Giáo án dạy thêm phương trình lương giác-2008

§ç ThÕ NhÊt Trêng THPT KÎ SÆt-B×nh Giang-H¶i D¬ng

Bµi 1: Ph¬ng tr×nh lîng gi¸c c¬ b¶n

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1. Ph ¬ng tr×nh c¬ b¶n

a. ( ) ( k Z,sin m)

x k2

x k2

sin x m m 1 ∈ α = 





= π − α + π

= α + π

= ≤ ⇔

b. ( ) ( k Z,cos m)

x k2

x k2

cosx m m 1 ∈ α = 





= − α + π

= α + π

= ≤ ⇔

c. tgx = m = tgα ⇔ x = α + kπ( k ∈ Z)

d. cot gx = m = cot gα ⇔ x = α + kπ( k ∈ Z)

2.LuyÖn tËp

Bµi1: Gi¶i c¸c ph¬ng tr×nh sau:

1) 2sin(3x- 6

π

)- 3 =0

2) cos 











 = −











x + 4x

5

2

cos

3

2

π π

3) 5

7 tan 2 21 0

6

x

  π

 ÷ − − =  

4) 2

cot 5 cot 3

3 6

x x

    π π

 ÷  ÷ − = −    

5) 2

tan cot 2

4 3

x x

    π π

 ÷  ÷ + = −     











< <

−

4

7

3

π π

x

6) ( x − ) = − ( − x)

0 0

sin 3 27 sin 54

7) sin 2 cos 3 0

4

x x

  π

 ÷ + + =  

8) sin 2x − 3 cos x =0

9) tan(6x).tan(11x)=1

*Bµi 2 :T×m tÊt c¶ c¸c nghiÖm nguyªn cña c¸c ph¬ng tr×nh

1) 2

cos( (3 9 160 800)) 1

8

x x x

π

− + + =

2) 2

cos( (3 9 80 40)) 1

10

x x x

π

− + − =

*Bµi 3:T×m nghiÖm d¬ng nhá nhÊt cña c¸c ph¬ng tr×nh:

1) 2 2 1

cos( ( 2 )) sin( ) 0

2

π π x x x + − − =

2) ( ) ( )

2 2 cos x cos (x+1) 0 π π − =

Sè 1

§ç ThÕ NhÊt Trêng THPT KÎ SÆt-B×nh Giang-H¶i D¬ng

Bµi 2: Ph¬ng tr×nh bËc hai ®èi víi mét

hµm sè lîng gi¸c

A.C¸c ph ¬ng tr×nh c¬ b¶n

(1):asin2

(u(x))+bsin(u(x))+c=0

§Æt t=sin(u(x)) §k : − ≤ ≤ 1 1 t

(2):acos2

(u(x))+bcos(u(x))+c=0

§Æt t=cos(u(x)) §k : − ≤ ≤ 1 1 t

(3):atan2

(u(x))+btan(u(x))+c=0

§Æt t=tan(u(x))

(4):acot2

(u(x))+bcot(u(x))+c=0

§Æt t=cot(u(x))

B.Chó ý:

2 2 2 2

2 2

cos 1 sin ; sin 1 cos

cos 2 1 2sin 2cos 1

x x x x

x x x

= − = −

= − = −

C.LuyÖn tËp:

Bµi1:Gi¶i c¸c ph¬ng tr×nh sau:

1) 3sin 2sin 1 0

2

x + x + =

2) 2

3tan 4 3 tan 3 0 x x − + =

3) 2cos 2 cos 2 0

2

x + x − =

4) cos sin 1 0

2

x + x + =

5) cos 2 sin 2cos 1 0

2

x + x − x + =

6) 3cos4x −sin 2x +1 = 0

7) 2

cos 2 sin 2cos 1 0 x x x + + + =

8) 3cos 2 2(1 2 sin ) sin 3 2 0 x x x + + + − − =

9) 2 2 cos (3 ) cos (3 ) 3cos( 3 ) 2 0

2 2

x x x

π π

+ − − − + =

10) 2

3

3cot 3

sin

x

x

= +

Bµi 2:Gi¶i c¸c ph¬ng tr×nh sau:

1)

2 2 4 sin 2 6sin 9 3cos 2 0

cos

x x x

x

+ − −

=

2) cos (cos 2sin ) 3sin (sin 2) 1

sin 2 1

x x x x x

x

+ + +

=

−

Bµi 3:Gi¶i ph¬ng tr×nh sau: 5sin 1 2 cos x x − =

§ç ThÕ NhÊt Trêng THPT KÎ SÆt-B×nh Giang-H¶i D¬ng

Bµi tËp vÒ nhµ

Gi¶i c¸c ph¬ng tr×nh sau:

