I. BiÕn ®æi ®a vÒ c¸c ph¬ng tr×nh quen thuéc:
1. a. §H Quèc gia (D+G). 97: 2cos2x - 3cosx + 1 = 0
b. C§SP Qu¶ng Ninh (D). 98: cos2(x + ) + 4sin(x + ) = 1 - 2sin2(x + ) + 4sin(x +
) =
4sin2(x + ) - 8sin(x + ) + 3 = 0
c. C§SP Qu¶ng Ninh (F). 98: sinx + cosx = sin5x sin(x + 450) = sin5x
d. §H KTQD. 97: cos7x - sin7x = sin300cos7x - cos300sin7x = sin450 sin(300 - 7x) = sin450 . B¹n tù gi¶i tiÕp. e. §H Ngo¹i ng÷. 98: sin3x + cos2x = 1 + 2sinxcos2x 3sinx - 4sin3x + 1 - 2sin2x = 1 + 2sinxcos2x sinx(4sin2x + 2cos2x + 2sinx - 3) = 0. B¹n tù gi¶i tiÕp. f. §H Ngo¹i th¬ng. 98: sinx + sin2x + sin3x + sin4x = cosx + cos2x + cos3x + cos4x sinx - cosx + (sinx - cosx)( sinx + cosx) + (sinx - cosx)(1 + sinxcosx) + (sinx - cosx)(sinx + cosx) = 0
(sin x - cosx)[2 + 2(sinx + cosx) + sinxcosx] = 0
(1) tgx = 1 x = + k. (2) §Æt: sinx + cosx = t, - t . 4 + 4t
+ t2 - 1 = 0
t2 + 4t + 3 = 0 t = - 1; - 3 sin(x + ) = - /2
g. §H D©n lËp §«ng §«. 97: cos2x + 2cosx + sin2x + 1 = 0 cos2x + 2cosx + 1 = 0 cosx = - 1.h. TT§T Bu chÝnh viÔn th«ng I. 97: Cho y = 4x - sin2x + 4 cosx. T×m x ®Ó y ‘ = 0 Ycbt 4 - 2cos2x - 4 sinx = 0 2sin2x - 2 sinx + 1 = 0 sinx = /2 B¹n tù gi¶i tiÕp.
i. TT§T Bu chÝnh viÔn th«ng I. 98: 2(1 + cos2x)sinx = sin2x B¹n tù
gi¶i tiÕp.
k. §H NNghiÖp 1 (B). 98: (1) . §K: 2sin2x - sinx - 1 0. x +
2k; . . .
(1) cos2x + sin2x = cosx - sinx cos2x + sin2x = cosx - sinx sin(2x +
600) = sin(300 - x)
l. §H NNghiÖp 1 (A). 98: . m. 3 - sinx + tgx = 5cos4
2
x (§¹i sè
ho¸)2.a. 63.II.1: 16(sin8x + cos8x) = 17cos22x (1 - cos2x)4 + (1 + cos2x)4 = 17cos22x 2cos42x - 5cos42x + 2 = 0
64
cos22x = cos4x = 0 x = +
b. 52.II: cos = cos2x cos = 4cos2 - 2 = 1 - 3cos + 4cos3
(cos - 1)(4cos2 - 3) = 0 B¹n tù gi¶i tiÕp.
c. 15.III.1: 2cos2 + 1 = 3cos cos + 2 = 6cos2 - 3 - 3cos + 4cos3 + 2
= 6cos2 - 3 4cos3 - 6cos2 - 3cos + 5 = 0 (cos - 1)( 4cos2 - 2cos -
5) = 0 B¹n tù gi¶i tiÕp .
d. 48.II.2: sin22x - cos28x = sin( + 10x) = cos10x
2cos10x + cos16x + cos4x = 0 2cos10x + 2cos10xcos6x = 0 cos10x(cos6x + 1) = 0B¹n tù gi¶i tiÕp .e. 68.II.1: 2cos3x + cos2x + sinx = 02cos3x + cos2x - sin2x + sinx = 0 cos2x(2cosx + 1) + sinx(1 - sinx) =0 (1 - sin2x)(2cosx + 1) + sinx(1 - sinx) =0 (1 - sinx)[(1 + sinx)(2cosx + 1) + sinx] = 0
x k
x x x x
2 2
2 1 2 2 0cos sin cos sin (*) (*) 2(sinx + cosx) + 2sinxcosx + 1 = 0
§Æt: sinx + cosx = t, - 2 t 2 th× cos: t = 0 x = - 4
+ k.
f. 69.III.2: 4cosx - 2cos2x - cos4x = 1 4cosx - 2cos2x - 2cos22x = 0 4cosx - 2cos2x(1 + cos2x) = 0
4cosx - 4cos2xcos2x = 0 cos
cos cos
x
x x
0
2 1 B¹n tù gi¶i tiÕp .
g. 72.II.1: cos4x - cos2x + 2sin6x = 0 cos4x - 1 + 2sin2x + 2sin6x = 0 (cos2x - 1)(cos2x + 1) + 2sin2x(1 + sin4x) = 0 sin2x[2(1 + sin4) - cos2x - 1] = 0
x k
x x
2 1 04 2sin cos (*)
(*) 2sin4x - sin2x = 0 sin2x(2sin2x + 1) = 0 x = k.
h. 76.II.1 §H TCKT.98: cos10x + 2cos24x + 6cos3xcosx = cosx + 8cosxcos33x cos10x + cos8x + 1 = cosx + 2cos(4cos3x - 3cos3x) 2cos9xcosx + 1 = cosx + 2cos9xcosx cosx = 1 x = 2k.
