Bài tập phương trình lượng giác

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 th 
ng ng
n nhi th sin
Anh mt 
Bng lc gia khong 
t em cha bao n s
 m 


  
hi t mt n hoa bi ri
i! anh ng- 
Sinh nht em anh t
t 
 n em thn thc

i c

nghim duy nht 

I. 
1. i s 
a. 

 b. 



2. i s 
a. 
b.


3. Cho m  

a.  b. 

4. Cho m  c 
a. 4m b. 8m c. 12m
5. biu di 
a.



b. -30
0
c. -315
0
d.



e. -60
0

m ngn c

6. Biu di
a. 
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b.



c. 
d. -


 
e.


f.






7. nh bi
(0 )
2
sd AM



  
. Gi M
1
,
M
2
, M
3
li xng ca M qua trc t 
c cung:
1 2 3
,,AM AM AM
.
II.  


1. nh du c sau:
a.
0 0 0 0
sin145 , cos198 ,tan( 31 ), cot777

b.
sin( ), cos( ) ,tan( ), cot( ) (0 )
3 6 2 2 2
a a a a a
    
     

2. Biu di 

000
7
390 , 225 , 750 ,
2


3.  

trong mng hp sau:
1
. cos , sin 0
8
a



13
. sin ,
3 2 2
b



  

. cot 6, 2
2
c


  
  

4. biu thc sau:
4 4 2 2
sin cos 2sin cosA x x x x  

6 6 2 2
sin cos 3sin cosB x x x x  

4 4 6 6 2 2
sin cos (sin cos ) sin cosC x x x x x x    

4 2 2
cos sin cosD x x x

2 2 2 2
22
1 sin cos 1 sin cos
sin cos
x x x x
E
xx



22
2
22

1 cos 1 sin cos 1
cos
sin cos 1 cos
x x x
Fx
x x x

   


5. Chng minh rng:
4 4 2
. sin cos 2sin 1a x x x  

4
24
21
.1 tan 0
cos cos
bx
xx
   

2
2
2
1 cos
. 1 2cot
1 cos
x

cx
x




2 tan 1
.1
cot 1 tan 1
x
d
xx

  


22
. tan cot sin tan cos cot 2sin cose x x x x x x x x   

2
22
1
. (tan sin ) (1 cos ) 1
cos
f x x x
x

    



6. u thc sau:
sin( ) cos( ) sin( ) cos( )
22
A x x x x


       

tan( ) cot( ) tan( ) cot( )
22
B x x x x


       

sin( ) cos( ) sin( ) cos( )
2
C x x x x


       

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tan( ) cot( ) tan(5 ) cot(7 )
22

D x x x x


        

7. n u thc sau:
22
5
sin cos
12 12
A



2 0 2 0 2 0
cos 10 cos 20 cos 80B    

2 0 2 0 2 0
sin 10 sin 20 sin 80C    

0 0 0 0
sin10 sin20 sin30 sin350D     

0 0 0 0
cos5 cos10 cos15 cos175E     

2 0 2 0 2 0
1 1 1

1 tan 1 1 tan 2 1 tan 89

F    
  

0 0 0
tan2 tan 4 tan88H 


III.  
8. n:
3sin cos , sin cosy x x y x x   

9. a m sau:
. 4sin 2 3a y x

. 2cos( ) 1
3
b y x

  

. 2 sin 3c y x  

2
. sin 2sin 5d y x x  


IV. BI
10. 













11. Chng minh rng:
22
cos sin cos( )cos( )y x x y x y   
, t 

2 0 0 0
sin 3 cos33 cos27Q 

2 0 0
sin cos( 60 )cos( 60 )P x x x   

12. ng minh rng:
222
sin sin sin 2 2cos cos cosA B C A B C   

13. CMR:
22
. sin sin sin( )sin( )a x y x y x y   

. sin( ) sin cos cos sin cos cos sin cos cos sin sin sinb a b c a b c b a c c a b a b c     

