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