i. §H D©n lËp HP. 97: 2cos2x + cos2 x
2 - 10cos(
5
2
- x) +
7
2 =
1
2cosx 4cos2x + 1 +
cosx - 20sinx + 7 = cosx 4 - 8sin2x - 20sinx + 8 = 0 2sin2x + 5sinx - 3 = 0 sinx = 1
2
x = 300 + k3600 Hay x = 1500 + k3600
k. §H Mü thuËt c«ng nghiÖp. 96: cos7xcos5x - 3 sin2x = 1 - sin7xsin5x cos2x - 3sin2x = 1. B¹n tù gi¶i tiÕp l. sin4x + cos4x = cos4x sin4x + cos4x = cos22x - sin22x sin4x + cos4x = sin4x + cos4x - 2cos2xsin2x - 4sin2xcos2x sin2x = 0 x k900.m. §H Më. 97: cosx(cos4x + 2) + cos2xcos3x = 0 cosx(cos4x + 2 - 3cos2x + 4cos2xcos2x) = 0 cosx = 0
HoÆc 2cos22x - 1 + 2 - 3cos2x + 2cos2x(1 + cos2x) = 0 cosx = 0 x = 2
+ k.
n. §H Th¸i Nguyªn (A).98: 2cos2(2
cos2x) = 1 + cos(sin2x) 2cos2(2
cos2x) - 1 =
cos(sin2x) 65
cos(cos2x) = cos(sin2x) cos2x = sin2x + 2k cos sin
cos sin
2 2 2 4 1
2 2 2 4 1
x x k
x x k
Cã nghiÖm 12 + 22 16k2 - 8k + 1 4k2 - 2k - 1 0 1 5
4
k 1 5
4
k = 0. B¹n tù
gi¶i tiÕp. o. § KTQD. 98: 16cosxcos2xcos4xcos8x = 1 Nh©n 2 vÕ víi sinx. B¹n tù gi¶i tiÕp .p. §HD©n lËp §«ng §«. 97: cos2x - 7sinx + 8 = 0.q. C§ Lao ®éng x· héi: 1 + sin2x + cosx + sinx + cos2x = 0 (cosx + sinx)2 + cosx + sinx+ cos2x - sin2x=0 (cosx + sinx)(2cosx + 1) = 0. B¹n tù gi¶i tiÕp .r. §H Thuû lîi. 98: (1 + sinx)2 = cosx (1 + sinx)4 = cos2x (1 + sinx)4 =1 - sin2x. B¹n tù gi¶i tiÕp. C¸ch 2: Nh©n hai vÕ víi cosx. C¸ch 3: §æi vÒ gãc x/2.s. §H Hµng h¶i. 98: sin2x = 1 + 2 cosx + cos2x 2sinxcosx = 1 + 2 cosx + 2cos2x - 1. B¹n tù gi¶i tiÕp. II. Ph¬ng tr×nh cã Èn sè ë mÉu sè:1. a. tg(1200 + 3x) - tg(1400 - x) = sin(800 + 2x) tg3(x + 400) + tg(x + 400) = sin2(x + 400). §Æt: X = x + 400. §K: cos3X 0; cos(x + 400) 0. PT tg3X + tgX = sin2X sin4X = sin2Xcos3XcosX sin2X(cos3XcosX - 2cos2X) = 0. B¹n tù gi¶i tiÕp.b. §H KiÕn Tróc. 92: 2tg3x - 3tg2x = tg22xtg3x. §K: cos3xcos2xcosx 0. PT 2(tg3x - tg2x) = tg2x(1 + tg3xtg2x) 2tgx = tg2x (Chia hai vÕ cho: 1 + tg3xtg2x).B¹n tù gi¶i tiÕp.c. 84.II.1: tg2x - tg3x - tg5x = tg2xtg3xtg5x. §K: cos5xcos3xcos2x 0. PT tg2x - tg5x = tg3x(1 + tg2xtg5x) tg(-3x) = tg3x (Chia hai vÕ cho: 1 + tg2xtg5x). B¹n tù gi¶i tiÕp.d. 34.II1: tg22xtg23xtg5x = tg22x - tg23x + tg5x. §K: cos5xcos3xcos2x 0.
PT tg23x - tg22x = tg5x(1 - tg23xtg22x) tg5x = ( )( )
( )( )
tg x tg x tg x tg x
tg xtg x tg xtg x
3 2 3 2
1 3 2 1 3 2
tg5x =
tg5xtgx.
e. - 100.III.1: tg2x = 1
1
3
3
cos
sin
x
x. §K: cosx 0. PT
1
1
2
2
cos
sin
x
x =1
1
3
3
cos
sin
x
x
1
1
1
1
1
1
2
2
cos
sin.(
cos
sin
cos cos
sin sin)
x
x
x
x
x x
x x= 0
cos
(cos sin )(sin cos sin cos )
x
x x x x x x
1
0
f. §H KiÕn tróc. 93: tg2x = 1
1
3
3
cos
sin
x
x. g - 61.III.1: tg2x =
1
1
cos
sin
x
x. B¹n tù gi¶i tiÕp.
h. §H Ngo¹i ng÷. 96: tgx + tg2x - tg3x = 0. §K: cos3xcos2xcosx 0. PT tgx = sin
cos cos
x
x x3 2
sin
cos cos cos
x
x x x
0
3 2
sin
cos cos
x k
x x
5
B¹n tù gi¶i tiÕp vµ kiÓm tra §K.
i. §H Quèc gia (B). 95: tgx - tg2x = sinx. B¹n tù gi¶i tiÕp vµ kiÓm tra §K. 2. a. §H Giao th«ng vËn t¶i. 97: 3(cotgx - cosx) - 5(tgx - sinx) = 2. §K: x k900.
PT 3cos cos sin
sin
sin sin cos
cos
x x x
x
x x x
x
5 = 2 3cos2x -3cos2xsinx -5sin2x + 5sin2xcosx =
2sinxcosx 3(cos2x - cos2xsinx - sin2x + sin2xcosx) - 2(sinxcosx + sin2x - sin2xcosx) = 0
66
3(cosx - sinx)(cosx + sinx - sinxcosx) - 2sinx(cosx + sinx - sinxcosx) = 0 (cosx + sinx - sinxcosx)(3cosx - 3sinx - 2sinx) = 0. B¹n tù gi¶i tiÕp vµ kiÓm tra §K.
b - 106.I.2: 2(tgx - sinx) +3(cotgx - cosx) + 5 =0. §K:x k900 2(sin
cos
x
x- sinx + 1) + 3(
cos
sin
x
x
- cosx + 1) = 0
2
cos x(sinx+cosx-sinxcosx) +
3
sin x (sinx+cosx-sinxcosx) = 0
sin cos sin cos
/
x x x x
tgx
0
3 2
c - 97.II2: 6tgx + 5cotg3x = tg2x. §K: x k900. PT 5(tgx + cotg3x) = tg2x - tgx
5cos
sin cos
sin
cos cos
2
3 2
x
x x
x
x x 5cos22x = sin3xsinx 10cos22x = cos2x - cos4x. B¹n tù gi¶i
tiÕp. d - 143.II.1: 3sinx + 2cosx = 2 + 3tgx. §K: cosx 0. PT 2(1 - cosx) + 3tgx(1 - cosx) = 0
(1 - cosx)(2 + 3tgx) = 0 cos x
tgx
123
B¹n tù gi¶i tiÕp vµ kiÓm tra §K.
e. §H KTQD. 96: ( cos ) ( cos )
( sin )sin
sin1 1
4 1
1
2
2 22 2
x x
xtg x x
xtg x . §K: cosx 0.