0 0 0 0 0 0 0 0 0 0 0 0
. ính:sin3 cos7 cos50 sin7 cos3 cos50 sin50 cos3 cos7 sin3 sin7 sin50cT 

sin( )
. tan tan tan tan tan tan
cos cos cos
abc
d a b c a b c
abc

   

e.
2 2 2
sin sin sin ( ) 2 2cos cos cos( )x y x y x y x y     

14.  m:
2
. sin sin 0a x x m  

   
2
. sin cos sin cos 2 1 0b x x x x m     

15. Chng minh rng:

. cot tan 2cot2a x x x

cos( ) 2cos( ) 3cot tan

.
sin( ) sin( ) 2
a b a b b a
e
a b a b
   

  


3
. sin3 3sin 4sinb x x x

3
. cos3 4cos 3cosc x x x


22
22
tan tan
. tan( )tan( )
1 tan tan
ab
d a b a b
ab

  


1 cos2

. cot tan 2
2 2 sin2 sin2
a a a
f
aa

  



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cos cos3 cos5
. cot3
sin sin3 sin5
a a a
ga
a a a




12
.
cos2
( ) ( )

44
h
a
co a co a




16. u thc sau:
a.    b.   
c. 









 










d. 

  



  

 

  

  
e.    
f. 

  

 
g.   
h.   
i.   
17. Cho    


   


   



Chng minh rng
a.


 
b.


 
18. Chng thc sau (vu kin c hai v ng th
a.     
b.    
c.     
d.     
e.     
f.   



g.





 
h.



 



 


i.













j. 





 
19. u thc sau:

a. 
b. 



 




c. 

  
d. 
e. 
















f.    
g.

  


 
h.





 

 



 


 
i.






j. 

  
k.   




l.  

  


m. 

  
n.   
o. 


 



p.   
20. Cho   


 






      



a. 

  


b. cos  )
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21. Cho  




  

 



   


   



22. Bi
a.



 
b.     
c.   
d. 

  

  


e.     
f.         
g.   
h.   
i. 

  


   
j. 

  

 
k. sin(a+b+c)    
l. (a+b+c)    
m.






n.



o.



p.


23. Bing:
a. 







b. 
c. 

  

  
d. 
e. 
f. 

 





 




g. 

  




  

  
24. n:
a. 








b. 








c.   

 



 


 



 

 



 

 



d.







e.









f.







g.






25. Chng minh rng:
a.  



b.  




c.  





d.  




e.  



f.  



26. Chng minh rng:
a.


   
b.       




c. 



 


 


 











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d.


 



e.


 


 
f.


 




 
g.





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
1.sinx=m


| | 1m 

1:sin 1 2
2
1:sin 1 2
2
0:sin 0
m x x k
m x x k
m x x k





     
        
    




sin

:
2
sin sin sin
2
xk
x m x

xk


  


   

  



arcsin 2
sin
arcsin 2
x m k
xm
x m k






  


VD1:
2
1

6
) sin sin sin( )
7
26
2
6
arcsin1/3 2
) sin 1/3
arcsin1/3 2
xk
a x x
xk
xk
b x x
xk








  


    








  

  


2.cosx=m

| | 1m 

1:cos 1 2
1:cos 1 2
0:cos 0
2
m x x k
m x k
m x x k




    
       
     





cos

:
cos cos cos 2x m x x k
  
      


cos arccos 2x m x m k

    

VD2:
3
) cos cos cos 2
2 6 6
) cos 1/3 arccos1/3 2
a x x x k
b x x k



      
    

3. tanx=m


tan


:
tan tan tanx m x x k
  
     


tan arctanx m x m k

   


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VD3:
) tan 1 tan tan
44
) tan 6 arctan6
a x x x k
b x x k



     
   

4. cotx=m



cot

:
cot cot cotx m x x k
  
     
.

cot arccotx m x m k

   
.
VD4:
1
) cot cot cot
33
3
) cot 8 arccot8
a x x x k
b x x k



     
   

  





2
2
2
2
sin sin 0
cos cos 0
tan tan 0
cot cot 0
a x b x c
a x b x c
a x b x c
a x b x c
  
  
  
  



-
2 2 2
a b c

-
22
ab


tan
2
x
t 



22
sin sin cos .cos 0a x b x x c x d   


2

2



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3 2 2 3
sin sin cos sin cos cos sin cos 0a x b x x c x x d x m x n x     


3


3



(sin cos ) sin cos 0 (1)a x x b x x c   



c 1. 
cos sin 2 cos( )
4
t x x x

  

22t  


2
1
sin cos
2
t
xx




c:   





    (2)
c 3. Gii (2)
i xng v
Dng: 



  



 

  

   
 t      

  

 





 




     .


