x. cos
cos
x.
sin
cos
sin
x
sin
2
x
2
k2
x
2
2
k2
,k
x
2
k2
x
2
k2
Nghim phng tr?nh: x 2 k2,x k2,k
2
2
Group:
- T?I LIU HC TP MIN PH?
C?u 2: Gii c?c phng tr?nh sau:
3 3 cos 2x
1).
cos x .
2 sin x
3). 2 cos4 x sin4 x sin x cos x
2). tan .sin x 2 cos2 x 2
7
2
4).
3
cos
2x
sin
2x
2
sin
2x
6
2
2
5).
3
sin
7x
cos
7x
2
sin
5x
6
6).
sin
2
2x
3 sin 2x 2
7). cos x
3
sin
x
2
cos
2x
3
0
8). 2 cos 2x 1 3 cos x sin x
9). 3 1 sin x 3 1 cos x 1 3 . 10). 3 sin 3x 3 cos 9x 1 4 sin3 3x
LI GII
1).
3
3 cos 2x cos x
1 . iu kin sin x 0 x k
2 sin x
1 3 3 cos 2x 2 sin x cos x
3 cos 2x sin 2x 3
3
1
3
cos 2x sin 2x
2
2
2
3
cos 2x.cos sin 2x.sin
6
62
cos
2x
6
cos
6
2x 2x
6 6
6
6
k2 k2
x
x
6
k
k
,
k
So vi iu kin th? nghim x k loi.
Vy nghim phng tr?nh: x k,k
6
Group:
- T?I LIU HC TP MIN PH?
2).
tan .sin x 2 cos2
x
2
tan .sin x 1 cos x 2
7
2
7
sin 7 sin x cos x 1
cos
7
sin sin x cos cos x cos
7
7
7
cos x
7
cos
7
x x
7 7
k2
7
k2
7
x
x
2
7
k2
k2
Nghim phng tr?nh: x 2 k2, x k2,k
7
3). 2 cos4 x sin4 x sin x cos x
2 cos2 x sin2 x cos2 x sin2 x sin x cos x
2 cos 2x 2 cos x
4
cos
2x
cos
x
4
2x 2x
x 4
x
k2
k2
4
x x
k2 4
k2
12 3
,
k
Nghim ca phng tr?nh:
x k2,x
k2 ,k
4
12 3
4).
3
cos
2x
sin
2x
2
sin
2x
6
2
2
3
1
2
cos
2x
2
sin
2x
sin
2x
6
2
cos
2x.
cos
6
sin
2x. sin
6
sin
2x
6
2
cos
2x
6
sin
2x
6
2
1 1
2
cos
2x
6
2
sin
2x
6
1
cos 2x
6
. cos
4
sin 2x
6
sin
4
1
Group:
- T?I LIU HC TP MIN PH?
5
cos
2x
6
4
1
2x k2 x k .
64
24
5 Nghim ca phng tr?nh: x k .
24
5).
3
sin
7x
cos
7x
2
sin
5x
6
3
1
2
sin
7x
2
cos
7x
sin
5x
6
sin 7x cos
6
cos 7x sin
6
sin 5x
6
sin 7x
6
sin 5x
6
7x 7x
6 6
5x
k2 6
5x
6
k2
x x
k
9
k 6
6).
sin
2
2x
3 sin 2x 2 cos 2x
3 sin 2x 2
cos 2x.cos sin 2x.sin 1
3
3
cos
2x
3
1
1
3
cos 2x sin 2x 1
2
2
2x k2 x k,k
3
6
7). cos x
3
sin
x
2
cos
2x
3
0
1
3
cos x 2
2
sin
x
cos
2x
3
cos
x
cos
3
sin
x
sin
3
cos
2x
3
cos x
3
cos 2x
4 3
2x 2x
4 3 4 3
x k2
3
x
3
k2
x x
5
3
k2
k2 3
8). 2 cos 2x 1 3 cos x sin x
2 cos2 x sin2 x 1 3 cos x sin x
Group:
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