PT ( cos ) ( cos )
( sin )
sin sin1 1
4 1
1
22
1
2
2 22
x x
x
xtg x
x
( cos ) ( cos )
( sin )
sin( )
1 1
4 1
1
21 2
2 22
x x
x
xtg x 2 + 2cos2x = 2cos2x(1 + 2tg2x) 2sin2x = 1 x
= 4
+k2
3. a. §H Quèc gia (A). 98: 2tgx + cotg2x = 2sin2x + 1
2sin x. §K: sin2x 0.
PT 2sin sin cos
cos
x x x
x
2+ cos
sin
2 1
2
x
x
= 0 2sinx 1 2 2 cos
cos
x
x + 2
2
2sin
sin
x
x = 0
1
cos x(sinx - 4sinxcos2x) = 0 4cos2x - 1 = 0 2 + 2cos2x - 1 = 0. B¹n tù gi¶i tiÕp.
b. §H C«ng ®oµn. 98: sin
sin cos
2
2 22
2
4
x
x x
= tg2 x
2. V×: sin2x = 4cos2 x
2sin2 x
2 Nªn §K: cos
x
2 0
PT 4cos2 x
2sin2 x
2- 2 = - 4cos2 x
2sin2 x
2 sin2x = 1 x =
2
+ k
c. §H Giao th«ng vËn t¶i. 98: tgx + cotgx = 2(sin2x + cos2x). §K: x k900.
PT 1
sin cosx x = 2(sin2x + cos2x) sin22x + cos2xsin2x = 1 cos2xsin2x - cos22x = 0
cos2 0
2 1
x
tg x
x k
x k
45 90
22 5 90
0 0
0 0,B¹n tù kiÓm tra ®iÒu kiÖn.
d. §H LuËt. 98: tgx - sin2x - cos2x + 2(2cosx - 1
cos x) = 0. §K: x
2
+ k
PT sinx(1
cos x - 2cosx) - cos2x + 2cos2x
1
cos x = 0 -
cos sin
cos
2x x
x - cos2x + 2cos2x
1
cos x
= 0
cos2x(2
cos x - sin
cos
x
x- 1) = 0
cos
sin cos ( )
2 0
2
x
x x VN
x =
4
+ k
2
e. §H An ninh (A). 98: 3 sinx + cosx = 1
cos x. §K: x
2
+ k. PT 3 sinxcosx + cos2x =
1 67
3 sinxcosx - sin2x = 0 sin x
tgx
0
3
x k
x k
3
g. §H B¸ck khoa. 98: 1 2
1tgx gx
x x
gx
cot
(cos sin )
cot. sinx.cosx 0; cosx - sinx > 0
PT sinx cosx = 2
1
(cos sin )
cot
x x
gx
cosx(cosx - sinx) = 2(cos sin )x x
cos sinx x ( 2 - cosx cos sinx x ) = 0 cosx cos sinx x = 2
cos
cos cos sin
x
x x x
0
23 2 V« nghiÖm. (§¸nh gi¸)
III. Ph¬ng tr×nh ®¼ng cÊp:1. a. HV Kü thuËt qu©n sù. 97: 2cos3x = sin3x 2cos3x = - 4sin3x + 3sinx 2cos3x = - 4sin3x + 3sinx(sin2x + cos2x) sin3x - 3sinxcos2x + 2cos3x = 0 tg3x - 3tgx + 2 = 0
(tgx - 1)2(tgx + 2) = 0 x k
x arctg k
4
2( )
b. §H LuËt. 96: 4sin3x + 3cos3x - 3sinx - sin2xcosx = 0 4sin3x + 3cos3x - 3sinx(sin2x + cos2x) - sin2xcosx = 0 sin3x - sin2xcosx - 3sinxcos2x + 3cos3x = 0 tg3x - tg2x - 3tgx + 3 = 0 (tgx - 1)(3tg2 - 1) = 0.c. §H Ngo¹i th¬ng. 96: cos3x - 4sin3x - 3cosxsin2x + sinx = 0 3tg3x + 3tg2x - tgx - 1 = 0 (tgx + 1)(3tg2 - 1) = 0 d. §H Quèc gia. 96: 1 + 3sin2x = 2tgx cosx + 6sinxcos2x = 2sinx (tgx + 1)(3tg2 - 1) = 0 6sinxcos2x + (cosx - 2sinx)(cos2x+sin2x) = 0 2sin3x - sin2xcosx - 4sinxcos2x - cos3x = 0 2tg3x - tg2x - 4tgx - 1 = 0 (tgx + 1)(2tg2x - 3tgx - 1) = 0 B¹n tù gi¶i tiÕp. C¸ch 2: §Æt tgx = t.
e. §H X©y dùng (T¹i chøc). 96: sin2x(tgx + 1) = 3sinx(cosx - sinx) + 3 sin2x
sin cos
cos
x x
x
= 3(1 - sin2x + sinxcosx) sin3x + sin2xcosx = 3cos3x + 3sinxcos2x tg3x + tg2x - 3tgx - 3 = 0 (tgx + 1)(tg2x - 3) = 0 B¹n tù gi¶i tiÕp. f. §H Th¸i Nguyªn. 97: 4cos2x - cos3x = 6cosx + 2(1 + cos2x) 4cos2x - cos3x = 6cosx - 4cos2x
cosx(4cos2x - 8cosx - 3) = 0 x = 2
+ k. HoÆc: x = 3
+ 2k.
