Phuong Trinh Luong Giac Va Ung Dung1

7/25/2019 Phuong Trinh Luong Giac Va Ung Dung1

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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on

MUC LUC Trang

#$ng thc lng gac c%n n&' vng *

A Phng trinh lng gc c bn +

, t- a dung +/0ng d1n g2 3 t- a dung 4, t- r5n l67-n *8

B Phng trinh bc ha v bc cao ! v" #$t h# lng gc 9*

, t- a dung 99/0ng d1n g2 3 t- a dung 9+, t- r5n l67-n +:

# Phng trinh bc nh%t th&o 'n v co' +8

, t- a dung +8/0ng d1n g2 3 t- a dung :*, t- r5n l67-n 4;

( Phng trinh lng gc )ng c%* 4*, t- r5n l67-n ;>C C BS;

L binhI Ty v-c U6Qt h-n 3 c6ngx/2x/3x O g[ t ln tng SRn v-c S ch[ng vJ c@ng '$-tc6ng. Nhng S vJ c6ng x h7 c6ng 2x #ac 3\n cK th tr2 lW c6 h SK dX vo k6nn-' s6I ? Trong hng trinh lng gac tfn t\ 3 c6ng x/2x/3x O t nn S vJ c6ngtr6ng gn 2x nR6 trong 36 thc cK ch sn*U (ho!-c cos*U). #^n Hh$ng ch sn*U (ho!-ccos*U)O nn S vJ c6ng x ?.

, g2 th' Hh2o

( ) ( ) ( )3 2 3 24cos x 3cosx 4 2cos x 1 3cosx 4 # 4cos x !cos x #? ' - - - + - = ' - =

( ) ( )

( ) ( ) 2

cosx # C4cos x cosx 2 # x $ / $

cosx 2 D 2

= %' - = ' ' = + % = .

#/ $ 3/ 3

Eo x #14 /$ # $ 14 x $2 2 2 2 2

7 7 F8- " " G 8 8% % % % %8 8 8 > ' " + % " ' ; 9 9 H A B 8 8 88 8 8: I:

.

, g2 th' Hh2o

( ) ( ) ( )2cosx 1 2sinx cosx 2sinxcosx sinx? ' - + = -

( ) ( ) ( ) 2cosx 1 2sinx cosx sinx 2cosx 1 #' - + - - =

( ) ( ) ( ) ( ) 2cosx 1 2sin x cosx sin x # 2cosx 1 sin x cosx # >' - + - = ' - + = A B

( )x $22cosx 1 # cosx cos 3 $J3

sinx cosx # tanx 1 x J4

% % = + % - = = ' ' ' + = % = - = - + %

.

L binhI Ty v-c U6Qt h-n cac c6ng3x v 2x O ch[ng t ngh_ ng7 SRn v-c S ch[ng vJ c@ng'$-tc6ng U 3Yng c$ng thc nhn 3 v c$ng thc nhn S$ c h' cos

, g2 th' Hh2o

( ) 3 2 3 24cos x 3cosx 2cos x 1 cosx 1 # 2cos x cos x 2cosx 1 #? ' - + - - - = ' + - - =

? #%n c@ 3@ th$ng 'nhAAAA? PgB 8

B W. p2 hng trinhI ( )cos3x 4cos2x 3cosx 4 # / x #14- + - = ? @ TrHch d th t12en 'nh f= hc 5h! ( n/# X``X

B X. p2 hng trinhI ( ) ( ) ( )2cosx 1 2sinx cosx sin2x sinx- + = - ?TrHch d th t12en 'nh f= hc 5h! ( n/# X``Z

B Y. p2 hng trinhI ( )cos3x cos2x cosx 1 #+ - - = ?TrHch d th t12en 'nh f= hc 5h! ( n/# X``\


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B Z. p2 hng trinhI ( )sinx cosx 1 sin2x cos2x #+ + + + = ?TrHch d th t12en 'nh f= hc 5h! B n/# X``[

Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)

( ) ( ) ( )( ) 2 2cos x 2cosx 1 2cosx 1 # 2cosx 1 cos x 1 #' + - + = ' + - =

( ) ( ) 2sinx # x $

2cosx 1 sin x # $J1 2cosx x J2

2 3

= = % ' - + = ' ' % = - = + %

.

, g2 th' Hh2o

( ) ( ) 2sinx cosx 2sinxcosx 2cos x #? ' + + + =

( ) ( ) sinx cosx 2cosx sinx cosx #' + + + =

( ) ( ) sinx cosx 1 2cosx #' + + =

( )sinx cosx tanx 1 x $

4 $J1 2 2cosx cosx cos x J22 3 3

% = - = - = - + % ' ' ' % %= - = = + %

.

L binhI Ty v-c U6Qt h-n c c6ng2x v c6ng x ' t ngh_ SRn v-c ch67n c6ng 2x vJ c6ng x3Yng c$ng thc nhn S$ c h' sn v cosO ty SK U6Qt h-n nhn t ch6ng h vR

( ) ( )2sinx 1 2cos x 1 2sinxcosx 1 cosx? ' + - + = +

( ) ( ) 22sinxcos x 2sinxcosx 1 cosx 2sinxcosx cosx 1 1 cosx #' + = + ' + - + =

( ) ( ) ( )21 x $2cosx 3cosx 1 sin2x 1 # $/J2

sin2x 1 x J4

% = + % = - ' + - = ' ' %= = + %

.

L binhI Ty v-c U6Qt h-n h c6ng3

x2%

- v

x4%

- g[ t s67 ngh_ SRn v-c S h c6ng

Hhac nh6 n7 vJ c@ng '$-t c6ng ch6ng l x . " l' Sc SJ6 SKO t cK th d@ng c$ngthc c$-ng c6ng ho!-c d@ng c6 th%n ch[ ?cos S b sn 3@ b hu chojj. T thXc h-n h ztng SK k6 h cach g2 s6 S7

PgB ;> ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B [. p2 hng trinhI ( ) ( )sinx 1 cos2x sin2x 1 cosx+ + = + ?


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"MI

sin x #13

sin x sin x # cos 2x # cos 2x #3 6 2 6 6

sin x #6

7 8 %8 + &8 8 % % % % 8 8 ' + - & ' - & ' - & 9 8 % 8 - &8 8 88:

.

( ) ( )2 21 1 1 $

1 sin 2x sin 2x 1 cos4x x / $2 ! 4 2 12 2

% %? ' - = ' = ' - = ' = + .

, g2 th' Hh2o

"MIcos x # 14

cos x cos x # cos2x cos # cos2x #4 4 2 2

cos x #4

7 8 % 8 - &8 8 % % % 8 8 ' - + & ' + & ' & 9 8 % 8 + &8 8 88:

.

T cKI tan x tan x tan x tan x tan x cot x 14 4 4 2 4 4 4

> % % % % % % % A - + = - - + = - - = A A B.

( ) ( )2 4 2 4 4 21 1

1 sin 4x cos 4x 1 1 cos 4x cos 4x 2cos x cos 4x 1 #2 2

? ' - = ' - - = ' - - =

( )( )

2

2 22

2

t 1 C2t t 1 # 1 cos 4x 1 sin 4x # sin4x #t Dt cos 4x # 2

t cos 4x #

7 8 =887 88 - - = 888 8' ' ' = ' = ' =9 9 = -8 8= K 8 88: 88 = K88:

( )( )

( )sin2x # C $

x / $cos2x # D 2

78 = %88' ' = 98 =88: .