1) cos 8 cos 0

4 8

x x

− = HD: 2

cos 2cos 1

4 8

x x

= −

§¸p sè: 2 2 8arccos 16 ,

2

x k k π

−

= ± + ∈Z

2) 3

17 sin( ) 17 cos(3 ) 0

2

x

− = x HD: 2 3

cos 3 1 2sin

2

x

x = −

§¸p sè: 2 5 17 4 2 2 5 17 4 arcsin , arcsin ,

3 4 3 3 3 4 3

x k k x k k π π

π

− −

= + ∈ = − + ∈ Z Z ;

3) 2

4cos 3 cos sin 2 0 x x x + =

HD:

2

2 2

2cos 3 cos cos 2 cos 4 cos 4 2cos 2 1

sin 2 1 cos 2

x x x x x x

x x

= + = −

= −

§¸p sè: 1 1

, arccos ,

2 2 3

x k k x k k π

= + ∈ = ± + ∈ π π Z Z

4) 1 sin 2 cos 4 1 2sin 2 − − = − x x x HD: 2

( ) 0

( ) ( )

( ) [ ( )]

g x

f x g x

f x g x

 ≥

= ⇔ 

 =

§¸p sè: 12

5

12

x k

k

x k

π

π

π

π



= + 

 ∈

 = + 

Z

5) 3 3 5 2 3

1 2 2

cos x sin x sinx cos x

sin x

  +

+ = +  ÷   +

(KA 2002)

HD: 3 3

cos

1 2 2

cos x sin x sinx x

sin x

+

+ =

+

§¸p sè: 2 ,

3

x k k π

= ± + ∈ π Z

6) ( )

1

1 sin 2x

cos x 2sin x 3 2 2cos x 1

2

=

+

+ − −

HD:sin 2 2sin cos x x x =

§¸p sè: 2 ,

4

x k k π

= + ∈ π Z ( 2 , sin 2 1

4

x k k x π

= − + ∈ ⇒ = − π Z n(kh«ng tm®k ))

7) T×m nghiÖm 











π

π

∈ ;3

2

x : 1 2sin x

2

7

3cos x

2

5

sin 2x  = +









 π

 − −









 π

+

Sè 2

§ç ThÕ NhÊt Trêng THPT KÎ SÆt-B×nh Giang-H¶i D¬ng

Bµi 3: Ph¬ng tr×nh bËc cao ®èi víi mét hµm

sè lîng gi¸c

A .C¸c ph ¬ng tr×nh c¬ b¶n

(1): asin3

(u(x))+bsin2

(u(x))+csin(u(x))+d=0 ( a ≠ 0 )

§Æt t=sin(u(x)) §k : − ≤ ≤ 1 1 t

(2): acos3

(u(x))+bcos2

(u(x))+ccos(u(x))+d=0 ( a ≠ 0 )

§Æt t=cos(u(x)) §k : − ≤ ≤ 1 1 t

(3): atan3

(u(x))+btan2

(u(x))+ctan(u(x))+d=0 ( a ≠ 0 )

§Æt t=tan(u(x))

(4): acot3

(u(x))+bcot2

(u(x))+ccot(u(x))+d=0 ( a ≠ 0 )

§Æt t=cot(u(x))

B.C hó ý

3 3

3 2

2 3

1) cos 3 4cos 3cos 2) sin 3 3sin 4sin

3tan tan 3cot 1 3) tan 3 4) cot 3

1 3tan cot 3cot

x x x x x x

x x x

x x

x x x

= − = −

− −

= =

− −

C .luyÖn tËp:

Bµi1: Gi¶i c¸c ph¬ng tr×nh sau:

3 2

2

1) 4cos (6 2 3)cos (4 3 3)cos 2 3 0

2) 4cos cos3 6cos 2(1 cos 2 )

3) 4(sin 3 cos 2 ) 5(sin 1)

4) cos3 3cos 2 2(1 cos )

x x x

x x x x

x x x

x x x

+ − − + + =

− = + +

− = −

+ = +

Bµi 2:Gi¶i c¸c ph¬ng tr×nh sau:

3 2

3 2

1) 6tan (3 2 3)tan (3 3)tan 3 0

2) cot 2cot 3cot 6 0

3) tan 3 tan 2

4) cot 3 cot 2 0

x x x

x x x

x x

x x

+ − − + + =

+ − − =

− =

− + =

Bµi 3:Gi¶i c¸c ph¬ng tr×nh sau:

1

1) 2cos 2 8cos 7

cos

3

2) cos 6 2sin( 4 ) 3

2

x x

x

x x

π

− + =

= + +

2 3 4 3) 2cos 1 3cos

5 5

x x

+ =