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Truy cp ti t?i liu hc tp, b?i ging min ph?
PHNG TR?NH LNG GI?C C? C?CH GII C BIT
Gii c?c phng tr?nh sau:
1). cos x cos 2x cos 3x cos 4x cos 5x 1 2
2). 4 sin 3x cos 2x 1 6 sin x 8 sin3 x
3). cos x cos 2x cos 4x cos 8x 1 16
4).
sin3
x
4
2 sin x 1
5).
sin4
x cos4
x
7 8
cot
x
3
cot
6
x
6).
tan
x
3
tan
x
6
sin
3x
sin x
sin 2x
7).
sin
3x
4
sin
2x
sin
x
4
8).
8
cos3
x
3
cos
3x
9).
2
sin3
x
4
2
sin
x
10). cos 4 x cos2 x 3
LI GII
1). cos x cos 2x cos 3x cos 4x cos 5x 1 2
Ta thy sin x 0 x m, m ? kh?ng phi l? nghim ca Nh?n hai v ca phng tr?nh cho sin x 0 x m, m ? , c:
sin x cos x cos 2x cos 3x cos 4x cos 5x 1 sin x 2 sin x cos x sin x cos 2x sin x cos 3x sin x cos 4x sin x cos 5x 1 sin x
2 sin 2x sin x sin 3x sin 2x sin 4x sin 3x sin 5x sin 4x sin 6x sin x sin 5x sin 6x 0 sin 6x sin(5x) x k2 x h2, k, h ?
11
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
k2
m
11
h2 n
k
h
11m 2
n1 2
k h
11m n
vi
m
2 m, n
2n 1
v?
m, n, m , n ?
Kt lun nghim phng tr?nh x k2 , x h2; k 11m , h n , k, h ? 11
2). 4 sin 3x cos 2x 1 6 sin x 8 sin3 x
4 sin 3x.cos 2x 1 2 3 sin x 4 sin3 x
2 sin x 2 sin 5x 1 2 sin 3 x
Ta thy cos x 0 x m, m ? kh?ng phi l? nghim ca 2
Nh?n hai v ca phng tr?nh cho cos x 0 x m, m ? , c: 2
2 sin x 2 sin 5x cos x 1 2 sin 3 x cos x
sin 2 x sin 4 x sin 6 x cosx sin 2 x sin 4 x
sin
6x
cos
x
sin
2
x
x
14
k2 7
hoc
x h2 ; k,h ? 10 5
iu
kin
14
10
k2 7
h2 5
2
2
m n
k h
3 7m 2
2 5n 2
vi m l? s nguy?n l, v? n l? s nguy?n chn.
3). cos x cos 2x cos 4x cos 8x 1 16
Ta thy sin x 0 x m, m ? kh?ng phi l? nghim ca
Nh?n hai v ca phng tr?nh cho sin x 0 x m, m ? , c:
16 sinx cos x cos 2x cos 4x cos 8x sin x
8 sin 2x cos 2x cos 4x cos 8x sin x
4 sin 4x cos 4x cos 8x sin x 2 sin 8x cos 8x sin x sin 16x sin x
x k2 x h2 ; k,h ?
15
17 17
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
k2
15 17
m h2 17
n
k h
15m 2
17n 1 2
vi
m
l?
s
nguy?n
chn
v?
n
l?
s
nguy?n l.
4).
sin3
x
4
2 sin x 1
t t x x t
4
4
1 sin3 t
2
sin
t
4
sin3
t
sin
t
cos
t
sin3 t sin t cos t 0 sin t sin2 t 1 cos t 0
cos2 t.sin t cos t 0 cos t 1 sin t.cos t 0 cos t 0 1 sin t.cos t 0
Vi: cos t 0 t k x k x k k ?
2
42
4
Vi 1 sin t.cos t 0 1 sin 2t 1 sin 2t 2 (v? nghim). 2
Vy nghim ca phng tr?nh: x k k ?
4
5).
sin4
x cos4
x
7 8
cot
x
3
cot
6
x
()
K:
sin sin
x 6
3
x
0 0
sin
x
3
sin
6
x
0
cos
2x
6
0
x
3
k 2
C?:
cot
x
3
cot
6
x
cot
x
3
cot
2
x
3
cot
x
3
tan
x
3
1.