L6 zO t cK th thXc h-n 3Rn Se '16 s 3Yng c$ng thc c$-ng thBo tn nh s6

tan tanx tan tan x 1 tanx 1 tanx4 4tan x .tan x . . 14 4 1 tanx 1 tanx

1 tan tanx 1 tan tanx4 4

% %- + % % - + - + = = = % % + - + -.

L binhI Nhin vo hng trinh n7O t ngh_ d@ng c$ng thc c$-ng c6ng thBo snAAO ho!-c Utteng c6ng c ch[ngO AA. nhng Syng v$- l' nh thRO nK s HhK S SRn HRt k62. T h7

UB' g h c6ng3 x1# 2

%- v

3x1# 2%

+ cK ' ln h-gi h7 Hh$ng Th-t v-7I

PgB ;* ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B ^. p2 hng trinhI( )

4 44sin 2x cos 2x cos 4x

tan x tan x4 4

+= ?

% % - + TrHch d th t12en 'nh f= hc 2 (7ng n/# W__]

B _. p2 hng trinhI ( )3 x 1 3x

sin sin 11# 2 2 1# 2

% % - = + TrHch d th t12en 'nh f= hc Thj2 L n/# X``W


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3x 3x 3x 3 xsin sin sin sin3

1# 2 1# 2 1# 2 1# 2

> % % % % A + = % - + = - = - A A B. Ty SKO t s

S!-t3 x

t1# 2

%= - v s dung c$ng thc nhn 3 l t 6 nhQt.

, g2 th' Hh2o

T cKI3x 3x 3x 3 x

sin sin sin sin31# 2 1# 2 1# 2 1# 2

> % % % % A + = % - + = - = - A A B.

( ) ( )3 x 1 3 x

1 sin sin3 21# 2 2 1# 2

% % ' - = - .

"!-t3 x

t1# 2

%= - . ( ) ( ) ( )3 2

1 12 sint sin3t sint 3sint 4sin t sint 1 sin t #

2 2' = ' = - ' - =

( )3 x 3t $ $ x $2

sin t # 1# 2 $/Jcost # 3 x 2t J J x J22 1# 2 2

% % = % - = % = - % = ' ' ' ' % = % % %= + % - = + % = - %

.

, g2 th' Hh2o

T cKI3

sin 3x sin 3x sin 3x sin 3x sin3 x4 4 4 4 4

> > % % % % % A A - = - - = - %- - = - + = - + A A A B B

"!-t t x x t4 4

% %= + ; = - . L[c SK ( )1 sin3t sin 2t .sint

2

% ' - = -

32 2 2

sint # sint #4sin t 3sint cos2tsint #

4sin t 3 1 2sin t # sin t 1

= = ' - + = ' ' - + - = =

( )t $ x $

sin t # 4 x m / $/J/mcost # 4 2t J x J2 4

% = % = - + % = % % ' ' ' ' = - + % = %= + % = + %

.

, g2 th' Hh2o

T cKI ( )cos3x cos 3x cos 3 x3

% = - % + = - + .

? #%n c@ 3@ th$ng 'nhAAAA? PgB ;9

B WW. p2 hng trinhI ( )3!cos x cos3x 13

% + = TrHch d th t12en 'nh f= hc k1!c 4a 8 ;$ n/# W___

B W`. p2 hng trinhI ( )sin 3x sin2xsin x 14 4

% % - = + TrHch d th t12en 'nh 8c D3n B1 ChHnh Dn Th$ng n/# W___


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( ) tanx 1 x $ / $4%

' = ' = + % .

Cch g Y.

( ) ( )3 3

1 11 2 . 2sin x 2sinx 2 sinx cosx 2sinx

42 2

> > % A A' + = ' + = A A A B B

( ) ( )3

sin x cosx 4sin x 2' + =

i ( )cosx # a sinx 1= = Hh$ng h2 l ngh-' c hng trinh ( )2 nn ch h vR c

hng trinh ( )2 cho 3cos x O t ScI ( ) ( ) ( )3 22 tanx 1 4tanx. 1 tan x' + = +

p2 hng trinh thBo tnU t Sc ngh-'I ( )tanx 1 x $ / $4

%= ' = + % .

, g2 th' Hh2oCch g W.

"!-t t x x t4 4% %

= - ; = + . L[c SKI ( ) ( )3 31 sin t 2sin t 4 sin t sint cost' = + ' = +

( )( ) 3 2 2sin t sint cost sin t cos t' = + +

( ) 3 3 2 2 3sin t sin t sintcos t costsin t cos t cost sintcost 1 #' = + + + ' + =

( ) ( )

cost # 3t $ x $ $ / $

sin2t 2 D 2 4 4

= % % %' ' = + % ' = + % O - + % = - .

Cch g X v cch g Y(tng tX v du ;9). ,\n Sc tX g2

L binhI , toan cK cac c6ng Hhac nh6 thBo '$-t h' 3-c nhQt lng gac cos (ho!-c sn ho!-c c2sn v cos) d\ng teng (ho!-c h-6). T nn gh cac s h\ng n7 thnh c!- so cho h-6(ho!-c teng) cac c6ng c ch[ng 3Yng nh6O tc l trong trWng h n7 S z

( )x 4x x+ = v ( )2x 3x x+ = . T\ so h2 gh nh v-7 Lz do rQt Sn g2nOch[ng t c%n nhng ?thy s ch6ng? S nhK' r ngoO S 3 toan vJ d\ng hng trinhtch s.

, g2 th' Hh2o

( ) ( ) ( ) x 3x x x

cosx cos4x cos2x cos3x # 2cos cos 2cos cos #2 2 2 2

? ' + + + = ' + =

x 3x x x x

2cos cos cos # 4cos cosxcos #2 2 2 2 2

' + = ' =

? #%n c@ 3@ th$ng 'nhAAAA? PgB ;+

B WZ. p2 hng trinhI ( )cosx cos2x cos3x cos4x #+ + + = ?

B WY. p2 hng trinhI ( )3sin x 2sinx 14

% - = TrHch d th t12en 'nh f= hc k1!c 4a T*8CM n/# W__^


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( )

x $2x $ x

cos # 2 2 2

cosx # x J x J $Jm2 2

x x x 2mcos # m2 2 2

% % % = + % = + = % % ' = ' = + % ' = + % % = % + %= = + %

.

L binhI 0 nhng hng trinh cK nhng h\ng t 3-c h thBo sn v cosO t thWng d@ng c$ngthc h\ 3-c S 3 toan tr nn Sn g2n hn.

, g2 th' Hh2o

( ) ( ) ( ) ( ) ( )1 1 1 3

1 cos2x 1 cos4x 1 cos6x cos2x cos6x cos4x #2 2 2 2

? ' - + - + - = ' + + =

( ) 2cos4xcos2x cos4x # cos4x 2cos2x 1 #' + = ' + =

( )$cos4x # 4x $ x

2 ! 4 $/J2 2cos2x cos 2x J2 x J3 3 3

% % % = = + % = + ' ' ' % % %= = + % = + %

.