g. §H Má. 97: sin
sin
5
5
x
x = 1 . §K: sinx 0. PT sin5x = 5sinx sin3xcos2x + sin2xcos3x =
5sinx (3sinx - 4sin3x)(cos2x - sin2x) + 2sinxcosx(-3cosx + 4cos3x) = 5sinx(cos2x + sin2x)2 12sin5x + 20sin3xcos2x = 0 3sin2x + 5cos2x = 0 x . C¸ch 2: PT sin3xcos2x + sin2xcos3x = 5sinx (3 - 4sin2x)cos2x + 2cosxcos3x = 5 . . . V« nghiÖm.C¸ch 3: PT sin5x - sinx - 4sinx = 0 4cos3xcosxsinx - 4sinx = 0. B¹n tù gi¶i tiÕp. h. §H Ph¬ng §«ng. 98: sin2x - 3sinxcosx = - 1 sin2x - 3sinxcosx = - sin2x - cos2x 2sin2x - 3sinxcosx + cos2x = 0i. §H HuÕ. 98: cos3x + sinx - 3sin2xcosx = 0. k. Quèc Gia (B). 98: cos3x + sin3x = 2(cos5x + sin5x)
68
l. sin(3x + 4
) = sin2xsin(x - 4
)
IV. Ph¬ng ph¸p ®¸nh gi¸ : * sinx + 2sin2x = 3 + sin3x 2sin2x - 2sinxcos2x = 3. VN V×: 22
+ 22sin2x < 32
1. a. §H Th¬ng m¹i. 97: cos2x + cos3
4
x = 2. V×: cos2x 1; cos
3
4
x 1 Nªn: PT
cos
cos
2 1
134
xx
B¹n tù gi¶i tiÕp.
b. C§SP Qu¶ng Ninh (T). 97: 2cos2x + 3cosx - 5 = 0 2cos2x + 3cosx = 5
V×: 2cos2x 2; 3cosx 3 Nªn: PT cos
cos
2 1
1
x
x
x = 2k.
c. §H Thuû lîi. 97: sin6x + cos6x = 1. V×: sin6x sin2x; cos6x cos2x. Nªn: sin6x + cos6x sin2x + cos2x = 1
VËy: PT sin sin
cos cos
6 2
6 2
x x
x x
x = k.
d. 94.III.2: sin14x + cos13x = 1 sin sin
cos cos
14 2
13 2
x x
x x
sin
sincos
cos
x
xx
x
0
10
1
x = k2
e. 109.II.2: sin cos sin cos
cos sin
10 10 6 6
2 24 4 2 2
x x x x
x x
. Ta cã: VP = sin cos
cos sin
6 6
2 24 2 2
x x
x x
=
= (sin cos ) sin cos (sin cos )
sin
2 2 3 2 2 2 2
2
3
4 3 2
1
4
x x x x x x
x
. VËy PT sin10x + cos10x = 1 B¹n tù gi¶i tiÕp.
f. §H An Ninh. 97: (cos2x - cos6x)2 = 6 - 2sin3x. VT 4; VP 4 B¹n tù gi¶i tiÕp V« nghiÖm.g. 74.II.1: (cos4x - cos2x)2 = 5 + sin3x. B¹n tù gi¶i tiÕp. 74.II.2: Gi¶i vµ biÖn luËn (cos4x - cos2x)2 = (a2 + 4a + 3)(a2 + 4a + 6) + 7 + sin3x. Gi¶i: VT 4. VP = [(a + 2)2 - 1][(a + 2)2 + 2] + 7 + sin3x = t4 + t2 + 5 + sin3x = t2(t2 + 1) + 5 + sin3x 4.
VËy: PT VT
VP
4
4
t
x
x x
0
3 1
4 2 2
sin
cos cos
B¹n tù gi¶i tiÕp.
h. §H KiÕn tróc. 97: sin3x(cosx - 2sin3x) + cos3x(1 + sinx - 2cos3x) = 0 sin3xcos3x - 2sin23x + cos3x + cos3xcosx - 2cos23x = 0 cos2x + cos3x = 2. B¹n tù gi¶i tiÕp. i. 35.II: sinx + cosx = 2 (2 - sin3x). VT = 2 sin(450 + x) 2 . VP 2 . B¹n tù gi¶i tiÕp. 2. a. §H Quèc gia. 96: 8 sin2 x + 8 cos2 x= 10 + cos2y 8 sin2 x + 8 1 2 sin x = 9 + 2cos2y.
§Æt: 8 sin2 x = t. §K: 1 t 8. PT t t
t
2 9 8 = 2cos2y. VP 0. VT 0 V× : t n»m trong
hai nghiÖm
VËy: PT t t
y
2 9 8 0
0
cos
B¹n tù gi¶i tiÕp.
b. §H Giao th«ng vËn t¶i. 98: 6 - 4x - x2 = 5
|sin cos |yx
yx
. VP = 102
|sin |yx
10. VT = - (x + 2)2 +
10 10
69
VËy: PT x
yx
2
12
sin
x
y k
2
22
c. DL §«ng §«. 98: cos2x + 8 = 7sinx. VP 7; VT 7. HÖ cos
sin
2 1
1
x
x
x = 2
+ 2k
d. 105.II: sin2x + sin2y + sin2(x + y) = 9
4
1 2
2
1 2
2
cos cosx y+ 1 - cos2(x + y) =
9
4
cos2(x + y) + cos(x + y)cos(x - y) + 1
4 = 0 [cos(x + y) +
1
2cos(x - y)]2 +
1
4[1 - cos2(x -
y)] = 0
sin( )
cos( )
x y
x y
012
y x l
x k l
6 2
y k l
x k l
6 2
6 2
e. 99.V: tg2x + tg2y + cotg2(x + y) = 1 (*). V× Cotg(x + y) = 1
tgxtgy
tgx tgy (tgx + tgy)cotg(x
+ y) = 1 - tgxtgy tgxtgy + tgy(cotg(x + y) + tgx(cotg(x + y) = 1 (**). LÊy (**) trõ (*): tg2x - tgxtgy + tg2y + cotg2(x + y) - tgxcotg(x + y) - tgycotg(x + y) = 0 Nh©n víi 2: (tgx - tgy)2 + [tgx - cotg(x + y)]2 + [tgy - cotg(x + y)]2 = 0
tgx tgy
tgx g x y
cot ( )
x y k
x x y l
2
B¹n tù gi¶i tiÕp.
