PHƯƠNG TRÌNH LƯỢNG GIÁC
Truy cp website: ti t?i liu thi min ph?
PHNG TR?NH LNG GI?C
A. T?M TT L? THUYT Dng to?n 1: Phng tr?nh lng gi?c c bn
1. Phng tr?nh: sin x = m (1)
* Nu: m 1 Phng tr?nh v? nghim
* Nu:
m
1
-
2
;
2
sin
=
m
x = + k2 (1) sin x = sin x = - + k2
( k
).
Ch? ? : * Nu
tha m?n
-
2
2
th? ta vit
= arcsin
m.
sin = m
*C?c trng hp c bit:
1. sin x = 1 x = + k2 2
2 sin x = -1 x = - + k2 2
3. sin x = 0 x = k
2. Phng tr?nh: cos x = m (2)
* Nu: m 1 phng tr?nh v? nghim
* Nu: m 1 [0; ] : cos = m
(2) cos x = cos
x = + k2 x = - + k2
( kZ ).
Ch? ? : * Nu
tha m?n
0 - cos = m
th? ta vit
= arccos m .
* C?c trng hp c bit:
1. cos x = 1 x = k2
2. cos x = -1 x = + k2
3. cos x = 0 x = + k 2
3. Phng tr?nh : tan x = m (3)
Vi
m
-
2
;
2
:
tan
=
m
Group:
Truy cp website: ti t?i liu thi min ph?
(3) tan x = tan x = + k .
Ch? ? : * Nu
tha m?n
- 2
2
th? ta vit
= arctanm .
tan = m
* C?c trng hp c bit:
1. tan x = 1 x = + k 4
2. tan x = -1 x = - + k 4
3. tan x = 0 x = k
4. Phng tr?nh: cot x = m (4)
Vi m (- ; ) : cot = m 22
(4) cot x = cot x = + k .
Ch? ? : * Nu
tha m?n
-
2
2
th? ta vit
= arccot m .
cot = m
* C?c trng hp c bit:
1. cot x = 1 x = + k 4
2. co t x = -1 x = - + k 4
3. cot x = 0 x = + k 2
Ghi ch?:
*
sin u
=
sin v
u u
= =
v + k2 - v + k2
(k
)
*
cos u = cos v u = v + k2 (k )
u = v + k
*
tan u
= tan v
u,
v
2
+ n
(k,n )
*
cot
u
=
cot
v
u = v + k u,v n
(k, n
)
Dng 2. Phng tr?nh bc nht i vi sinx v? cosx
L? phng tr?nh c? dng: a sin x + bcos x = c (1) ; vi a,b,c v?
a2 + b2 0 .
Group:
Truy cp website: ti t?i liu thi min ph?
C?ch gii: Chia hai v cho a2 + b2 v? t
cos = a ;sin = b .
a2 + b2
a2 + b2
(1) sin x.cos + cosx.sin = c sin(x + ) = a2 + b2
(2).
Ch? ?:
? (1) c? nghim (2) c? nghim a2 + b2 c2 .
c a2 + b2
? sin x
3
cos
x
=
2
1 2
sin
x
-
3 2
cos
x
=
2
sin(x
-
) 3
?
3 sin x cos x = 2
3 2
sin
x
1 2
cos x
=
2
sin(x
) 6
? sin x cos x =
2
1 sin x 2
1 2
cos
x
=
2 sin(x ) . 4
Dng 3. Phng tr?nh bc hai cha mt h?m s lng gi?c
sin u(x) 2 sin u(x)
L?
phng
tr?nh
c?
dng
:
a
cos tan
u(x) u(x)
+
b
cos tan
u(x) u(x)
+
c
=
0
cot
u(x)
cot
u(x)
sin u(x)
C?ch gii: t
t
=
cos tan
u(x) u(x)
ta c? phng tr?nh :
at2
+ bt + c = 0
cot
u(x)
Gii phng tr?nh n?y ta t?m c t , t ? t?m c x
Khi
t
t
=
sin u(x) cos u(x)
,
ta
co
iu
kin:
t -1;1
Dng 4. Phng tr?nh ng cp L? phng tr?nh c? dng f(sin x,cos x) = 0 trong ? lu tha ca sinx v? cosx
c?ng chn hoc c?ng l. C?ch gii: Chia hai v phng tr?nh cho cosk x 0 (k l? s m cao nht) ta c phng tr?nh n l? tanx . Dng 5. Phng tr?nh i xng (phn i xng) i vi sinx v? cosx L? phng tr?nh c? dng: a(sin x + cos x) + bsin xcos x + c = 0 (3)
gii phng tr?nh tr?n ta s dng ph?p t n ph
Group:
Truy cp website: ti t?i liu thi min ph?