, g2 th' Hh2o

( ) ( ) ( ) ( )1 1 1

1 cos2x 1 cos4x 1 cos6x 22 2 2

? ' - + - + - =

( ) ( ) ( )1 1

cos2x cos4x cos6x cos2x cos6x cos4x 1 #2 2

' - + + = ' + + + =

( )2

2cos4xcos2x 2cos 2x # 2cos2x cos4x cos2x #' + = ' + =

( )

x $2cosx #

4cos2x cos3x cosx # cos2x # x J $/J/m4 2

cos3x #x m

6 3

%= + % = % %' = ' = ' = + = % % = +

.

, g2 th' Hh2o

PgB ;: ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B W[. p2 hng trinhI ( )2 2 2 3

sin x sin 2x sin 3x2

+ + = ? .

TrHch d th t12en 'nh Cao )ng Ph=# 8ng m3n 5h! A n/# X```

B W\. p2 hng trinhI ( )2 2 2sin x sin 2x sin 3x 2+ + = ? .TrHch d th t12en 'nh f= hc Ph=# Th1t T* 8CM 5h! A n/# X``W

B W]. p2 hng trinhI ( )2 2 2 2sin x sin 3x cos 2x cos 4x+ = + ?

TrHch d th t12en 'nh f= hc nh tF k1!c (n n/# W___


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( ) ( ) ( ) ( ) ( )1 1 1 1

1 cos2x 1 cos6x 1 cos4x 1 cos!x2 2 2 2

? ' - + - = + + +

( ) cos2x cos6x cos4x cos!x 2cos4xcos2x 2cos6xcos2x' - + = + ' - =

( ) 2cos2x cos6x cos4x # 4cos2xcosxcosx #' + = ' =

( )cosx #

mcos2x # x $ x J x $/J/m

2 4 2 1# cosx #

= % % % % % ' = ' = + % = + = + =

.

, g2 th' Hh2o

( ) ( ) ( ) ( ) ( )1 1 1 1

1 cos6x 1 cos!x 1 cos1#x 1 cos12x2 2 2 2

? ' - - + = - - +

cos6x cos!x cos1#x cos12x 2cosxcosx 2cos11xcosx+ = + =

( ) ( )

x $2

cosx # Jcosx cosx cos11x # x $/J/m

cosx cos11x 2m

x

%= + % = %' - = ' ' = - = %

=

.

, g2 th' Hh2o

( ) U Ucos3x sinx 1 cos x 1 cos cos3x sinx sinx cos2

% ? ' + = - + - - ' + = -

cos3x cosx sinx sinx # 2cos6xcos3x 2cos6xsinx #' + + - = ' + =

( ) ( )

x $12 6cos6x #

cos6x cos3x sinx # x J $/J/mcos3x cos x 4

2 mx

! 2

% %= + = % ' + = ' ' = + % % = + % % = - +

.

, g2 th' Hh2o

? #%n c@ 3@ th$ng 'nhAAAA? PgB ;=

B W^. p2 hng trinhI ( )2 2 2 2sin 3x cos 4x sin x cos 6x- = - ?TrHch d th t12en 'nh f= hc 5h! B n/# X``X

B W_. p2 hng trinhI ( )sin2 2x x

cos3x sinx 2 2cos4 2 2

% + = + - ? TrHch d th t12en 'nh f= hc The (Ic The Thao n/# X``W

B X`. p2 hng trinhI ( )2 2 2sin x cos 2x cos 3x= + ?

TrHch d th t12en 'nh f= hc k1!c 4a 8 ;$ n/# W__^


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( ) ( ) ( )1 cos2x 1 cos4x 1 cos6x

cos2x cos4x 1 cos6x #2 2 2

- + +? ' = + ' + + + =

( ) 22cos3xcosx 2cos 3x # 2cos3x cosx cos3x #' + = ' + =

( )

$x

$6 3cosx # xJ 6 34cos3xcos2xcosx # cos2x # x $/J/mJ4 2 xcos3x #

4 2x m2

% %= + % %= = +% % ' = ' = ' = + ' % % = += % = + %

.

L binhI Ty v-c U6Qt h-n cac c6ng( ) ( ) ( )x / 2x / x v nh-n Ut x x 4x2

+= O t cK th SVnh

h0ng nhK' ( )sinx sinx- O ( )22sin 2x 1- l\ v0 nh6O S s6 Hh d@ng c$ng thcteng thnh tch v h\ 3-c nhY' U6Qt h-n nhn t ch6ng v c6 c@ng S t Sc

hng trinh tch s Sn g2n hn.

, g2 th' Hh2o

( ) ( ) ( )2sinx sinx 1 2sin 2x # 2cos4xsin3x cos4x #? ' - - - = ' - =

( ) ( )

$2cos4x # x1! 3

cos4x 2sin3x 1 # $/J1 J2sin3x x2 1! 3

% % = = +

' - = ' ' % %= = + .

, g2 th' Hh2o

( ) ( ) ( )sinx sin3x sin2x 1 cos2x cosx? ' + + = + +

( ) ( ) 22sin2xcosx sin2x 2cos x cosx sin2x 2cosx 1 cosx 2cosx 1 #' + = + ' + - + =

( ) ( ) ( ) ( ) 2cosx 1 sin2x cosx # 2cosx 1 2sinxcosx cosx #' + - = ' + - =

( ) ( ) ( )

x $2

cosx #x J21 6cosx 2sinx 1 2cosx 1 # sinx $/J/m/n

2 x m21 6cosx22

x n23

% = + % = % = + % ' - + = ' = ' % = + % = - % = + %

.

PgB ;4 ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B XX. p2 hng trinhI ( )sinx sin2x sin3x 1 cosx cos2x+ + = + + ?

B XY. p2 hng trinhI ( )3 3 3sin xcos3x cos xsin3x sin 4x+ = ?TrHch d th T12en 'nh f= hc ;go= Thng n/# W___

B XW. p2 hng trinhI ( )22sin 2x sinx 1 sinx+ - = ?TrHch d th t12en 'nh f= hc n/# 5h! A n/# X``]


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, g2 th' Hh2o

( ) ( ) ( )3 3 3 3 3sin x 4cos x 3cosx cos x 3sinx 4sin x sin 4x? ' - + - =

3 3 3 3 3 3 34sin xcos x 3sin x3cosx 3cos xsinx 4cos xsin x sin 4x' - + - =

( ) 2 2 33sinx cosx cos x sin x sin 4x' - =

3 33 3

sin2x cos2x sin 4x sin4x sin 4x2 4

' = ' =

( ) 3 $

3sin4x 4sin 4x # sin12x # 12x $ x / $12

%' - = ' = ' = % ' = .

,g2 th' Hh2o( ) ( ) ( )3cos1#x 1 cos!x cosx 2cosx 4cos 3x 3cos3x? ' + + = + -

( ) cos1#x cos!x 1 cosx 2cosx cosx' + + = + 2cosxcosx 1 cosx 2cosx cosx' + = +

( ) cosx 1 x $2 / $' = ' = % .

