PHÖÔNG TRÌNH LÖÔÏNG GIAÙC
H???ng da?n gia?i CDBT t?? ca?c ?TQG Toa?n ho?c ?
Chuye?n ?e? 2:
L???NG GIA?C
Va?n ?e? 1:
PH??NG TR?NH L???NG GIA?C
A. PH??NG PHA?P GIA?I
1. Ph??ng tr?nh l???ng gia?c c? ba?n
cosx = cos x = + k2
sinx = sin
x k2 x k2
tanx = tan x = + k
cotx = cot x = + k
(v??i k )
2. Ph??ng tr?nh ba?c hai ?o?i v??i mo?t ha?m so? l???ng gia?c asin2x + bsinx + c = 0. ?a?t t = sinx, t 1 acos2x + bcosx + c = 0. ?a?t t = cosx, t 1 atan2x + btanx + c = 0. ?a?t t = tanx acot2x + bcotx + c = 0. ?a?t t = cotx
3. Ph??ng tr?nh ba?c nha?t ?o?i v??i sinx, cosx
asinx + bcosx = c (*) ?ie?u kie?n co? nghie?m: a2 + b2 c2
Ca?ch 1: Chia hai ve? cho a2 b2 0
(*) a sinx + b cosx = c
a2 b2
a2 b2
a2 b2
a 2 b 2
Do
a2 b2
+
a2
b2
= 1
Ne?n co? the? ?a?t a = cos, b = sin
a2 b2
a2 b2
Khi ?o?:
(*) sinxcos + sincosx = c sin(x + ) = c
a2 b2
a2 b2
Ca?ch 2: Chia hai ve? cho a (gia? s?? a 0)
(*) sinx + b cosx = c
a
a
?a?t b = tan. Khi ?o?: (*) sinx + sin cosx = c
a
cos
a
70
TT Luyn Thi i Hc VNH VIN
sinx cos + sin cosx = c cos sin(x + ) = c cos
a
a
Ca?ch 3: ?a?t a?n so? phu?.
Xe?t x = (2k + 1) v??i (k ) co? la? nghie?m 0
Xe?t x (2k + 1) v??i (k )
?a?t t = tan x 2
Khi
?o?:
(*)
a
1
2t t
2
+
b
1 1
t2 t2
= c (b + c)t2 ? 2at + c ? b = 0
4. Ph??ng tr?nh ?o?i x??ng: a(sinx + cosx) + bsinxcosx + c = 0
?a?t t = sinx + cosx =
2
cos
x
4
?ie?u kie?n t 2
Khi ?o?: t2 = 1 + 2sinxcosx sinxcosx = t2 1 2
Thay va?o ph??ng tr?nh ta ????c ph??ng tr?nh ?a?i so? theo t. Chu? y?: a(sinx cosx) + bsinxcosx + c = 0
?a?t t = sinx ? cosx (v??i t 2 )
5. Ph??ng tr?nh ?a?ng ca?p ba?c 2 ?o?i v??i sinx, cosx asin2x + bsinxcosx + ccos2x = 0
Xe?t cosx = 0 x = + k (k ) co? la? nghie?m kho?ng? 2
Xe?t cosx 0. Chia 2 ve? cho cos2x ta thu ????c ph??ng tr?nh ba?c 2 theo tanx.
Chu? y?: Ne?u la? ph??ng tr?nh ?a?ng ca?p ba?c k ?o?i v??i sinx, cosx th? ta xe?t cosx = 0 va? xe?t cosx 0 chia 2 ve? cu?a ph??ng tr?nh cho coskx va? ta thu ????c mo?t ph??ng tr?nh ba?c k theo tanx.
B. ?E? THI
Ba?i 1: ?A?I HO?C KHO?I A NA?M 2011
Gia?i
ph??ng
tr?nh:
1 sin 2x cos2x 1 cot2 x
2 sin x.sin 2x .
Gia?i
?ie?u kie?n: sinx 0. Khi ?o?:
(1)
1
sin 2x 1
cos 2x
2 sin x. 2sin x cosx
sin2 x
71
H???ng da?n gia?i CDBT t?? ca?c ?TQG Toa?n ho?c ?
sin2 x 1 sin2x cos2x 2 2 sin2 x.cosx
1 sin2x cos2x 2 2 cosx (v? sinx 0)
2cos2 x 2sin x cosx 2 2 cosx 0
cosx 0 cosx sin x 2
cos x
0
sin
x
4
1
x k x k2 (k Z) (Tho?a ?ie?u kie?n sinx 0).
2
4
Va?y nghie?m cu?a (1) la? x k x k2 (k Z).
