Chuyªn ®Ò :ph­¬ng tr×nh l­îng gi¸c



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a/kiÕn thøc cÇn nhí vµ ph©n lo¹i bµi to¸n

d¹ng 1 Ph­¬ng tr×nh bËc nhÊt vµ bËc hai , bËc cao víi 1 hµm sè l­îng gi¸c

§Æt HSLG theo t víi sinx , cosx cã ®iÒu kiÖn [pic] [pic]1

Gi¶i ph­¬ng tr×nh ……….theo t

NhËn t tho¶ m·n ®iÒu kiÖn gi¶i Pt l­îng gi¸c c¬ b¶n

Gi¶i ph­¬ng tr×nh:

1/ [pic] 2/ 4sin3x+3[pic]sin2x=8sinx

3/ 4cosx.cos2x +1=0 4/ [pic]

5/ Cho 3sin3x-3cos2x+4sinx-cos2x+2=0 (1) vµ cos2x+3cosx(sin2x-8sinx)=0 (2).

T×m n0 cña (1) ®ång thêi lµ n0 cña (2) ( nghiÖm chung sinx=[pic])

6/ sin3x+2cos2x-2=0 7/ a/ tanx+[pic] -2 = 0 b /[pic]+tanx=7

c* / sin6x+cos4x=cos2x

8/sin([pic])-3cos([pic])=1+2sinx 9/[pic]

10/ cos2x+5sinx+2=0 11/ tanx+cotx=4 12/ [pic]

13/[pic] 14/ cos2x+3cosx+2=0

15/[pic] 16/ 2cosx-[pic]=1

d¹ng 2: Ph­¬ng tr×nh bËc nhÊt ®èi víi sinx vµ cosx : asinx+bcosx=c

|C¸ch 1: asinx+bcosx=c |

|§Æt cosx=[pic] ; sinx= [pic] |

|[pic] |

| |

|C¸ch : 2 [pic] |

|§Æt [pic] |

|[pic] |

| |

C¸ch 3: §Æt [pic][pic] ta cã [pic] [pic]

§¨c biÖt :

1. [pic]

2. [pic]

3. [pic]

§iÒu kiÖn Pt cã nghiÖm : [pic]

gi¶i ph­¬ng tr×nh :

1/ 2sin15x+[pic]cos5x+sin5x=k víi k=0 vµ k=4 víi k=0

2/ a : [pic] b: [pic]

c: [pic]

3/ [pic] *t×m nghiÖm [pic]

4/( cos2x-[pic]sin2x)- [pic]sinx-cosx+4=0 5/ [pic]

6/ [pic]

D¹ng 3 Ph­¬ng tr×nh ®¼ng cÊp ®èi víi sin x vµ cosx

|§¼ng cÊp bËc 2: asin2x+bsinx.cosx+c cos2x=0 |

|C¸ch 1: Thö víi cosx=0 Víi cosx[pic]0 .Chia 2 vÕ cho cos2x ta ®­îc: |

|atan2x+btanx +c=d(tan2x+1) |

|C¸ch2: ¸p dông c«ng thøc h¹ bËc |

|§¼ng cÊp bËc 3: asin3x+b.cos3x+c(sinx+ cosx)=0 hoÆc |

|asin3x+b.cos3x+csin2xcosx+dsinxcos2x=0 |

|XÐt cos3x=0 vµ cosx[pic]0 Chia 2 vÕ cho cos2x ta ®­îc Pt bËc 3 ®èi víi tanx |

Gi¶i ph­¬ng tr×nh

1/a/ 3sin2x- [pic]sinxcosx+2cos2x cosx=2 b/ 4 sin2x+3[pic]sinxcosx-2cos2x=4

c/3 sin2x+5 cos2x-2cos2x-4sin2x=0 d/ 2 sin2x+6sinxcosx+2(1+ [pic])cos2x-5-[pic]=0

2/ sinx- 4sin3x+cosx=0 2 c¸ch +/ (tanx -1)(3tan2x+2tanx+1)=0

[pic] + sin3x- sinx+ cosx- sinx=0 [pic](cosx- sinx)(2sinxcosx+2sin2x+1)=0

3/ tanx sin2x-2sin2x=3(cos2x+sinxcosx)

4/ 3cos4x-4sin2xcos2x+sin4x=0 5/ 4cos3x+2sin3x-3sinx=0

6/ 2 cos3x= sin3x 7/ cos3x- sin3x= cosx+ sinx

8/ sinx sin2x+ sin3x=6 cos3x 9/sin3(x-[pic]/4)=[pic]sinx

Dang 4 Ph­¬ng tr×nh vÕ tr¸i ®èi xøng ®èi víi sinx vµ cosx

|* a(sin x+cosx)+bsinxcosx=c ®Æt t= sin x+cosx [pic] |

|[pic] at + b[pic]=c [pic]bt2+2at-2c-b=0 |

|* a(sin x- cosx)+bsinxcosx=c ®Æt t= sin x- cosx [pic] |

|[pic] at + b[pic]=c [pic]bt2 -2at+2c-b=0 |

Gi¶i ph­¬ng tr×nh

1/ a/1+tanx=2sinx + [pic] b/ sin x+cosx=[pic]-[pic]

2/ sin3x+cos3x=2sinxcosx+sin x+cosx 3/ 1- sin3x+cos3x= sin2x

4/ 2sinx+cotx=2 sin2x+1 5/[pic]sin2x(sin x+cosx)=2

6/ (1+sin x)(1+cosx)=2 7/[pic](sin x+cosx)=tanx+cotx

8/1+sin3 2x+cos32 x=[pic]sin 4x 9/* a* 3(cotx-cosx)-5(tanx-sin x)=2

9/b*: cos4x+sin4x-2(1-sin2xcos2x) sinxcosx-(sinx+cosx)=0

10/ [pic] 11/ cosx+[pic]+sinx+[pic]=[pic]

12/ sinxcosx+[pic]=1

dang 5 Gi¶i ph­¬ng tr×nh b»ng ph­¬ng ph¸p h¹ bËc

| C«ng thøc h¹ bËc 2 | C«ng thøc h¹ bËc 3 |

|cos2x= [pic] ; sin2x= [pic] |cos3x= [pic] ; sin3x= [pic] |

Gi¶i ph­¬ng tr×nh

1/ sin2 x+sin23x=cos22x+cos24x 2/ cos2x+cos22x+cos23x+cos24x=3/2

3/sin2x+ sin23x-3 cos22x=0

4/ cos3x+ sin7x=2sin2([pic])-2cos2[pic]

5/ sin24 x+ sin23x= cos22x+ cos2x víi[pic]

6/sin24x-cos26x=sin([pic]) víi[pic] 7/ cos4x-5sin4x=1

8/4sin3x-1=3-[pic]cos3x 9/ sin22x+ sin24x= sin26x

10/ sin2x= cos22x+ cos23x 11/ (sin22x+cos42x-1):[pic]=0

12/ 4sin3xcos3x+4cos3x sin3x+3[pic] cos4x=3 [pic] 13/ 2cos22x+ cos2x=4 sin22xcos2x

14/ cos4xsinx- sin22x=4sin2([pic])-7/2 víi [pic] ................
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