, g2 th' Hh2o

( ) ( ) ( )2 2 2sinx 4sin x 3 cosx sin x 3cos x #? ' - - - =

( ) ( ) 2 2 2sinx 4sin x 3 cosx sin x 3 1 sin x # >' - - - - = A B( )( ) 24sin x 3 sin x cosx #' - - =

( ) ( ) 2 1 cos2x 3 sinx cosx # >' - - - = A B

( )21 2 2

x $cos2x cos 2x $2 3 $J2 3 3sinx cosx tanx 1 x J

4

% % % = + % = - = = + % ' ' ' %= = = + %

.

, g2 th' Hh2o

( ) ( ) ( ) ( )22sin x 1 3cos4x 2sin x 4 4 1 sin x 3 #? ' + + - + - - =

( ) ( ) ( ) ( ) 2sin x 1 3cos4x 2sin x 4 1 2sin x 1 2sin x #' + + - + - + =

( ) ( ) 2sinx 1 3cos4x 2sinx 4 1 2sinx #' + + - + - =

? #%n c@ 3@ th$ng 'nhAAAA? PgB ;8

B XZ. p2 hng trinhI ( )2 3cos1#x 2cos 4x 6cos3xcosx cosx !cosx cos 3x+ + = + ?

B X[. p2 hng trinhI ( )3 3 24sin x 3cos x 3sinx sin xcosx #+ - - = ?

B X\. p2 hng trinhI ( ) ( ) ( )22sinx 1 3cos4x 2sinx 4 4cos x 3+ + - + = ?


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( ) ( ) 3 cos4x 1 2sinx 1 #' - + =

( )

$x4x $2 2cos4x 1

x J2 x J2 $Jm1 6 6sin2x2

x m2 x m26 6

% = = % = % %' ' = - + % ' = - + % = - % % = + % = + %

.

, g2 th' Hh2o

( ) ( ) ( )6 ! 6 ! 6 2 6 2sin x 2sin x cos x 2cos x # sin x 1 2sin x cos x 2cos x 1 #? ' - + - = ' - - - =

( ) 6 6 6 6sin xcos2x cos xcos2x # cos2x sin x cos x #' - = ' - =

( ) 6 6

$xcos2x # cos2x # $4 2 x / $

tanx 1sin x cos x 4 2x $4

% % = += = % % ' ' ' ' = + = %= = + %

.

, g2 th' Hh2o

( ) ( ) ( )1# ! ! 1#

2cos x cos x sin x 2sin x cos2x #4

? ' - - - + =

( ) ( ) ! 2 ! 2

cos x 2cos x 1 sin x 1 2cos x cos2x #4

' - - - + =

! ! ! !

cos x.cos2x sin xcos2x cos2x # cos2x cos x sin x #4 4

' - + = ' - + =

( )( ) ! ! ! !

cos2x # 2x $ $2 x / $ 4 2cos x sin x # sin x cos x 1 MC4 4

% = = + % % %' ' ' = + - + = = + N

.

, g2 th' Hh2o Cch g W

( ) ( ) ( )3 3sin x 2sin x 2cos x cos x #? ' - - + =

( ) ( ) 3 2 3 2 3 3sin x 1 2sin x cos x 2cos x 1 # sin xcos2x cos xcos2x #' - - - = ' - =

PgB *> ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B X]. p2 hng trinhI ( ) ( )6 6 ! !sin x cos x 2 sin x cos x+ = + ?TrHch d th t12en 'nh f= hc k1!c k1a 8 ;$ h! B n/# W___

B X^. p2 hng trinhI ( ) ( )! ! 1# 1#

sin x cos x 2 sin x cos x cos2x4

+ = + + ?

TrHch d th t12en 'nh f= hc ;go= Thng T*8CM 5h! ( X```

B X_. p2 hng trinhI ( ) ( )3 3 sin x cos x 2 sin x cos x+ = + ?TrHch d th t12en 'nh f= hc k1!c ga 8 ;$ 5h! ( W__^


21/135


( ) ( ) 3 3 3cos2x # m

cos2x sin x cos x # x / mtan x 1 4 2

= % %' - = ' ' = + = .

Cch g X

( ) ( ) ( )3 3 2 2 sin x cos x sin x cos x 2sin x 2cos x #? ' + + - - =

( ) ( ) 3 2 3 2sin xcos x sin x cos x cos sin x #' - - - =

( ) ( ) ( ) 3 2 2 3 2 2 3 3sin x cos x sin x cos x cos x sin x # cos2x sin x cos x #' - - - = ' - =

( ) 3 3 3cos2x # cos2x # m

x / msin x cos x # tan x 1 4 2

= = % % ' ' ' = + - = = .

, g2 th' Hh2o

( )2 2

21 cos2x 1 cos2x3 sin 2x #2 2

+ - ? ' - + =

( ) ( ) ( ) 2 2 23 1 2cos2x cos 2x 4 1 cos 2x 1 2cos2x cos 2x #' + + - - + - + =

( ) 2!cos 2x 4cos2x # 4cos2x 2cos2x 1 #' + = ' + =

( )$

cos2x # x $4 2 4 $/m1cos2x x m2 3

% % % = = + O + % ' ' %= - = + %

.

Cch 5hc

]o cosx # a sinx 1= = Hh$ng l ngh-' c hng trinh ( )?

#h h vR c ( )? cho 4cos x O t ScI

( )2

2 42

t 4t 3 #3 4tan x tan x #

t tan x #

78 - + =88? ' - + = ' 98 = K88:

( )2

22

t 1 x $tanx 1tan x 1 4t 3 $/mtan x 3 tanx 3 x mt tan x 3

7 8 %=8 = + %= 8 = 8 =' ' ' ' 9 8 %= = 8 = + % 8 = 8:

.

, g2 th' Hh2o L binhI T nh-n thQ7 trong hng trinh cK ch cos3xl1n sin3x O nR6 t s dung c$ng thc

nhn 3 S Hh tr`n c{ng S SRn HRt k62 c6 c@ngO nhng nK tng S hc t\.

? #%n c@ 3@ th$ng 'nhAAAA? PgB *;

B Y`. p2 hng trinhI ( )4 2 2 43cos x 4cos xsin x sin x #- + = ?TrHch d th t12en 'nh f= hc k1!c k1a T*8CM W__^ W___ t W

B YW. p2 hng trinhI ( )3 3 2 3 2

cos3xcos x sin3xsin x!

-- = ?


22/135


#hnh vi thRO S7 t Hho lo hn tch S a dung c$ng thc tch thnh teng cK U6Qt

h-n s12

nhY' t g2n Sc v0 s 2 3 2!

- hc t\ 3n vR h2 c hng trinh.

( ) ( ) ( )2 2 2 3 2

cos3xcosx cos x sin3xsinx sin x!

-? ' - =

( ) ( ) 2 21 1 2 3 2cos4x cos2x cos x cos2x cos4x sin x2 2 !-' + - - =

2 2 2 2 2 3 2

cos4xcos x cos2xcos x cos2xsin x cos4xsin x4

-' + - + =

( ) ( ) 2 2 2 2 2 3 2

cos4x cos x sin x cos2x cos x sin x4

-' + + - =

( ) 2 2 3 2 1 2 3 2

cos4x cos 2x cos4x 1 cos4x4 2 4

- -' + = ' + + =

( ) ( ) 2 $4cos2x 2 1 cos4x 2 3 2 cos4x x / $2 16 2% %' + + = - ' = - ' = + .