f. 131.III.2: sin2x + 1
4sin23x = sinxsin23x sin2x +
1
4sin23x +
1
4sin43x = sinxsin23x +
1
4sin43x
sin2x - sinxsin23x + 1
4sin23x +
1
4sin43x +
1
4sin23x -
1
4sin43x = 0
[sinx - 1
2sin23x]2 +
1
4 sin23x(1 - sin23x) = 0
sin
sin
2
12
3 1x
x
cos
sin
3 012
x
x
x k
k
k
30 6030 360
150 360
0 0
0 0
0 0
g. 91.II.1: sin4xcos16x = 1
sin
cossin
cos
4 1
16 14 1
16 1
x
xx
x
x k
x k
x k
x k
8 4
32 16
8 16
16 8
2. a - 77.III.2: [tgx + 1
4cotgx]n = cosnx + sinnx (n = 2, 3, 4, . . .)
+ n = 2: VT = [tgx + 1
4cotgx]2 1; VP = 1 VËy: PT tgx =
1
4cotgx tg2x =
1
4 x =
arctg 1
2 + k.
+ n > 2: VT 1 VP. Nhng vÕ ph¶i = 1 x = k2
lóc ®ã VT kh«ng x¸c ®Þnh PT v«
nghiÖm.
b. 136.II.2: (cos2x + 12cos x
)2 + (sin2x + 12sin x
)2 = 12 + 0,5siny. VP 12,5.
70
Theo B§T Bunhiacèpxki: a + b 2 2 2( )a b 1
2(a + b)2 a2 + b2
VT 1
2[cos2x + sin2x +
12 2sin cosx x
]2 = 1
2[ 1 +
4
22sin x]2
25
2 = 12,5. VËy PT
sin
sin
y
x
1
2 12
y k
x k
2
4 2
2.
c. 83.III.1: (cos3 x
2 +
1
23cosx )2 + (sin3 x
2 +
1
23sinx )2 =
81
4cos24x. VP
81
4.
VT = cos6 x
2 + sin6 x
2 +
162cos x +
162sin x + 4 = cos6
x
2 + sin6 x
2 +
sin cos
sin cos
62
62
62
62
x x
x x
+ 4 =
= (cos2 x
2 + sin2 x
2)3 - 3 cos2
x
2sin2 x
2( cos2
x
2 + sin2 x
2) + 4 +
1 3 22
22
164
6
sin cos
sin
x x
x =
= 5 - 4
3sin2x +
64 1 34
2
6
( sin )
sin
x
x 5 -
4
3.1 +
64 1 1
1
34( . )
= 81
4. PT
cos
sin
2
2
4 1
1
x
x
x = 2
+ k.
d - 101.II.1: sinx - 2sin2x - sin3x = 2 2 . VT = -2cos2xsinx - 2sin2x [( cos ) ( sin ) ](sin ) 2 2 2 2 12 2 2 2x x x = 2 sin2 1x 2 2 .
VËy: PT
1
2sin
sin
2cos1sin 2x
x
xx
V« nghiÖm.
e - 146.III: sinx + 2 22 2 sin sin sinx x x = 3. Ta cã: sinx + 2 2 sin x ( )(sin sin )1 1 22 2 2 2 x x = 2
sinx 2 2 sin x | sinx 2 2 sin x | = |sinx|.| 2 2 sin x | (|sin | sin |
)x x 2
2
22 1
Céng hai B§T thøc cïng chiÒu cã: VT 3. VËy PT sinx = 2 - sin2x sinx = 1 x =900+k3600
f * . 2cosx + 2 sin10x = 3 2 + 2cos28xsinx 2cosx - 2cos28xsinx + 2 sin10x = 3 2
4 4 282 cos x [2
4 4 282
cos
cos
x
x -
2 28
4 4 282
cos sin
cos
x x
x] + 2 sin10x = 3 2
2 1 1 282 cos x [cos
cos
x
x1 282 -
cos sin
cos
28
1 282
x x
x] + 2 sin10x = 3 2
§Æt:1
1 282 cos x = cos ;
cos
cos
28
1 282
x
x = sin. PT 2 1 1 282 cos x cos(+x) + 2 sin10x=3 2
VT 2 2 + 2 = 3 2 . VËy PT cos
cos( )
sin
2 28 1
1
10 1
x
x
x
x k
x k
x k
28
20 5
2
g. sin8x + cos8x = 32(sin12x + cos12x). HD: VT 1; VP 1 VN.V. Ph¬ng tr×nh chøa c¨n vµ GTT§ :
1. a. §H B¸ch Khoa. 97: ( 1 cos cosx x )cos2x = 1
2sin4x
cos ( )
cos cos sin ( )
2 0 1
1 2 2
x
x x x
71
B¹n tù gi¶i tiÕp (1). Cßn (2) cos ; sin
cos cos cos cos sin
x x
x x x x x
0 2 0
1 2 22 2
2 cos cosx x 2 = - cos22x cos22x = 0 (Theo trªn).. KL: x = 4
+ 2k.
b. 108.II.2: 4sinx = 1 1 cos cos
cos
x x
x 4sinxcosx = 2 (|cos
x
2| + |sin
x
2|) (1)
Ta thÊy: NÕu x0 lµ nghiÖm cña (1) th× x0 + còng lµ nghiÖm. Nªn ta t×m nghiÖm x [0, ]. Lóc ®ã:
(1) 4sinxcosx = 2 ( cosx
2 + sin
x
2) 2sin2x = 2sin(
x
2 +
4
)
x
x
6
36
x k
x k
6
36
c. C§SP Qu¶ng Ninh (A, B). 97: 4 2 x (sin2x + 3cosx) = 0. §K: - 2 x 2. x = 2 HoÆc:
cosx(2sinx + 3) = 0 cos x = 0 x = 2
+ k x = 1
2 + k.
