PHƯƠNG TRÌNH LƯỢNG GIÁC

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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

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(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 .

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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

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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

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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)

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4x 4x

= =

- 3x + k2 - + 3x + k2

x x

= =

+ k 2 77 - + k2

4.

Phng

tr?nh

1 2

sin

5x

- sin x

=

1 2

sin 11x

- sin x

sin 5x = sin11x x = k hoc x = + k

6

16 8

5. Phng tr?nh (sin x + sin 3x) + sin 2x = (cos x + cos 3x) + cos 2x

2 sin 2x cos x + sin 2x = 2 cos 2x cos x + cos 2x

(2cos x

+ 1)(sin 2x - cos 2x)

=

0

cos

x

=

-

1 2

sin 2x = cos 2x

x x

= =

2 + k2 3

+k 82

.

6. ?p dng c?ng thc h bc, ta c?:

Phng tr?nh 1 - cos6x - 1 + cos8x = 1 - cos10x - 1 + cos12x

2

2

2

2

cos 6x + cos 8x = cos10x + cos12x

2 cos7x cos x

=

2 cos11x cos x

cos x = 0 cos11x =

cos 7 x

x x

= =

+ k 2 k; x=

2

k

9

.

7. Phng tr?nh (1 + cos6x)cos 2x - 1 - cos 2x = 0

cos 6x.cos 2x - 1 = 0 cos 8x + cos 4x - 2 = 0 2cos2 4x + cos 4x - 3 = 0 cos 4x = 1 x = k .

2 Nhn x?t: * cos 6x.cos 2x - 1 = 0 ta c? th s dng c?ng thc nh?n ba, thay cos6x = 4cos3 2x - 3cos 2x v? chuyn v phng tr?nh tr?ng phng i vi h?m s lng gi?c cos 2x . * Ta cng c? th s dng c?c c?ng thc nh?n ngay t u, chuyn phng tr?nh ? cho v phng tr?nh ch cha cosx v? t t = cos2 x Tuy nhi?n c?ch c tr?nh b?y tr?n l? p hn c v? ch?ng ta ch s dng c?ng thc h bc v? c?ng thc bin i t?ch th?nh tng .

V? d 3 Gii c?c phng tr?nh sau:

1. 3sin x + 4 cos x = 0

2. sin 2x + 3 cos 2x = 1

3. 2sin 3x + 5 cos 3x = 5

4. 3cos x + 3 sin x = 1

5. sin7x - cos 2x = 3(sin 2x - cos7x) 6. sin 3x - 3 cos 3x = 2sin 2x

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7. sin x + cos xsin 2x + 3 cos 3x = 2(cos 4x + sin3 x)

Li gii.

1.

Phng tr?nh

3sin x = -4cos x tan x = - 4 3

x

=

arctan

-

4 3

+

k

.

2. Phng tr?nh 2sin(2x + ) = 1 sin(2x + ) = 1 = sin

3

32

6

2x

+

3

=

6

+

k2

x

=

- 12

+

k ,

k

.

2x

+

3

=

5 6

+

k2

x

=

4

+

k

( ) 3. Ta c? 22 + 5 2 = 9 52 phng tr?nh v? nghim.

4. Phng tr?nh 3 cos x + sin x = 1 cos(x - ) = 1

3

6 23

x = arccos 1 + k2 , k .

6

23

5. Phng tr?nh sin7x + 3 cos7x = 3 sin 2x + cos 2x

cos(7x

-

)

=

cos(x

-

)

7x

-

6

=

x

-

3

+

k2

x

=

-

36

+

k

3

,

k

.

6

3

7x

-

6

=

-x

+

3

+

k2

x

=

16

+

k

4

6.

Phng

tr?nh

sin(3x

-

)

=

sin 2x

3x

-

3

=

2x

+

k2

3

3x

-

3

=

-

2x

+

k2

x

=

3

+

k2

, k

.

x

=

4 15

+

k

2 5

7. Phng tr?nh 3 sin x + 1 sin 3x + 3 cos 3x = 2cos 4x + 3 sin x - 1 sin 3x

2

2

2

2

sin 3x +

3 cos 3x = 2cos 4x

cos(3x

-

)

=

cos

4x

x

=

-

6

+

k2

.

3

x

=

42

+

k

2 7

V? d 4. Gii c?c phng tr?nh sau:

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1. cos(sin x) = cos(3sin x)

2.

tan

4

(sin

x

+

1)

=1

Li gii.

1.

Phng

tr?nh

3sin x 3sin x

= =

sin x + k2 -sin x + n2

sin x sin x

= =

k n 2

? X?t phng tr?nh sin x = k . Do k v? -1 sin x 1 n?n ta c? c?c gi? tr

ca k : -1,0,1

T ? ta c? c?c nghim: x = m,x = + m, m 2

? X?t phng tr?nh sin x = n . Ta c? c?c gi? tr ca n l?: n = 2,n = 1,n = 0 2

T ? ta t?m c c?c nghim l?: x = + l,x = l,x = + l, l

2

6

Vy nghim ca phng tr?nh ? cho l?:

x = m,x = + m,x = + m m .

2

6

2. Phng tr?nh (sin x + 1) = + k

4

4

sin x + 1 = 1 + 4k sin x = 4k sin x = 0 x = m , m .

V? d 5. Gii c?c phng tr?nh sau:

( ) ( ) 1. 3 - 1 sin x + 3 + 1 cos x = 2 2 sin 2x

2. 3sin2 x + 5cos2 x - 2cos 2x = 4sin 2x

3. 5 sin x - 2 = 3(1 - sin x) tan2 x

4.

sin2

x 2

-

4

tan2

x

-

cos2

x 2

=

0

Li gii.

1. Phng tr?nh 3 sin x + cos x + 3 cos x - sin x = 2 2 sin 2x

sin(x + ) + cos(x + ) = 2 sin 2x sin(x + 7) = sin 2x

6

6

12

2x

=

x

+

7 12

+

k2

x

=

7 12

+

k2

.

2x

=

-

x

-

7 12

+

k2

x

=

5 36

+

k

2 3

2. Phng tr?nh ? cho tng ng vi

3sin2 x + 5cos2 x - 2(cos2 x - sin2 x) = 8sin xcos x

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