t = sin x + cos x =
2
sin
x
+
4
t2 - 1
2
t -
= sin x cos
2;
2
x
Thay v? (5) ta c phng tr?nh bc hai theo t.
Ngo?i ra ch?ng ta c?n gp phng tr?nh phn i xng c? dng
a(sin x - cos x) + bsin xcos x + c = 0 (3')
gii phng tr?nh n?y ta cng t
t = sin x - cos x =
2
sin
x
-
4
t - 2 sin x cos
; x
2
=1
- t2 2
Thay v?o (3') ta c? c phng tr?nh bc hai theo t.
B.PHNG PH?P GII TO?N.
Vn 1. Gii c?c phng tr?nh lng gi?c c bn
C?c v? d
V? d 1. Gii c?c phng tr?nh sau:
1. sin x - cos 2x = 0 3. 2sin(2x - 350 ) = 3
2. cos2 x - sin 2x = 0 4. sin(2x + 1) + cos(3x - 1) = 0
Li gii.
1. Phng tr?nh cos 2x = sin x = cos( - x) 2
2x
=
2
-
x
+
k2
x
=
6
+
k
2 3
, k
.
2x
=
-
2
+
x
+
k2
x
=
-
2
+
k2
2. Phng tr?nh cos2 x - 2sin xcos x = 0
cos
x(cos
x
-
2
sin
x)
=
0
cos x = 0 2 sin x = cos
x
cos tan
x x
= =
0 1 2
x
=
2
+
k
,k .
x
=
arctan
1 2
+
k
Group:
Truy cp website: ti t?i liu thi min ph?
3. Phng tr?nh sin(2x - 350 ) = 3 = sin 600 2
2x - 350 = 600 + k3600
x
=
950
+
k.1800
2
.
2x - 350 = 1800 - 600 + k3600
x
=
1550
+
k.1800
2
4.
Phng tr?nh
cos(3x
-
1)
=
sin(-2x
-
1)
=
cos
2
+
2x
+
1
3x
-
1
=
2
+
2x
+
1
+
k2
x
=
2
+
2
+
k2
.
3x
-
1
=
-
2
-
2x
-
1
+
k2
x
=
-
10
+
k
2 5
V? d 2. Gii c?c phng tr?nh sau:
1. cos x - 2 sin 2x = 0
2. sin3 xsin 3x - cos3 xcos 3x = - 5 2
3. sin2 2x = cos2 2x + cos 3x
4. sin 2x.cos 3x = sin 5x.cos 6x
5. sin x + sin 2x + sin 3x = cos x + cos 2x + cos 3x
6. sin2 3x - cos2 4x = sin2 5x - cos2 6x
7. cos2 3xcos 2x - cos2 x = 0
Li gii. 1. Phng tr?nh cos x - 4sin xcos x = 0 cos x(1 - 4sin x) = 0
cos x sin x
= =
0 1 4
x x
= =
+ k 2 arcsin 1
4
+
k2, x
=
-
arcsin
1 4
+
k2
2. Ta c? sin3 x = 3sin x - sin 3x ; cos3 x = cos 3x + 3cos x
4
4
N?n phng tr?nh ? cho tng ng vi
sin 3x(3sin x - sin 3x) - cos 3x(cos 3x + 3cos x) = - 5
2
3(sin 3xsin x - cos 3xcos x) - 1 = - 5
2
-3cos 4x = - 3 cos 4x = 1 x = + k , k .
2
2
12 2
3. Phng tr?nh sin2 2x - cos2 2x = cos 3x
cos4x = -cos3x = cos( - 3x)
Group:
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