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
Phn I: n
p 1: gi
1.  




2. 

  


 




3. 

  




4. 

  


 




5. 

  






vi



   


6. 

 



 




7. 

  

 
8.



 



 
9. 


  


 


10. 

  

 



11. 

0
45
 






12. 

  





vi    

13. 

  

 

 vi



  



14. 

  

   
15.   
16. 

  

   
17.     

18.  

  
19. 

  

  
20.   
21. 


 



 




 
p 2: Gi
1. Sin(3x+20
0
) + sin4x = 0.
2. 2sin2x + 5cosx = 0.
3. sin
2
(2x + 1) =

4
32 
.
4. (tg2x - 1)cos2x = 0.
5.
0
2cos1
2sin

 x
x
.
6.
0
3sin

tgx
x
.
7. Sinx.(1 + tg
2
2
x
) = 0.
8. sinx + sin2x = 2.
9. sin
2
(x -
4


) + tg
2
2x = 0.
10. tg4xtgx = -1.
11. cosxcos2xcos4xcos8x =
16
1
.
12.



1801693
4
cos
2
 xxx

.
13.



180016093
8
cos
2
 xxx

.

14.

 
0sin12cos
22
 xxx

.


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Phn II:
c 2
p 1: Gi
1. 

      6. 

     
2.       7.       
3. 

    8. 

 


  
4. 

 


   9. 

 


  
5.     10. 

  

   

  
p 2:
1.
22
2 4 2 0sin x cos x sinx   

2
22
(3 ) 3 3 ( 3 ) 2 0
22
cos x cos x cos x


     

3.
2 3 2 4
2
22
cos x cotg x sin x
cotg x cos x




4.
22
4 2 6 9 3 2
0
sin x sin x cos x
cosx
  


5.
2 (2 1) 1 0cos x m cosx m    

a) Gii
3
2
m 

 m

3
( ; )
22
x



6.
2
2
(1 )tan 1 3 0a x a
cosx
    

a) Gii
1
2
a 

 m
(0; )
2
x




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    
(BC NHT I VI sinx cosx)
 Gin lu
     (1)
P
Ta  la chn m
: Thc hic:
Bước 1. Kim tra:
1. Nu 

 

 

m
2. Nu 

 

 

 m cc
hin tic 2.
Bước 2. Chia 2 v 




 

c:




 

 




 

 




 












 








 n t







 






 
ng:
   







 

  









 n ca cos.



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 Thc hic:
Bước 1: Vi 



       
Bước 2: Vi 


        t   


, suy ra
 



 





ng: 



 




 



  



        (2)
Bước 3. Gi.
P
1.     
2.

   


3. 



 

   

4. 

     


5. 

  





6.

     


7.   
  
8.  


 


9.   

 

10.   

    
11.

22
3 2 1cos x sin x sin x  

12.

44
4( ) 3 4 2sin x cos x sin x  

ng cp bc 2: 

    

   (1)
: Chia hai Thc hic:
Bước 1. Vi cosx=0   


   
ng a=d
- Nn  


m.

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- Nn  


m.
Bước 2. V  



   
Chia hai v c

  c


       


ng:


 



        (2)
Bước 3. Gi
: S dc:


 
  



 
  


 



c :
b.sin2x+(c-a)cos2x=d-c-a (3)
c nht c
P
1. 

  

  

  
2. 

  

  
3. 

    
4.    



5. 


      
6. 