Do §K: x = - 3
2; x = -
1
2; x =
1
2; x =
3
2; = - 2; x = 2.
d. C§SP Qu¶ng Ninh (D). 97: x 2 1 (cos22x - 2cos2x + 1) = 0. B¹n tù gi¶i tiÕp. e. HVQH Quèc tÕ. 97: sin x + sinx + sin2x + cosx = 1 sin x + sinx + cosx - cos2x = 0
§Æt: sin x = U 0; cosx = V. Ta cã U + U2 + V - V2 = 0 U V
U V
1
sin cos ( )
sin cos ( )
x x
x x
1
1 2
(1) cos
sin cos
x
x x
02 sinx = 1 5
2 (KÕt hîp ®iÒu kiÖn): x = - arcsin 1 5
2+ 2k
(2) sinx = 0 Vµ cosx = 1 x = 2k.
f. 37.II.1: 1 1 sin sinx x = 2cosx cos
cos cos
x
x x
0
2 2 42 2 cosx = 1 x = 2k.
37.II.2: Gi¶i vµ biÖn luËn: 1 1 sin sinx x = kcosx k x
xk
k
cos ( )
|cos | ( )
0 1
1 1 21 2
2
2
Tõ (2) cã: 1 2 2 k k2 - 1 k - 2 HoÆc k 2.
+ NÕu k -2. Th×: cosx = - 1 1 2 2
2
k
k x = arccos(- 1 1 2 2
2
k
k) + 2k.
+ NÕu k 2. Th×: cosx = 1 1 2 2
2
k
k x = arccos 1 1 2 2
2
k
k + 2k.
+ NÕu - 2 < k < 2. Th×: PT V« nghiÖm.g. 1 1 cos cosx x= 4sinxcosx ( 1 1 cos cosx x )( 1 1 cos cosx x ) = 4sinxcosx( 1 1 cos cosx x )
- 2cosx = 4sinxcosx( 1 1 cos cosx x ) cos
sin ( cos cos )
x
x x x
0
2 2 1 1
+ x = 2k1800.HoÆc sin
sin ( sin )
x
x x
0
4 2 2 12 2 4sin2x(2 - 2sinx) = 1 (2sinx - 1)(4sin2x -
2sinx - 1) = 0
sinx = 1 5
4
= sin (-180) x k
x k
18 360
198 360
0 0
0 0
h. 64.II.1: cos sin sin cos2 1 2 2x x x x
72
(cos sin )(cos sin ) (sin cos ) sin cosx x x x x x x x 2 2 (1). §K: cosx+sinx 0; cos2x - sin2x 0.
+ NÕu: cosx + sinx = 0 Th× PT cã nghiÖm tgx = - 1 x = - 4
+ k .
+ NÕu: cosx + sinx > 0 Th× §K: cosx - sinx 0 vµ (1) cos sin cos sinx x x x = 2 (cos sin )(cos sin )x x x x = 2 - cosx cos2x + 4cosx - 5 = 0 cosx = 1 x = 2k.i. 111.II.1: cos sin sin cos2 1 2 2x x x x . B¹n tù gi¶i tiÕp.
k. §H SP II. 97: 5 2cos cosx x + 2sinx = 0 5 2cos cosx x = - 2sinx sin
cos cos
x
x x
0
2 5 3 02
l. §H V¨n ho¸. 97: 1 2 cossin
x
x = 2 (cosx -
1
2)
2.a. §H Quèc gia (A). 97: cosxsinx + |cosx + sinx| = 1. §Æt: |cosx + sinx| = t; §K: 0 t 2 .
PT 1
2(t2 - 1) + t = 1 t2 + 2t - 3 = 0 t = 1 cosxsinx = 0 sin2x = 0 x = k
2
.
b. 51.II.1: |cosx - sinx| + 4sin2x = 1. B¹n tù gi¶i tiÕp.
c. §H C«ng ®oµn. 96: |tgx| = cotgx + 1
cos x. §K: x k900 .
+ NÕu tgx > 0 Th× ta cã: sin2x = cos2x + sinx 2sin2x - sinx - 1 = 0 sinx = - 1
2 x =
2100 + k3600
+ NÕu tgx < 0 Th× cã: - sin2x = cos2x + sinx sinx = - 1 (Lo¹i).
d. 46.I.2: |cotgx| = tgx + 1
sin x . B¹n tù gi¶i tiÕp. e. 57.III.2: Gi¶i víi k = 2, 3:
3cosx + 2|sinx| = k
+ k = 2: 2|sinx| = 2 - 3cosx cos
sin cos cos
x
x x x
23
2 24 4 12 9 cosx = 0 x =
2
+ k.
+ k = 3: 2|sinx| = 3 - 3cosx 4sin2x = 9 - 18cosx + 9cos2x cos
cos
x
1513
x k
x k
2
2513
arccos
e. 59.III: |cosx| + sin3x = 0:+ NÕu cosx 0 cosx = cos(900 + 3x). + NÕu cosx 0 cosx = cos(900 - 3x). f. 86.III.2: |cosx + 2sin2x - cos3x| = 1 + 2sinx - cos2x |2sin2xsinx + 2sin2x| = 2sin2x + 2sinx
|2sin2x(sinx + 1)| = 2sinx(sinx + 1) sin
| sin | sin
x
x x
1
2 2 sinx = - 1 Hay sinx = 0 Hay cosx
= 1
2Ph¬ng tr×nh chøa tham sè:1. a: §H KiÕn Tróc. 88: Gi¶i vµ biÖn luËn: 2msinxcosx - (sinx + cosx) + 1 = 0 (1). m 0. §Æt: sinx + cosx = t. |t| 2 (*). Th× (1) f(t) = mt2 - t + 1 - m = 0 (2)
C¸ch 1: + (1) V« nghiÖm
0
2 21 2t t(§· cã ac < 0)
mf
mf
( )
( )
2 0
2 0 B¹n tù gi¶i tiÕp.