, g2 th' Hh2o

L binhI Trong 3 toan U6Qt h-n 3n c6ng x/2x/4x/!xHhac nh6O g[ t ln tng SRn v-cS ch[ng vJ c@ng '$-t c6ng. " l' v-c n7 t s s67 ngh_ SRn v-c d@ng c$ng thc

2 2cos2x 2cos x 1 1 2sin x= - = - O nhng nK thi Hh$ng Hh2 k6n cho 'Q7O 3 thRhng trinh s tr thnh hng trinh 3-c coO v-c g2 s g7 HhK Hh!n. Nhng S zrYngO cac c6ng n7 l%n lt gQ S$ nh6O t cht nh0 SRn c$ng thc nhn S$ c sin O

3Yng cach nhn th' h vR c ( )? cho sinx . " S2' trong h nhnO t nn H'tr UB' sin x #= cK h2 l ngh-' h7 Hh$ng tr0c Hh nhn.

C Nh-n thQ7I ( )sin x # x $ a cosx 1 cos2x cos4x cos!x 1= ' = % = ' = = = nn

( ) 11

16? ' = (v$ ngh-') nn sinx # x $= ' = % Hh$ng l ngh-' c ( )?

C Nhn c2 * vR c hng trinh ( )?

cho 16sinx #& O t ScI

( ) 16sinxcosx cos2xcos4xcos!x sinx !sin2xcos2x cos4xcos!x sinx

sinx # sinx #

7 78 8= =8 8? ' '9 98 8& &8 8: :

4sin4xcos4xcos!x sinx 2sin!xcos!x sinx sin16x sinx

sinx # sinx # sinx #

7 7 78 8 8= = =8 8 8' ' '9 9 98 8 8& & &8 8 8: : :

$2x $2

x11J

x Jx1 1 1 1x m

78 %8 =8 %8 =8 88 % %' '9 = +8 % % 8 = +8 8 8 & %88:

v0 ( )1% 1

$ 1n J $/J/m/n/%

2

-& & .

PgB ** ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B YX. p2 hng trinhI ( )1

cosxcos2xcos4xcos!x16

= ?

TrHch d th t1 en 'nh f h c nh tF k1!c (n n/# W__^


23/135


, g2 th' Hh2o

( ) ( )34sin3xcos2x 1 2 3sinx 4sin x 4sin3xcos2x 1 2sin3x? ' = + - ' = +

( ) ( ) ( ) 22sin3x 2cos2x 1 1 2sin3x 4cos x 3 1' - = ' - = o

]o ( )cosx # x $ / $2%

= ' = + % Hh$ng l ngh-' hng trinh ( )o O nn nhn h vR ( )o cho

cosx #& O t ScI ( ) ( )32sin3x 4cos x 3cosx cosx 2sin3xcos3x cosx' - = ' =o

( )J2

x14 sin6x cosx cosx cos 6x J/$

m22 x1#

% %= + % ' = ' = - ' % % = +

.

, g2 th' Hh2o

C Mh ( )x $2 / $= % thi ( ) ( )1

2

? ' = - ; ? Hh$ng cK ngh-' ( )x $2 / $= % .

C Mh ( ) x

x $2 / $ sin #2

& % ; & . Nhn h vR c ( )? cho x2sin #2

& O t ScI

( ) x x x x x x2sin cosx 2sin cos2x 2sin cos3x 2sin cos4x 2sin cosx sin2 2 2 2 2 2

? ' + + + + = -

3x x x 3x x x x x 11x x x

sin sin sin sin sin sin sin sin sin sin sin2 2 2 2 2 2 2 2 2 2 2

' - + - + - + - + - =- .

( )11x 11x 2m

sin # m x / m 11/ m2 2 11

%' = ' = % ' = & .

, g2 th' Hh2o

L binhI Mh g2 hng trinh lng gac cK ch tn ho!-c cotO cK |n '16 h7 c!n 3-c chZnOA t h2 S!-c SJ6 H-n S hng trinh Uac SVnh. "!-c 3-t S v0 nhng 3 toan cK

ch tn (ho!-c cot)O t h7 th7 thR ch[ng 3Yngsin cos

/cos sin

nhY' 'uc Sch ? Sn

g2n hK ? v ch c^n l\ h ga trV lng gac l sn v cos ' th$.T s d@ng cac cach s6 S7 S H' tr UB' cK nh-n ngh-' h7 Hh$ng

Th7 cac ga trV U ti' Sc vo SJ6 H-n UB' cK th Hh$ng. NR6 th thi ghnh-n ngh-' Q7O nR6 Hh$ng th thi lo\.

? #%n c@ 3@ th$ng 'nhAAAA? PgB *9

B YY. p2 hng trinhI ( )34sin3xcos2x 1 6sinx !sin x= + - ?

B YZ. p2 hng trinhI ( )1

cosx cos2x cos3x cos4x cosx2

+ + + + = - ?

B Y[. p2 hng trinhI ( )sin2x 2cosx sinx 1

#tanx 3

+ - -= ?

+TrHch d th t12en 'nh f= hc 5h! ( n/# X`WW


24/135


/o!-c 36 d`n cac ngn c6ng SJ6 H-n v ngn c6ng ngh-' trn c@ng '$-tSWng tr^n lng gac. T s lo\ 3 ngn c6ng c ngh-' Hh cK tr@ng v0 ngnc6ng c SJ6 H-n.

/o!-c so v0 SJ6 H-n trong k6a trinh g2 hng trinh.

C "J6 H-nItanx 3

cosx #

78 & -889

8 &88:( ) ( ) ( )sin2x 2cosx sin x 1 # 2cosx sin x 1 sin x 1 #? ' + - - = ' + - + =

( ) ( ) ( )1 x $2cosx 3sinx 1 2cosx 1 # $/J2

sinx 1 x J22

% = + % = ' + - = ' ' %= - = - + %

.

C o v0 SJ6 H-nO h ngh-' c hng trinh l ( )x $2 / $3%

= + % .

, g2 th' Hh2o

C "J6 H-nI sin x #&

( ) 2 2sin x(1 sin2x cos2x) 2 2sin xcosx 1 sin2x cos2x 2 2cosx? ' + + = ' + + =

( ) 22cos x 2cosxsinx 2 2cosx # 2cosx cosx sinx 2 #' + - = ' + - =

( )cosx # x $cosx #

2 $/Jcos x 1cosx sin x 2 x J24

4

%= = + %= ' ' ' % %- =+ = = + %

.

C o v0 SJ6 H-nO h ngh-' hng trinh l ( )x $ x J2 / $/J2 4% %

= + % = + % .

, g2 th' Hh2o

C "J6 H-nI ( )sin x #

2sinxcosx # sin2x # 2x $ x $ / $cosx # 2

78 & %8 ' & ' & ' & % ' & 98 &8: .

( ) ( ) ( )2 2sinx cosx sin x cos x

2 sin2x cos2x 2 sin2x cos2xcosx sinx sinxcosx

+? ' + = + ' = +

( ) ( )1 1

2 sin2x cos2x sin2x cos2xsinxcosx sin2x

' = + ' = +

( ) 2sin2x sin2x cos2x 1 sin 2x sin2xcos2x 1 #' + = ' + - =( ) 2sin2xcos2x cos 2x # cos2x sin2x cos2x #' - = ' - =

PgB *< ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B Y\. p2 hng trinhI ( )21 sin2x cos2x 2sinxsin2x

1 cot x+ + = ?