2
4
Ba?i 2: ?A?I HO?C KHO?I B NA?M 2011
Gia?i ph??ng tr?nh: sin2xcosx sinxcosx cos2x sinx cosx
Gia?i sin2xcosx sinxcosx cos2x sinx cosx 2sinx.cos2x + sinx.cosx = 2cos2x ? 1 + sinx + cosx sinx.cosx(2cosx + 1) = cosx(2cosx + 1) + sinx ? 1 cosx (2cosx + 1)(sinx ? 1) = sinx ? 1 sinx ? 1 = 0 hoa?c cosx (2cosx + 1) = 1 sinx = 1 hoa?c 2cos2x + cosx ? 1 = 0
sinx = 1 hoa?c cosx = ?1 hoa?c cosx = 1 2
x k2 hoa?c x k2 hoa?c x k2
2
3
x k2 hoa?c x k 2 (k Z)
2
33
Ba?i 3: ?A?I HO?C KHO?I D NA?M 2011
Gia?i ph??ng tr?nh: sin 2x 2 cosx sin x 1 0 tan x 3
Gia?i
sin 2x 2 cosx sin x 1 0 . ?ie?u kie?n: tanx 3 va? cosx 0. tan x 3
sin2x 2cosx sinx 1 0 2sin x cosx 2cosx sin x 1 0
72
TT Luyn Thi i Hc VNH VIN
2cosxsinx 1 sinx 1 0 sinx 12cosx 1 0
sin x 1 (Loa?i v? khi ?o? cosx = 0)
cos
x
1 2
x k2 (k Z). 3
So v??i ?ie?u kie?n ta ????c nghie?m cu?a ph??ng tr?nh la? x k2 (k Z). 3
Ba?i 4: CAO ?A?NG KHO?I A, B, D NA?M 2011 Gia?i ph??ng tr?nh: cos4x + 12sin2x ? 1 = 0.
Gia?i cos4x + 12sin2x ? 1 = 0 2cos22x ? 1 + 6(1 ? cos2x) ? 1 = 0 cos22x ? 3cos2x + 2 = 0 cos2x = 1 hay cos2x = 2 (loi) 2x = k2 x = k (k Z). Ba?i 5: ?A?I HO?C KHO?I A NA?M 2010
Gia?i ph??ng tr?nh:
(1
sin
x
cos
2x)
sin
x
4
1
cos x
1 tan x
2
Gia?i
?ie?u kie?n: cosx 0 va? tanx ? 1
V??i ?ie?u kie?n tre?n, ph??ng tr?nh ?a? cho t??ng ???ng:
(1 sin x cos2x).(sin x cosx) cosx 1 tan x
(1 sin x cos2x).(sin x cosx) cosx cosx sin x cosx
1 sin x cos2x 1 sin x cos2x 0
2sin2 x sin x 1 0 sin x 1(loa?i) hay sin x 1 2
x k2 hay x 7 k2 (k Z)
6
6
Ba?i 6: ?A?I HO?C KHO?I B NA?M 2010
Gia?i ph??ng tr?nh (sin 2x + cos 2x) cosx + 2cos2x ? sin x = 0
Gia?i
Ph??ng tr?nh ?a? cho t??ng ???ng:
(2sinxcosx + cos2x)cosx + 2cos2x ? sinx = 0 cos2x (cosx + 2) + sinx (2cos2x ? 1) = 0
cos2x (cosx + 2) + sinx.cos2x = 0
73
H???ng da?n gia?i CDBT t?? ca?c ?TQG Toa?n ho?c ?
cos2x (cosx + sinx + 2) = 0
cos2x 0 cosx sin x 2 0
(vn)
2x = k (k ) x = k (k ) .
2
42
Ba?i 7: ?A?I HO?C KHO?I D NA?M 2010
Gia?i ph??ng tr?nh sin2x cos2x 3sinx cosx 1 0 Gia?i
Ph??ng tr?nh ?a? cho t??ng ???ng: 2sin x cos x 1 2sin2 x 3sin x cos x 1 0
cos x(2sin x 1) 2sin2 x 3sin x 2 0
cos x(2sin x 1) (2sin x 1)(sin x 2) 0
(2sin x 1)(cos x sin x 2) 0
sin
x
1 2
cos x sin x 2 (VN)
x x
k2 6 5 k2 6
(k
).
Ba?i 8: CAO ?A?NG KHO?I A, B, D NA?M 2010
Gia?i ph??ng tr?nh 4 cos 5x cos 3x 2(8sin x 1)cosx 5 . 22
Gia?i
Ph??ng tr?nh ?a? cho t??ng ???ng: 2(cos4x cosx) 16sin x cosx 2cosx 5
2cos4x 8sin2x 5 2 4sin2 2x 8sin2x 5
4sin22x ? 8sin2x + 3 = 0 sin 2x 3 (loa?i ) hay sin 2x 1
2
2
2x k2 hay 2x 5 k2
6
6
x k hay x 5 k (k ) .
12
12
Ba?i 9: ?A?I HO?C KHO?I A NA?M 2009
Gia?i
ph??ng
tr?nh:
1 2sin x cosx 1 2sin x1 sin x
3.