      
7.
22
(2 2) ( 1)sin x m sinxcosx m cos x m    

a) Gii m = 2
 m

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8.
22
3 (2 1) ( 1)sin x m sinxcosx m cos x m    

 m
(0; )
4
x



4. ng cp bi vi 
 Gin lu



  

  

  

   (1)

Thc hic:
Bước 1. Vi      


   
ng a=0.
- Nn  


  m.
- Nn  


  m.
Bước 2. Vi      


   
Chia hai v c

c



 

     
ng:


 

      (2)
Bước 3. Gi
P
1. 

 

    
2.


   
3.   


4.     
5. 

 


    
6. 

   

  
7.     


8.  







9.



 


  
10. 

  

  


  
11.       



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i xng vi sinx, cosx:
: Gi


  

     


cos sin 2 cos( )
4
t x x x

  

22t  


2

1
sin cos
2
t
xx




c:   




    (2)
c 3. Gii (2) theo t.
P
16.
2
(1 )( )
2
sinxcosx sinx cosx  

17.
1 1 10
3
cosx sinx
cosx sinx
   

18.
2 2 ( ) 1
4
sin x sin x

  

19.
| | 1 4 2sinx cosx sin x  

20.
33
sin x cos x m

 
nghit
(0; )x



21.
2 4( )sin x cosx sinx m  

 m
Phn III.
1. S dc h bc
a.
2 2 2
3
23

2
cos x cos x cos x  

b.
2 2 2 2
3 4 5 6sin x cos x sin x cos x  


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c.
33
4 3 4 3 3 3 4 3cos xsin x sin xcos x cos x  

d.



 


e. 



   




f. 

 

 




  
g. 

  

  

    
h.
66
7
16
sin x cos x

i.
66
4sin x cos x cos x

  

a.   

  

  

 

  


b.          
c. 

 

    

  


d.
33
2 2 2sin x sinx cos x cosx cos x   

3. Chia 2 v 

vng cp bc n.
a. 


     
b. 

 

  

 

  
c.

  



  

  

  
d. 

 


 




a.    b.     
m
a. 

 

      
b. 

  

    

  





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c. 

      

  
d.


  


 
e. 

          
f.
3
2 2 0sin x cosx cos x  

g.
10 8 6 1 0cos x cos x cos x   

h.
2
(2 1)(2 2 1) 3 4sinx sin x cos x   
6. X 
a)   

 



b)  



     




c)   







  







 





 
t   
a)      
b)       
c)  



 
d)
6tan tan2xx

e)
tan 2tan2cotx x x

8. X a 
a)
2( ) tansinx cosx x cotx  

b)
tancotx x sinx cosx  

c)
2 3 2 3
tan tan tan 6x x x cotx cot x cot x     

d)
tan2 tan3 tan5 tan2 tan3 tan5x x x x x x  


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e)

2 2 2 2 2 2
tan .tan 3 .tan 4 tan tan 3 tan 4x x x x x x  

f)
2
2
3
3tan (tan ) 1 0x m x cotx
sin x
    
 m
 u kin
a)      
b)

 


   
c)






   
10. Hp nghi ng. m thuc [1; 100] c

a) 


      
b)   

 









11. PP gi)
a) 

  

 




b) 

 

 




12 tuyi:
a.
3 2| | 2cosx sinx

b.
| | 3 0cosx sin x

c.
| | | | 2sinx cosx sinx cosx   

13. H 
a.
2
3
sin 3
xy
sinx y









b.
22

3
sin
2
5
sin
4
sinx y
sin x y











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c.
3
sin
2
1
cos

2
sinx y
cosx y










d.
1
cos
4
3tan tan
sinx y
xy








e.
7cos 0
5 cos 6 0
sinx y
sinx y



  


f.
3
3
cosx acos y
sinx asin y






 h m
g.
2
22
1 | |
tan 1
ax a y sinx
xy

   





 h m duy nht

NG HP
1. - 1999]:
3
sin 4sin cos 0x x x  
.
2. - 1998]:
2 2 2
sin cos 2 cos 3x x x
.
3. - 2001]:
22
2cos2 sin cos cos sin 2(sin cos )x x x x x x x   

4. [HVKTQS - 1999]:
33
2sin sin 2cos cos cos2x x x x x   
.
5. - 1996]:
3 3 2 5
(sin sin3 cos cos3 ) cos 2x x x x x
.
6. - 1961]:

2 2 2
sin sin 2 sin 3 2x x x
.
7. 
2
1 2cos (3 /5) 3cos(4 /5)xx
.
8.
2 2 0 2 2 0
sin (2 3 ) cos (45 2 ) cos (2 5 ) sin (45 6 )x x x x      

9.
6 3 4
8 2 cos 2 2sin sin3 6 2 cos 1 0x x x x   
.
10.
22
cos sin ( 3) 1xx
.
11. [HVNH - 2000]:
23
sin sin cos 0x x x  
.
12. - 2000]:
1 3tan sin2xx
.
13.
3
sin 3sin
4 2 4 2

xx

   
  
   
   
.
14. [-2001]:
2sin2 cos2 7sin 2cos 4x x x x   
.
15. - 1999]:
3
(sin cos ) 4sinx x x
.
16. [BCVT - 1999]:
sin 3 sin2 sin
44
x x x

   
  
   
   
.
17. [QHQT - 2000]:
cos cos3 2cos5 0x x x  
.

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18. - 1999]:
2
sin (tan 1) 3sin (cos sin ) 3x x x x x   
.
19. - 1982]:
22
3
7sin 2sin2 3cos 3 15 0x x x   
.
20.
*****

3 2 3
sin 3sin cos cos 0x x x x  
.
21.  m thuc [1, 70] c

23
2
2
cos cos 1
cos2 tan
cos
x
xx
x




22. [HVQY - 1997]:
88
1
sin cos
8
xx

23. - 2001]: 3tanx + 2cot2x = tan2x
24. - 1999]:
22
sin 4 cos 6 sin(10,5 10 )x x x

  

25. - 2000]:
tan 3cot 4(sin 3cos )x x x x  

26. - 1998]:
1
2tan cot2 2sin2
sin2
x x x
x
  

27.  71]:
2
3tan3 cot2 2tan

sin4
x x x
x
  

28. [HVQHQT - 1998]:
3
5
sin 5cos sin
22
xx
x

29. [HVQS - 2002]:
1
cos cos2 cos3 cos4 cos5
2
x x x x x     

30. [HVBCVT - 1998]:
sin4 cos4 1 4(sin cos )x x x x   
.

p v  m thuc mi

31.  m thun
-
[ , ]
44


:
33
cos sinx x m

32. -  m:
66
cos sin sin 2x x m x

33. (HVQY -  m:
cos2x + (m + 1) sinx + m = 0
34.  m:

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sin2x + sinx + cosx + m = 0
35.  m:
2
2
1
tan 2(tanx+cotx) 0
sin
xm
x
   


36.   m

xk



m.tanx = sin4x
 mi nghim cm c

2
cos sin 1 (1)
sin cos (2)
m x x
m x x m



-  

2
sin3 cos2 1 2sin cos2 (1)
sin3 sin (4 2| |)sin (2)
x x x x
x m x m x
  
  

 m thuc min D
39.  m thuc
[0, ]:


2sin x+1
sin 2
m
x


.
40. m thuc khong
-3
( , )
4


c:

2 2 2 2
sin sin cos cos cos sina x a x a x a x x x    
.
41.  m:
66
sin cos |sin2 |x x m x
.
42.  

2
sin2 2 2cos 2 sin (1)
2sin cos2 sin2 2 sin cos 1 (2)
a x x a x
x x x b b x x

  
     

43.-  m thuc
[0, ]:



2
(2sin 1)(2cos2 2sin ) 3 4cosx x x m x    

44. m thuc
[0, ]:

2sin x+1
sin 2
m
x



p v min - max:

TCT
0976853538

23
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:
2 cos

cos sin 2
x
y
xx



.
- :
(2sin cos )(2cos sin )y x x x x  
.
t - :
20 20
sin cosy x x
.
 :
5cos cos5 , [- /4; /4]y x x x

  
.
 :
sin cosy x x
.
- :
2
cos , 0;
4
y x x x



  


.
- :
sin cos cos siny x x x x
.
- 2001): Cho
22
1
sin sin
2
xy
a
22
tan tanP x y
.