+TrHch d th t12en 'nh f= hc 5h! A n/# X`WW

B Y]. p2 hng trinhI ( ) ( )tanx cot x 2 sin2x cos2x+ = + ?TrHch d th t12en 'nh f= hc 4ao Th$ng Dn T T*8CM n/# W__^


25/135


26/135


, g2 th' Hh2o

C "J6 H-nI

cosx #

sinx # sin2x #

sin2x #

78 &888 & ' &988 &88:

.

( ) 2 2 2 2 2 2 21 1 1 11 1 1 1 2#

1 1 1 3 3cos x sin x sin 2x cos x sin x 4sin xcos x

? ' - + - + - = ' + + =

( )2 2

2

2 2 2

4sin x 4cos x 1 2# 2# 3 1 3sin 2x 1 cos4x

3 3 4 2 44sin xcos x sin 2x

+ +' = ' = ' = ' - =

( )1 2 $

cos4x cos x / $2 3 6 2

% % % ' = - = ' = + .

C Th7 vo ngh-' vo SJ6 H-nO th. -7 h ngh-' c hng trinh l

( )$

x / $6 2% %

= + .

, g2 th' Hh2o

"J6 H-nI cosx # sinx 1& & .

( ) ( ) ( )( )

( )22

2 2

1 sinx 1 cos x1 sin x 11 cos x 1 cosx # 1 cosx #

2 2 2cos x 1 sin x

> - - % A? ' - - - + = ' - + = A - A B

( ) ( )21 cos x 1 cosx

1 cosx # 1 cosx 1 #1 sinx 1 sinx

- - ' - + = ' + - = + +

( ) ( ) ( )

( ) ( )

x $2cosx 1 C1 cosx cosx sinx # $J

tanx 1 C x J24

= % + % = - ' + - - = ' ' % = - = - + %

.

, g2 th' Hh2o

"J6 H-nI 2

cosx 1sin x #sin x #2cos2x # 2cos x 1 # cosx

2

78 & 77 888 && 888 8 8' '9 9 98 8 8& - & & 8 8 8: 8: 88:

T cKIcosx sin2x cos2xcosx sin2xsinx cosx

cot x tan2xsin x cos2x sinx cos2x sin xcos2x

++ = + = = .

L[c SKI ( )

22 2 2sin2xcosx 2cos x 2

4cos x 4cos x # cos x 4 #sinxcos2x cos2x cos2x

? ' = ' - = ' - =

PgB *: ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B Z`. p2 hng trinhI ( )2 2 2x x

sin tan x cos #2 4 2

% - - = ? TrHch d th t12en 'nh f= hc 5h! ( n/# X``Y

B ZW. p2 hng trinhI ( ) ( )2sin2x cotx tan2x 4cos x+ = ?TrHch d th T12en nh f= hc Mp f0a ch%t n/# X```


27/135


( )

( ) ( )

cosx # cosx # C x $2 $J1 1

2 cos2x C x Jcos2x 2 6

% = = = + % ' ' ' %= = = + %

.

, g2 th' Hh2o

"J6 H-nIsin2x #

sin4x #cos2x #

78 &8 ' &98 &8:.

T cKI2 2 4 4

2 2

2 2 2 2 2

cos x sin x cos x sin x 4cos2xcot x tan x

sin x cos x sin xcos x sin 2x

-- = - = = .

( ) ( ) ( ) ( ) ( )224 16 1 cos4x 1 4 1 cos4x sin 2x 1 2 1 cos4x 1 cos4xsin 2x? ' = + ' = + ' = + -

( ) 2 2 2 1

1 2 1 cos 4x 1 2sin 4x sin 4x2

' = - ' = ' = (Nh-n do sin4x #& )

( ) ( )1 1 $

1 cos!x cos!x # x / $2 2 16 !

% %' - = ' = ' = + .

, g2 th' Hh2o

"J6 H-nIsin2x # sin2x #

cosx # cos2x 1

7 78 8& &8 8'9 98 8& & 8 8: :

( ) 2 22sinx cos2x 1

2sin2x 4sin x cos2x 2sin 2x 1cosx sin2x sin2x

? ' + = + ' + = +

( ) ( ) 2 2 2 2 2 24sin x 1 2sin x !cos xsin x 1 2sin x 1 4cos x #' + - = + ' - =

( ) ( )( )

( )2 sinx # D2sin x 1 2 1 cos2x # x $ / $1 3cos2x C2

= % > ' - + = ' ' = + % A B = -

.

, g2 th' Hh2o

"MI ( ) sin x #sinx 1 cosxsinx

tanx sinx # sinx # # cosx # sin2x #cosx cosx

cosx 1

78 &8- 88- & ' - & ' & ' & ' &988 &88:

? #%n c@ 3@ th$ng 'nhAAAA? PgB *=

B ZX. p2 hng trinhI ( ) ( )2 2cot x tan x 16 1 cos4xcos2x

- = + ?

B ZY. p2 hng trinhI ( )1

2tanx cot2x 2sin2xsin2x

+ = + ?

TrHch d th t12en 'nh f= hc k1!c 4a T*8CM n/# W__^ W___

B ZZ. p2 hng trinhI ( ) ( ) ( )3 sinx tanx

2 1 cosx #tanx sinx

+- + = ?

-TrHch d th t12en 'nh f= hc T ChHnh F Ton n/# X```


28/135


( ) ( )

( ) ( )

( ) ( )

3 sinx tanx 3 cosx 1cot x. 2 1 cosx # 2 1 cosx #

tanx sinx cot x 1 cosx

+ +? ' - + = ' - + =

- -

( )cosx 11 cosx #3

1 cosx 2 # 11 2cosx #1 cosx cosx2

= - + = ' + - = ' ' + =- = -

( )2

x $2 / $3%

' = + % .

, g2 th' Hh2o

"J6 H-nI sinx 1 cosx #& Q & .

( ) ( )

( ) ( )

2 3 2

2 2

2 1 cos x sin x 1 sin x1 sinx

21 sin x 1 sin x4 1 sinx

+? ' - = + +

- --

( )( ) ( )( ) 2 3 2 21 cos x 1 sin x 2sin x 1 sin x 1 sin x 2sin x' + + - = + - +

( )( ) ( ) ( ) 2 2 21 cos x 1 sinx 1 sinx cos x 2sin x 1 sinx' + + = + + +

( )2 2 2

1 sinx # sinx 1 Dcos2x #

1 cos x cos x 2sin x 1 1 2cos2x

+ = = -' ' ' = + = + = -

$

2x $ x2 4 2% % %

' = + % ' = + (nh-n do cosx #& )

, g2 th' Hh2o

C "J6 H-nI cosx #& .

( ) sinx

cos3x sinx sinxcos3x sinxcosxcosx? ' = ' =

( ) ( )1 1

sin!x sin2x sin12x sin2x sin!x sin12x2 2

' + = + ' =

( )$

x12x !x $2 2 $J12x !x J2 J

x2# 1#

%= = + % ' ' = % - + % % % = +

.

C o v0 SJ6 H-nI

0$

x 2%

= thi$ $

cosx cos cos2 2

% %= = lo\ nR6 H l}.