() sin4 x cos4 x 7 3 cos 4x 7 cos 4 x 1 x k : (tha k).
8
4
8
2
12 2
Kt lun: Tp nghim cn t?m ca phng tr?nh l? x k vi k ? . 12 2
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?
6).
tan
x
3
tan
x
6
sin
3x
sin x sin 2x
()
iu kin:
cos
x
3
0,
cos
x
6
0
x
k , 62
vi
k? .
Ta c?:
tan
x
3
tan
x
6
tan
2
6
x
tan
x
6
cot
6
x
tan
x
6
1.
() sin 3x sin x sin 2x (sin 3x sin x) sin 2x 0
2 sin 2x cos x sin 2x 0 sin 2x(2 cos x 1) 0 sin 2x 0 cos x 0, 5
x k hoc x 2 k2 vi k ? (tha m?n iu kin).
2
3
Kt lun: C?c tp nghim cn t?m ca phng tr?nh l? x k , x 2 k2.
2
3
7).
sin
3x
4
sin
2x
sin
x
4
()
Nhn
thy:
sin
3x
4
sin
4
3x
sin
4
3x
sin
3
x
4
t t x , suy ra: x t n?n phng tr?nh ? cho:
4
4
()
sin
3t
sin
2t
2
sin
t
4
sin3
t
3
sin
t
cos
2t
sin
t
0
4 sin3 t 3sin t (1 2 sin2 t)sin t 0 sin t 0 hoc sin2 t 1
sin t 0 hoc cos t 0 t k hoc t k vi k ? . 2
Suy ra: x k hoc x k x k vi k ? .
4
4
42
Kt lun: Tp nghim cn t?m ca phng tr?nh l? x k , vi k ? . 42
8).
8
cos3
x
3
cos 3x
()
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?
Nhn
thy:
cos 3x
cos( 3x)
cos
3
x
3
t
t
x
3
() 8cos3 t cos 3t 4 cos3 t 3cos t 12 cos3 t 3cos t 0
cos t 0 hoc 4 cos2 t 1 0 cos t 0 hoc cos 2t 1 2
t k hoc t k. Suy ra: x k hoc x k vi k ? .
2
3
2
Kt lun: C?c tp nghim cn t?m ca phng tr?nh l? x k, x k. 2
9).
2
sin3
x
4
2
sin
x
() . t t x x t
4
4
() sin3 t
2
sin
t
4
sin3
t
sin
t
cos
t
(sin
t
cos
t)(sin2
t
cos2
t)
cos t(sin2 t sin t cos t cos2 t) 0 cos t(sin 2t 2) cos t 0 sin 2t 2
t k. Suy ra: x k vi k ? .
2
4
Kt lun: Tp nghim cn t?m ca phng tr?nh l? x k vi k ? . 4
PHNG TR?NH LNG GI?C C BIT
Gii c?c phng tr?nh sau:
1). 3 tan2 x 4 sin2 x 2 3 tan x 4 sin x 2 0
2). sin 4x cos 4x 1 4
2
sin
x
4
3). sin 3x cos x 2 sin 3x cos 3x 1 sin x 2 cos 3x 0
4). cos 2x 3 sin 2x 3 sin x cos x 4 0 . 5). cos 2x cos2 x 1 tan x . 6). cos 3x 2 cos2 3x 2(1 sin2 2x) 7). sin1996 x cos1996 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?
8). sin4 x cos15 x 1
9).
2
cos2
2
cos
2
x
1
cos
sin
2x
LI GII
1). 3 tan2 x 4 sin2 x 2 3 tan x 4 sin x 2 0
3 tan2 x 2 3 tan x 1 4 sin2 x 4 sin x 1 0
3 tan x 1 2 2 sin x 12 0
3 tan x 1
2
sin
x
1
0
0
tan x sin x
3 3 1 2
x
6
k
x
6
k2
x
5 6
k2
Biu din nghim tr?n v?ng tr?n lng gi?c:
sin
5
6
6
cos
0
7 6
Nghim ca 3 tan x 1 0 c biu din hai u m?t l? , 7 . 66
Nghim ca 2 sin x 1 0 c biu din hai u m?t l? , 5 . 66
Kt lun nghim ca phng tr?nh: x k2 6
2). sin 4x cos 4x 1 4
2
sin
x
4
sin 4x 1 cos 4x 4sin x cos x
2 sin 2x cos 2x 2 cos2 2x 4 sin x cos x
cos x sin xcos x sin xsin 2x cos 2 x 2 sin x cos 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?