C¸ch 2: + NÕu m = 0 th× PT cã nghiÖm t = 1 x = 4
+ 2k
+ NÕu m 0 Th× (2) lu«n cã nghiÖm t = 1 vµ t = 1 mm
. B¹n tù gi¶i tiÕp.
b. §H Th¬ng m¹i. 96: T×m m ®Ó ph¬ng cã hai nghiÖm thuéc [0,]: 2 1
3
sin
sin
x
x
= m
73
§Æt: sinx = t. Th× Ycbt T×m m ®Ó ph¬ng tr×nh cã 1 nghiÖm [0,1]: 2 1
3
t
t
= m. B¹n
tù gi¶i tiÕp. c. §H Ngo¹i ng÷. 97: T×m m ®Ó f(x) = sin cos sin cos4 4 2x x m x x cã nghÜa x.
Gi¶i: Ycbt sin4x + cos4x - 2msinxcosx 0, x 1 - 1
2sin22x - msin2x 0, x
f(t) = X2 + 2mX - 2 , X [-1,1] X1 - 1 < 1 X2 f
f
( )
( )
1 0
1 0 -
1
2 m
1
2
d. 5.II - 56.II.2: T×m a ®Ó PT cã nhiÒu h¬n mét nghiÖm thuéc(0,2
): (1 - a)tg2x - 2
cos x + 1
+ 3a = 0
Gi¶i: PT (1 - a)12cos x
- 2
cos x+ 4a = 0. §Æt: X =
1
cos x VÝ: 0 < x <
2
0 < cosx < 1 1 < X
<
Nªn:Ycbt f(X) = (1 - a)X2 - 2X + 4a = 0 tho¶: 1 < X1 < X2 ( ) ( )
'
1 1 0
0
12
a f
S
a
a
12
13 1
e. 11.II.1: T×m a ®Ó PT cã nghiÖm: sin6x + cos6x = a|sin2x|
C1: PT a|sin2x| = 1 -4
3sin22x a =
1
2|sin |x- 4
3|sin2x| =
1
2|sin |x+ |sin2x| -
7
4|sin2x| 2 -
7
4|
sin2x| 1
4C2: §Æt |sin2x| = X, §K: 0 < X < 1. Kh¶o s¸t hµm sè KL.C3: §Æt |sin2x| = X. XÐt f(X) = 3X2 + 4aX - 4 = 0 cã nghiÖm (0,1).2.a. §H B¸ch khoa. 98: Gi¶i vµ biÖn luËn: 2 22 2 x x x xsin cos |a + 1| + |a - 1|Gi¶i: PT V« nghiÖm 4 < 2a2 + 2 a < - 1 HoÆc a > 1.
+ NÕu: a = 1. PT 2 22 2 x x x xsin cos 2 1
2( 2 22 2 x x x xsin cos ) = 1
§Æt: 2
2
2 x = cos; 2
2
2 x = sin. PT sin( + x) = 1 + x = 2
+ 2k x = - +2
+
2k
+ NÕu: - 1 < a < 1: Th× PT sin( + x) = | | | |a a 1 1
2 x = - + (-1)karcsin
| | | |a a 1 1
2 +
2k
b. §H X©y dùng. 98: Gi¶i vµ biÖn luËn: mcotg2x = cos sin
cos sin
2 2
6 6
x x
x x
mcotg2x = cos
sin cos
2
1 2 2
x
x x
+ Lu«n cã nghiÖm: x = 4
+ k
2
. Ngoµi ra: m
x xsin sin2
4
4 22
. §Æt sin2x = X, X 0 - 1 X
1. Cã
4m - mX = 4X (m + 4)X = 4m. NÕu: m = - 4 VN. m - 4 X = 4
4
m
m. B¹n tù gi¶i
tiÕp. c. T×m m ®Ó PT cã ®óng 4 nghiÖm (0,2): mcos2x + sinx = cosxcotgx. PT
sin ( )
cos ( sin ) ( )
x
x m x
0 1
2 1 0 2
74
+ NÕu m = 0. HÖ cos2x = 0 x1 = 4
; x2 = 3
4
; x3 = 5
4
; x4 = 7
4
. KL: m = 0 lµ mét gi¸
trÞ.
+ NÕu m 0. HÖ cos
sin
2 01
x
x m
Cã ®óng 4 nghiÖm sinx =
1
m V« nghiÖm |
1
m| > 1
|m| < 1 vµ m 0.
HoÆc sinx = 1
m cã nghiÖm c¸c nghiÖm cña cos2x = 0 |
1
m| = 2
2 |m| = 2 KL: |m| <
1 HoÆc |m| = 2 .
d. T×m m ®Ó: sin5x + cos5x - m(sinx + cosx) sinxcox(sinx + cosx), x [0, 4
].
Gi¶i: V× sin5x + cos5x = (sin2x + cos2x)(sin3x + cos3x) - sin2xcos2x(sinx + cosx). Nªn ®Æt : t = sinx + cosx Th×:
Ycbt T×m m ®Ó f(X) = X2 + 4X + 4m - 4 0, X (0;1) X1 0 < 1 X2 f
f
( )
( )
0 0
1 0
m - 1
4.
e. T×m m ®Ó: sin3x + msin2x + 3sinx 0 (*), x [0,2
].
Gi¶i: (*) -2sin3x + msinxcosx + 3sinx 0 - 2sin2x + mcosx + 3 0 (V× sinx 0) 2cos2x + mcosx + 1 . §Æt cosx = X Th×: 0 X 1
Vµ Ycbt f(X) = 2X2 + mX + 1, X [0;1]
0
1
01 2
1 2
X X
X X
m - 2 2
3. a. Gi¶i vµ biÖn luËn: (8a2 + 1)sin3x - (4a2 + 1)sinx + 2acos3x = 0
+ NÕu a = 0 Ta cã: sin3x - sinx = 0 x = k2
+ NÕu a 0. V× sinx = 0 kh«ng lµ nghiÖm nªn PT 2aCotg3x - (4a2 + 1)Cotgx + 8a2 + 1 = 0
(Cotgx - 2a)(2aCotg2x - Cotgx - 2a) = 0 x arc g a k
aCotg x Cotgx a
cot
(*)
2
2 2 02
(*) 4a = 2
12
cot
cot
gx
g x tg2x = 4a x =
1
2arctg4a + m
2
.
b. T×m m ®Ó ph¬ng tr×nh cã nghiÖm: sin2x + sin23x - mcos22x = 0.