PgB *4 ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B Z[. p2 hng trinhI

( ) ( )

( ) ( ) ( )

2 2

2 21 cosx 1 cosx 1

tan x sin x 1 sin x tan x24 1 sinx

- + +- = + + ?

-

B Z\. p2 hng trinhI ( )cos3xtanx sinx= ?TrHch d th T12en 'nh Cao )ng nh tF C$ng ;gh3* T* 8CM n/# X``]

(Lo\ do sin x #& )

(Nh-n)


29/135


30/135


1 cos2x 1 cos4x 1 cos2x 1 cos4x

cot3x . 11 cos2x 1 cos4x 1 cos2x 1 cos4x

- + - + ' - = - + - + -

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

cot3x 1 cos2x 1 cos4x 1 cos2x 1 cos4x

1 cos2x 1 cos4x 1 cos2x 1 cos4x

>' - + - + - A B= - - - + +

( ) ( ) cot3x 2cos4x 2cos2x 2 cos4x cos2x' - = - +

( )cos3x

4sin3xsinx 4cos3x cosxsin3x

' - = -

( )$

xcos3x # 6 3cos3xsinx cos3xcosx $/Jtanx 1

x J4

% %= + = ' = ' ' = % = + %

.

#ach HhacI ( ) ( )2 2 2 2cot 3x tan xcot 2x 1 tan x cot 2x? ' - = -

( ) ( )( ) ( )

2 2 2 2

2 2 2 2

1 tan2x tanx 1 tan2x tanxtan x cot 2x tan 2xtan x 1cot3x tan xcot 2x 1 tan x tan 2x tan2x tanx tan2x tanx

+ -- -' = = =- - - +

cos3x #

cot3x cot xcot 3xsinx cosx

=' = ' = (p2 tng tX nh trn)

, g2 th' Hh2o

"J6 H-nI ( )

sin x #sin2x #

x $cos # x / $x2 2cos #

2cosx #

78 &8 78 8 &8 8 %8 88 8& ' ' & 9 98 8 &8 88 88:&888:

.

( )

x x xsin cosxcos sinxsincosx sinx cosx2 2 2sinx 1 . 4 sinx 4

sinx cosx x sinx xcos cosxcos2 2

+ ? ' + + = ' + =

xcos x

2cosx cosx sinxsinx. 4 4 1 4sinxcosx

sinx x sinx cosxcosxcos

2

- ' + = ' + = ' =

( )2x $2 x $1 6 12sin2x $/J

2 2x J2 x J6 12

% % = + % = + % ' = ' ' % % = + % = + %

.

o v0 SJ6 H-nO h ngh-' c hng trinh l ( )

x $ x J $/J12 12% %

= + % = + % .

PgB 9> ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B [`. p2 hng trinhI ( )x

cotx sin x 1 tan x tan 42

+ + = ? TrHch d th t12en 'nh f= hc 5h! B n/# X``\


31/135


32/135


33/135


34/135


w sinx cosx 2sin x 2cos x4 4

% % = = m

w( ) ( )

2 2cosx cos x 1 sin x 1 sinx1 sinx cosxcosx 1 sinx cosx 1 sinx

- += = =

- - -(' ln h-g snU v cosU)

B9 TVP >P (U;4B [W. p2 hng trinhI ( )2cos4x 12sin x 1 #+ - = ?

B [X. p2 hng trinhI ( )4 4cos x sin x cos4x #- + = ?

B [Y. p2 hng trinhI ( ) ( )cos3x sin3x

sinx 3 cos2x / x #21 2sin2x

+ + = + ? @ % +

B [Z. p2 hng trinhI ( )sin3x sinx

3 = ?

B [[. p2 hng trinhI ( )sinx

1sinx

= ?

B [\. p2 hng trinhI ( )2 2cos 3xcos2x cos x # 1- =

B []. p2 hng trinhI ( )4 4 3

cos x sin x cos x sin 3x #4 4 2

% % + + - - - = ?

B [^. p2 hng trinhI ( )

sin 2x 3cos x 1 2sinx x 22 2 2

% % % + - - = + @ % ?

B [_. p2 hng trinhI ( ) ( )2

sinx 2 3 1 sinx tan x- = - ?

B \`. p2 hng trinhI ( ) ( )2

sin2x 3cos2x cos 2x6

% + - = - ?

B \W. p2 hng trinhI( )

( )6 62 cos x sin x sinxcosx

#2 2sinx

+ -= ?

-

B \X. p2 hng trinhI( )

( )1 sinx cos2x sin x

4 1cosx

1 tanx 2

% + + + = ?

+

B \Y. p2 hng trinhI ( )1 1

2sin3x 2cos3xsinx cosx

- = + ?

B \Z. p2 hng trinhI ( ) ( )2cosx 2sinx 3 2 2cos x 1

11 sin2x

+ - -= ?

+

B \[. p2 hng trinhI ( )x 3x x 3x 1

cosxcos cos sinxsin sin2 2 2 2 2

- = ?

B \\. p2 hng trinhI ( )4 4sin x cos x 1 1

cot2x

sin2x 2 !sin2x

+= - ?

B \]. p2 hng trinhI ( ) ( )2 23cot x 2 2sin x 2 3 2 cosx+ = + ?

B \^. p2 hng trinhI ( )6 23cos4x !cos x 2cos x 3 #- + + = ?

PgB 9< ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?


35/135


36/135


B _W. p2 hng trinhI 22

1 1sinx sin x

sinx sin x- = - ( ) ?

B _X. p2 hng trinhI2

2

1 1cos x 2 cosx 1

cosxcos x

+ - + = ( ) ?

B _Y. p2 hng trinhI1

2tanx cot x 2sin2xsin2x

+ = + ( ) ?

B _Z. p2 hng trinhI ( )2 2tan x cot x 2 1 tan x cot x #+ + + + = ( ) ?

B _[. p2 hng trinhI ( )4 4 11

sin x cos x sin2x!

+ = - ?

B _\. p2 hng trinhI 24sin xcosx 4cos xsinx sin 4x- = ( ) ?

B _]. p2 hng trinhI ( ) ( )cos 2x cos 2x 4sinx 2 2 1 sinx4 4

% % + + - + = + - ?

B _^. p2 hng trinhI2 2 2 2 3

cos x cos 2x cos 3x cos 4x 2+ + + = ( ) ?

B __. p2 hng trinhI6 6

2 2

sin x cos x 1tan2x

4cos x sin x

+=

- ( ) ?

B W``. p2 hng trinhI 6 6sin x cos x sin2x+ = ( ) ?

8;4 (; 49S9 P8;4 TQR;8 BVC 8A9 D BVC CA fu9D9 M?T 8M L;4 49>C

L binhI Trong 3 toan toan cK ch h c6ng x v 4xnn t S vJ c@ng '$-t c6ng l 2x3Yngc$ng thc nhn S$ c 2cos4x 2cos 2x 1= - v c$ng thc h\ 3-c

2 1 cos2xsin x2

-= . Ty SKO S t vJ hng trinh 3-c h thBo cos2x .

, g2 th' Hh2o

( ) ( )2 22cos 2x 1 6 1 cos2x 1 # cos 2x 3cos2x 2 #? ' - + - - = ' - + =

( )2 t 1 t 2t 3t 2 #

cos2x 1 x $ / $t cos2x/ t 1t cos2x/ t 1

7 8 = =- + =88 ' ' ' = ' = % 9 8 = "= " 8 8: .