74
TT Luyn Thi i Hc VNH VIN
Gia?i ?ie?u kie?n: sinx 1 va? sinx 1 (*)
2 V??i ?ie?u kie?n tre?n, ph??ng tr?nh ?a? cho t??ng ???ng:
(1 ? 2sinx)cosx = 3 1 2sin x1 sin x
cosx 3 sin x sin2x 3 cos2x
cos
x
3
cos
2x
6
x k2 hoa?c x k 2 (k )
2
18 3
Ke?t h??p (*), ta ????c nghie?m: x k 2 k
18 3
Ba?i 10: ?A?I HO?C KHO?I B NA?M 2009
Gia?i ph??ng tr?nh: sinx + cosxsin2x + 3 cos3x 2 cos4x sin3 x
Gia?i Ph??ng tr?nh ?a? cho t??ng ???ng:
(1 ? 2sin2x)sinx + cosxsin2x + 3 cos3x 2cos4x
sinxcos2x + cosxsin2x + 3 cos3x 2cos4x
sin3x +
3
cos
3x
2
cos
4x
cos
3x
6
cos
4x
4x = 3x k2 hoa?c 4x 3x k2 (k )
6
6
Va?y:
x
=
6
k2;
x
42
k
2 7
k
.
Ba?i 11: ?A?I HO?C KHO?I D NA?M 2009
Gia?i ph??ng tr?nh: 3 cos5x 2sin3x cos2x sin x 0 Gia?i
Ph??ng tr?nh ?a? cho t??ng ???ng:
3 cos5x sin5x sin x sin x 0
3 2
cos5x 1 sin 5x sin x 2
sin
3
5x
sin x
5x x k2 hay 5x x k2 (k )
3
3
Va?y: x = k hay x k k
18 3
62
75
H???ng da?n gia?i CDBT t?? ca?c ?TQG Toa?n ho?c ?
Ba?i 12: CAO ?A?NG KHO?I A, B, D NA?M 2009 Gia?i ph??ng tr?nh (1 + 2sinx)2cosx = 1 + sinx + cosx
Gia?i
Ph??ng tr?nh ?a? cho t??ng ???ng:
(1 + 4sinx + 4sin2x)cosx = 1 + sinx + cosx
cosx + 4sinxcosx + 4sin2xcosx = 1 + sinx + cosx
1 + sinx = 0 hay 4sinxcosx = 1
sinx = 1 hay sin2x = 1 2
x k2 hay x k hay x 5 k (v??i k ) .
2
12
12
Ba?i 13: ?A?I HO?C KHO?I A NA?M 2008
Gia?i ph??ng tr?nh:
1 sin x
sin
1 x
3 2
4
sin
7 4
x
Gia?i
Ta
co?:
sin
x
3 2
cos x
?ie?u kie?n:
sin x 0 cos x 0
sin2x 0
V??i ?ie?u kie?n tre?n, ph??ng tr?nh ?a? cho t??ng ???ng:
1 sin
x
1 cos
x
4
sin
x
4
cosx sin x 2 2 sin x cosxsin x cosx
cosx sin x 1 2 sin2x 0
cos x sin x 0 tan x 1
x
4
k
sin 2x
1
2
sin
2x
2
x
8
k
2
x
5
k
(k
) .
8
Ba?i 14: ?A?I HO?C KHO?I B NA?M 2008
Gia?i ph??ng tr?nh: sin3 x 3 cos3 x sin x cos2 x 3 sin2 x cosx
Gia?i
sin3 x 3 cos3 x sin x.cos2 x 3 sin2 x.cosx (1)
76
TT Luyn Thi i Hc VNH VIN
Ca?ch 1: Ph??ng tr?nh ?a? cho t??ng ???ng: sin x(cos2 x sin2 x) 3 cosx(cos2 x sin2 x) 0
cos2 x sin2 x sin x 3 cosx 0
cos2x 0 tan x 3
x
4
k 2
x
3
k
(k )
Nghie?m cu?a ph??ng tr?nh la?: x k va? x k
42
3
(k )
Ca?ch 2:
cosx = 0 kho?ng pha?i la? nghie?m cu?a ph??ng tr?nh (1). Chia hai ve? cu?a ph??ng tr?nh (1) cho cos3x ta ????c:
tan3 x 3 tan x 3 tan3 x
(tan x
3
)(tan2
x
1)
0
tan tan
x x
1
3
x x
3 4
k k
k
Ba?i 15: ?A?I HO?C KHO?I D NA?M 2008
Gia?i ph??ng tr?nh: 2sinx(1 + cos2x) + sin2x = 1 + 2cosx.
Gia?i
Ph??ng tr?nh ?a? cho t??ng ???ng: 4sinx.cos2x + sin2x ? 1 ? 2cosx = 0
2cosx(2sinxcosx ? 1) + (sin2x ? 1) = 0
(sin2x ? 1)(2cosx + 1) = 0
sin 2x 1hay cosx 1 x k hayx 2 k2 hay x 2 k2 (k )
2
4
3
3
Ba?i 16: CAO ?A?NG KHO?I A, B, D NA?M 2008
Gia?i ph??ng tr?nh: sin3x 3 cos3x 2sin2x .
Gia?i Ph??ng tr?nh ?a? cho t??ng ???ng:
1 sin3x 3 cos3x sin 2x cos sin3x sin cos3x sin 2x
2
2
3
3
sin
3x
3
sin
2x
77
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