Gi¶i: PT 1 2
2
1 6
2
cos cosx x - mcos22x = 0 4cos32x + 2mcos22x - 2cos2x - 2 = 0
§Æt: cos2x = X, - 1 X 1. Th× Ycbt T×m m ®Ó f X X X mX
X
( )
2 1
1 1
3 2
VÏ h×nh cã m
0.HÖ Ph¬ng tr×nh, BÊt ph¬ng tr×nh:
1. a. §H Më. 98: Cho sin
sin
2
2
x mtgy m
tg y m x m
. a. Gi¶i khi m = 1. b. T×m m ®Ó hÖ cã nghiÖm.
+ m = 1. §Æt: sinx = X, - 1 X 1; tgy = Y (x = 2
+ k). Cã: X Y
X Y
2
2
1
1
X2 - Y2 + Y - X =
0
75
(X - Y)(X + Y - 1) = 0 Y X
Y X
1
NÕu Y = X Th× X2 + X - 1 = 0 Y = X = 1 5
2
x k
y arctg
k
( ) arcsin11 5
21 5
2
NÕu Y = 1 - X X2 - X = 0. B¹n tù gi¶i tiÕp. + T¬ng tù: NÕu Y = X cã X2 + mX - m = 0 NÕu: Y = m - X cã: X2 - mX + m2 - m = 0. KL: m 0
b. Gi¶i: sin cos cos
cos sin sin
2
2
x x y
x x y
Céng vµ trõ hai ph¬ng tr×nh cã: cos( )
cos( ) cos
x y
x y x
1
2B¹n tù gi¶i tiÕp.
c. Gi¶i: x y
tgx tgy
2
1
33
32
( )
( ) (1) cos(x + y) = 0 tgxtgy = 1. H Ö
tgx
tgy
33
3
HoÆc tgx
tgy
3
33
d. Gi¶i: sin cos
sin cos
x y
y x
7
5 6 0 sin2x + cos2x = 49cos2y + (5siny + 6)2 siny = - 1 B¹n tù gi¶i
tiÕp.
e. T×m a ®Ó hÖ sau cã nghiÖm: cos cos
sin sin
x a y
x a y
3
3 cos2x + sin2x = a2(cos6y + sin6y)
sin22y = 4 1
3
2
2
( )a
a
cã nghiÖm 0
4 1
3
2
2
( )a
a
1 1 |a| 2. Ngîc l¹i ®óng.
f. T×m a ®Ó hÖ cã nghiÖm duy nhÊt: ax a y x
tg x y
2
2 2
1 1
1 2
|sin | ( )
( ) NÕu (x,y) lµ nghiÖm (-
x,y) còng lµ nghiÖm Ycbt x = 0 thay vµo hÖ cã a = 2 HoÆc a = 0.
Ngîc l¹i: Víi a = 0 cã
1
12 2
y x
tg x y
|sin | cã (0,-1) vµ (,-1) lµ nghiÖm . . . Lo¹i a = 0.
Víi a = 2: (1) y = 2x2 + 1 + |sinx| 1; (2) y2 = 1 - tg2x 1. VËy y = 1 x = 0. KL: a = 2
g. T×m m ®Ó hÖ cã nghiÖm: sin cos
sin cos
x y m
y x m
2
(Céng vµ trõ cã) sin( )
sin( )
x y m m
x y m m
2
2 Cã
nghiÖm
1 1
1 1
2
2
m m
m m - 1 - 2 m 1 - 2 HoÆc: - 1 + 2 m 1 + 2 .
h. Ngo¹i ng÷ - Tin hoc. 97: cos cos sin
sin sin cos
3
3
0
0
x x y
y y x
i. V¨n Lang. 97: sin / sin sin
cos / cos cos
x x y
x x y
1
1
2. a. §H Dîc. 97: T×m x (0, 2)mµ: cosx - sinx - cos2x > 0 (cosx - sinx)(1 - cosx - sinx) > 0. B¹n tù gi¶i tiÕp.b. cosx + 3 sinx < 1. c.cosx(1 - 2sinx) > 0 d. sinx + sin3x < sin2xe. QGTP. Hå ChÝ Minh. 97: 2cos2x + sin2cosx + sinxcos2x > 2(sinx + cosx)Ghi chó: Mét sè bµi to¸n chøa hµm lîng gi¸c ngîc:
1. TÝnh: a. A = cos(arsin 1
4). §Æt t = arsin
1
4 sint =
1
4 (0 t
2
) cost = 15
4 =
76
cos( arsin1
4)
b. A = sin(2arccos 1
3). §Æt: t = arccos
1
3 cost =
1
3 sint = 2 2
3 Ta tÝnh sin2t = 2sintcost
= 4 2
9
c. A = tg(arsin 1
6). §Æt: t = arsin
1
6 sint =
1
6 cost = 35
6. VËy A = tgt =
1
35.
d. A = arccos4
5 - arccos
1
4. §Æt: x = arccos
4
5 cosx =
4
5; sinx =
3
5; y = arccos
1
4 cosy =
1
4; siny = 15
4
V× A = x - y cosA = cos(x - y) = 15
3 15
20 A = arccos( 1
5
3 15
20 ).
e. A = arctg 1
3 - arctg
1
4. §Æt: arctg
1
3 = x; arctg
1
4= y tgx =
1
3; tgy =
1
4.
A = x - y tgA = tg(x - y) = tgx tgy
tgxtgy
1 =
1
13 A = arctg
1
13.
2. Gi¶i: a. arccos(x 3 ) + arccosx = 2
arccos(x 3 ) = 2
- arccosx x 3 = cos(2
-
arccosx)
x 3 = sin(arccosx) 3x2 = 1 - cos2(arccosx) 3x2 = 1 - x2 x = 1
2. Thö l¹i lo¹i x = -
1
2b. arcsinx = arccos 1 2 x cos(arcsinx) = 1 2 x . §Æt: y = arcsinx x = siny PT cosy = 1 2 x = 1 2 sin y §óng y. VËy nghiÖm lµ: x [-1,1].
77
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