cos x sin x cos x sin xsin 2x cos 2 x 2 0 cos x sin x 0 hoc cos x sin xsin 2x cos 2 x 2 0 (2)
Gii (1)
2
cos
x
4
0
x
4
k, k
?
Gii
2
2
cos
x
4
cos
2x
4
2
0
cos
x
sin
3x
2
cos x 1 sin 3x 1
x x
k2 m2 63
; m, k
?
.
H
(1)
c?
nghim
khi
v?
ch
khi
k2 m2 12k 4m 1 phng tr?nh n?y v? nghim v? v tr?i lu?n l? mt s nguy?n chn, c?n v 63
phi l? s nguy?n l. Kt lun (2) v? nghim.
Nghim phng tr?nh x k, k ? 4
3). sin 3x cos x 2 sin 3x cos 3x 1 sin x 2 cos 3x 0
sin 3x cos x cos 3x sin x 2 sin2 3x cos2 3x cos 3x 0
sin 4x 2 cos 3x 0
sin 4x
cos 3x
2
sin 4x 1 cos 3x 1
4x 3x
k2 2 m2
x x
k 82 m2
3
,k,m?
Nu phng tr?nh c? nghim th? tn ti c?c gi? tr k, m sao cho :
k m2 k 2m 1 3k 4m 1 12k 16m 3
82 3 2 3 8
6
8
2 6k 8m 3 (kh?ng tha). Phng tr?nh v? nghim v?: V tr?i l? mt s chn, c?n v phi l? mt s
l .
Kt lun phng tr?nh v? nghim.
4). cos 2x 3 sin 2x 3 sin x cos x 4 0 . Chia 2 v cho 2 ta c :
1 cos 2x 2
3 2
sin
2x
3 2
sin
x
1 2
cos
x
2
0
sin
2x
3
sin
x
6
2
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 sin
2x 3
x
6
1
1
2x 3
x
6
2
k2
l2 2
x x
k 12 l2
3
.
Phng
tr?nh
c?
nghim
khi
:
k l2 1 12k 4 24l 12k 24l 5 . V? nghim vi mi k, l ? v? VT l? mt s chn,
12
3
c?n VP l? mt s l .
5). cos 2x cos2 x
1 tan x
.
iu
kin
:
cosx tanx
0
-1
*
.
t t tan x , iu kin t 0
Khi ? phng tr?nh tr th?nh :
1 t2 1 1 t2 1 t2
1 t 1 t2
1 t , iu kin 1 t2 0 0 t 1 .
1 t2 2 1 t 1 t 1 t1 t2 1 0 .
1 t1 t2 1 0 t3 t2 t 0 t 0 t 1 5 t 1 5 .
2
2
So vi iu kin nhn t 0 tan x 0 x k,k ?
6). cos 3x 2 cos2 3x 2(1 sin2 2x) Ta c? nhn x?t sau :
Bt ng thc Bunhiacopxki: cos 3x 2 cos2 3x
12 12
cos2
3x
2 cos2 3x
2
cos 3x 2 cos2 3x 2
Suy ra v tr?i t gi? tr ln nht bng 2 khi cos 3x 2 cos2 3x VP 2(1 sin2 2x) 2
Vy
ch
xy
ra
khi
:
cos 3x sin2 2x
2 cos2 0
3x
cos2 3x 1 sin 2x 0
cos 6x sin 2x
1 0
x x
6
l
2
k 3
.
Nu phng tr?nh c? nghim th? tn ti k,l thuc Z sao cho :
k l 1 2k 3l k 3l 1 l l 1 .
63 2
2
2
k l? nguy?n th? ta chn : l 1 2n l 2n 1 .
Truy cp ti t?i liu hc tp, b?i ging min ph?
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