Phuong trinh luong giac

Ph¬ng tr×nh lîng gi¸c

A- Ph¬ng tr×nh bËc 2 ®èi víi mét hµm sè lîng gi¸c

1/. 6sin2x-13sinx+5=0 2/.2cos2x-(2- 3 )cosx- 3 =0 3/. 2tg2x-5tgx+2=0 4/.3cotg2x-4cotgx-7=05/.tgx+cotgx=2 6/.2cos2x+cosx=1 7/.3cos2x=2sin2x+4sinx

8/. 3/cosx=3+2tg2x 9/. 3sin2x-2sinx-2=0 **4cos2x-2cosx-1=0 **tg22x-tg2x-6=0**2sin2x+5cosx-4=0**1/sin2x=2 3 +(2+ 3 )/tgx**cos4x - 3 2 cos2x + 2=0**tg2x+(1-

3 )tgx- 3 =0**2sin2x+5cosx+1=0**sin2x-(2+ 3 )sinx+ 3 =0**16-15sin2x-8cosx=0**6-4cos2x-9sinx=0**6cos24x+11cos4x-2=0**4cos2x-2( 3 - 2 )cosx- 6

=0**4sin2x-2( 3 +1)sinx+ 3 =0**4cos22x-2( 2 +1)cos2x+ 2 =0 **4sin23x+2( 3 +1)sin3x+ 3 =0 **4cos24x+2( 3 + 2 )cos4x+ 6 =0**

B- Ph¬ng tr×nh bËc nhÊt ®èi víi sinx vµ cox

*cosx-sinx= 2 ** 3 sinx+cosx= 2 **2sinx-5cosx=4**5cos2x-12sin2x=13**sinx+cosx= 6 /2**cosx- 3 sinx=2** 3

cos2x+sin2x=1**3cos3x+5sin3x=4**3sin3x- 3 cos9x=1+4sin33x**2cos2x=(1+ 3 )(cosx-sinx)**sin2x+( 2 -1)cos2x=1. CMR tg

8

π

= 2 -1 **

=

+ −

−

2cos sin 1

cos 2cos .sin

2

x x

x x x

3

** 2 sin3x- 3 cos5x=sin5x+ 2 cos3x **

6

3cos 4sin 6

3cos 4sin 1

x x

x x

+ + =

+ +

** x x

x

x x

sin 2 cos 2

1 cos 2

sin 3 sin

= +

−

−

C- Ph¬ng tr×nh ®¼ng cÊp ®èi víi sinx vµ cosx

** 2 2 6 sin x 7 3 sin 2x 8 cos x 6 + − = **4sin2x+3 3 sin2x-2cos2x=4** 3 cos2x+2sinx.cosx- 3 sin2x- 2 =0**4 cos2x- 3 sin2x+2

sin2x=1**( 2 -1)cos2x+2sinx.cosx+( 2 +1)sin2x- 2 =0**5 sin2x+ 3 sinx.cosx+6 cos2x=5**2 sin2x+4sinx.cosx-4 cos2x-1=0 **sin2x-( 3 -1)sinx.cosx- 3 cos2x=0

**

2 sin2x+(1- 3 )sinx.cosx+(1- 3 )cos2x=1**2cos(2x+

6

π

)- 3 sinx=0** cos2x+2 3 sinx.cosx+5 sin2x=2**3 sin2x+8sinx.cosx+4 cos2x=0**2 sin2x+4sinx.cosx-4 cos2x￾1=0**2 cos3x=sin3x**cos3x+sinx-3 sin2x.cosx=0**3(tgx+cotgx)=2(2+sin2x)**tgx.sin2x-2.sin2x=3(cos2x+sinx.cosx)**sinx-4.sin3x+cosx=0**1+3tgx=2sin2x **2. 2

(sinx+cosx)cosx=3cos2x **sin2x(cotgx+tg2x)=4.cos2x ** 6sinx￾x

x x

2cos 2

sin 4 .cos

=2.cos3x**4.cos3x+2.sin3x-3sinx=0**6sinx-2.cos3x=5sin2x.cosx **4.cosx.cos2x=cosx

– sinx**3tgx+2cotgx=sin2x**sin2x+2tgx=3**sin2x+cos2x+tgx=2**

D- Ph¬ng tr×nh ®èi xøng - ph¶n ®èi xøng

*sin2x=sinx+cosx+1**3(sinx+cosx)+2xinx.cosx+3=0**|sinx-cosx|+4sin2x=1**(1+ 2 )(sinx+cosx)-2sinx.cosx-(1+ 2 )=0**2sinx.cosx-(sinx+cosx)+1=0

**2sin2x-4(sinx+cosx)+3=0**4sinxcosx-(2- 2 )(sinx+cosx-1)=0**sinx-cosx-sinxcosx+1=0**6(sinx-cosx)+sinxcosx+6=0**2sin2x-( 6 + 2 )(cosx-sinx)=2+ 3

**4-4(cosx-sinx)-sin2x=0** sin3x+cos3x=

2

2

** sin cos 1 x x + = ** 4 4 3

sin cos

4

x x + = **

1 1 10 cos sin

cos sin 3

x x

x x

+ + + = **

3

2

3

1 cos

1 sin

x

tg x

x

−

=

−

**

2 1 cos

1 sin

x

tg x

x

+

=

−

**cotgx-tgx=sinx+cosx**

3 3 sin cos cos 2

2 cos sin

x x

x

x x

+

=

−

E: C¸c ph¬ng tr×nh kh¸c

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

3 3 1

sin x cos x 1 sin 2x

2

+ = − ** 2

sin x cos x 1 0 + + = ** 4 2 (sin 2x 3) sin x (sin 2x 3) sin x 1 0 + − + + = **

4 4 1

sin x cos x

4 4

  π

+ − =  ÷   ** 4 4 5

sin x cos x 1

3

+ = ** 4 6 sin x cos x cos 2x + = − ** 6

32cos x cos 6x 4 − = **

2 2 2 cos x cos 2x cos 3x 1 + + = **

1

cos x cos 2x cos 3x

2

− + = ** 3

sin x 2 sin x

4

  π

 ÷ + =   **T×m nghiÖm x∈(0, 2 ) π cña pt: sin 3 sin sin 2 cos2

1 cos 2

−

= +

−

x x

x x

x

**T×m nghiÖm ( ,3 )

2

x

π

∈ π cña pt:

5 7 sin 2 3cos( ) 1 2sin 2 2

  π π

 ÷  

x x x + − − = + **2cosx-|sinx|=1** 2 2 17 sin 2 cos 8 sin 10 2

  π

 ÷  

x x x − = + **2cos3x+cos2x+sinx=0** 2 3 4 2cos 1 3cos 2 3 + = x x **

sin cos 4sin2 1 x x x − + = ** 2 (2sinx-1)=4(sinx-1)-cos 2

4

x

  π

+  ÷  

-sin 2

4

x

  π

+  ÷  

** 2 2sin 3 1 8sin 2 cos 2 4

  π

 ÷   x x x + = + **tgx+cotgx=4**

3 3

2 3

1 2 sin cos

cos 1 sin

x x

x x

− −

=

−

**2tgx+cotgx= 3 +

2

sin 2x