PgB 9: ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B [W. p2 hng trinhI ( )2cos4x 12sin x 1 #+ - = ?xTrHch d th t12en 'nh Cao )ng 5h! AO BO ( n/# X`WWy

B [X. p2 hng trinhI ( )4 4cos x sin x cos4x #- + = ?

xTrHch d th t12en 'nh Cao )ng 2


37/135


L binhI Trong v du n7O c{ng tfn t\ h c6ng Hhac nh6 x v 4x nn t S vJ c@ng '$-t c6ngl 2x O nhng l%n n7 c%n h2 HRt h g hYng SZng thc v c$ng thc nhn S$I

( ) ( ) ( ) ( )2 2

2 2 2 2 2 2cos x sin x cos x sin x sin x cos x cos2x- = - + = . #^n cos4x t s

a dung c$ng thc nhn S$ nh trn S Sc hng trinh 3-c h thBo cos2x .

, g2 th' Hh2o( ) ( ) ( )2 2 2 2 2cos x sin x cos x sin x 2cos 2x 1 #? ' - + + - =

( ) 2cos2x 1 x $

22cos 2x cos2x 1 # $/J1cos2x x J2 6

% = - = + % ' + - = ' ' %= = + %

.

LW 3inhI Trong 3 toan n7O cK ch Sfng thW 3 c6ng x/2x/3xv t Hh$ng th S c6ng x csinx vJ c6ng2x Sc (Hh$ng cK c$ng thc lng gac no)O do SK ch c^n cach d67nhQt l S 3 c6ng n7 vJ c@ng c6ng x . Nh-n thQ7 rYngO trong vR tra hng trinh cKch cos3x sin3x+ O t nn hn tch h thnh h%n n7 tr0c S tranh l- l\ v dd^ng Hh g2 hng trinh. #^n cos2x tQt nhn S vJ c6ng x3Yng c$ng thc nhnS$I 2 2 2 2cos2x cos x sin x 2cos x 1 1 2sin x= - = - = - O nhng trong 3 c$ng thcSKO t s a dung c$ng thc no #6 tr2 lW l ?dX vo sX 3Rn Se c vR tra S chnc$ng thc h@ h?.

, g2 th' Hh2o

C "J6 H-nI1

1 2sin2x # sin2x2

+ & ' & - .

T cKI ( ) ( )3 3cos3x sin3x 4cos x 3cosx 3sinx 4sin x+ = - + -

( ) ( ) 3 33 cosx sinx 4 cos x sin x= - - + -

( ) ( ) 3 33 cosx sinx 4 cos x sin x= - - + -

( ) ( )2 2

cosx sinx 3 4 cos x cosxsinx sin x

= - - + + +( ) ( ) cosx sinx 1 2sin2x= - + .

( ) ( ) ( )cosx sinx 1 2sin2x

sinx 3 cos2x1 2sin2x

- + ? ' + = + +

( ) 2 sinx cosx sinx 2cos x 1' + - = -

( )

( ) 21

cosx2cos x cosx 1 # x $2 / $2

3cosx 2 D

= %' - - = ' ' = + %

=

.

C MRt h v0 SJ6 H-nO t Sc h ngh-' l ( )x $2 / $3%

= + % do 3 1sin2x2 2

= & - .

? #%n c@ 3@ th$ng 'nhAAAA? PgB 9=

B [Y. p2 hng trinhI ( ) ( )cos3x sin3x

sinx 3 cos2x / x #21 2sin2x

+ + = + ? @ % + xTrHch d th t12en 'nh f= hc 5h! A n/# X``Xy


38/135


39/135


( ) ( ) 4sinx cos3xcosx 1 # 2sinx cos4x cos2x 1 #' - = ' + - =

( ) ( )2

2

sinx # Dsinx 2cos 2x cos2x 3 #

2cos 2x cos2x 3 #

=' + - = ' + - =

( ) ( ) 2cos2x 1

1 cos2x # 2sin x # sinx # Dcos2x 1/ D

=

' ' - = ' = ' = = --7 hng trinh v$ ngh-'.

C Cch g X

( ) ( )sin 2x 3x sinx sin2xcos3x sin3xcos2x sinx? ' + = ' + =

( ) ( ) ( ) ( )2

3 3 2 2 2 22sinxcosx 4cos x 3cosx 3sinx 4sin x cos x sin x sinx sin x cos x' - + - - = +

( ) 2 3 3 2 212sin x 2#cos xsin x # sin x 12sin x 2#cos x #' + = ' + =

( ) ( )( )

33 22

sin x # Dsin x 12 !cos x ## 12 !cos x # D =' + = ' R + =

-7 hng trinh v$ ngh-'.

, g2 th' Hh2o

( ) ( )1 cos6x 1 cos2x1 .cos2x # cos6xcos2x 1 # 22 2+ +' - = ' - =Cch g W

( ) ( )3 4 22 4cos 2x 3cos2x cos2x 1 # 4cos 2x 3cos 2x 1 #' - - = ' - - =

( ) ( )

2

2

cos 2x 1$

sin2x # 2x $ x / $1 2cos 2x D4

= %' ' = ' = % ' = = -

.

Cch g X

( ) ( ) 212 cos!x cos4x 1 # cos!x cos4x 2 # 2cos 4x cos4x 3 #2

' + - = ' + - = ' + - =

( ) ( )

cos4x 1$

4x $2 x / $3 2cos4x D2

= %' ' = % ' = = -

.

Cch g Y. Phng trinh lng gac Hh$ng '16 'Xc

( ) ( )cos6x cos2x 1 $

2 x / $cos6x cos2x 1 2

= = %' ' = = = - .

Cch g Z. Phng trinh lng gac Hh$ng '16 'Xc

( ) ( )$

2 cos!x cos4x 2 # cos!x cos4x 1 x / $2%

' + - = ' = = ' = .

? #%n c@ 3@ th$ng 'nhAAAA? PgB 98

B [\. p2 hng trinhI ( )2 2cos 3xcos2x cos x # 1- =xTrHch d th t12en 'nh f= hc 5h! A n/# X``[y


40/135


, g2 th' Hh2o

( ) 21 1 31 sin 2x sin 4x sin2x #2 2 2 2

> % A? ' - + - + - = A A B

( ) 21 1 1

sin 2x cos4x sin2x #2 2 2

' - - - + - =

( ) 2 21 1 1 1

sin 2x 1 2sin 2x sin2x #2 2 2 2

' - - - + - =

( ) ( ) 2

sin2x 1sin 2x sin2x 2 # x $ / $

sin2x 2 D 4

= %' + - = ' ' = + % = - .

L binhI Ty v-c U6Qt h-n cac s

/2 2% % l nhng 3$- s c

2% O l' t ln tng SRn c6

?cos S b sn 3@ b hu cho?O th-t v-7O cac 3\n S z cach g2 s6I

, g2 th' Hh2o

( ) ( ) ( )sin 2x 2 3cos x 4 1 2sinx2 2

> >% % A A? ' - - - % - - - + % = + A A B B

( ) ( ) cos 2x 2 3sin x 4 1 2sinx cos2x 3sinx 1 2sinx >' - + % - - + % = + ' + = + A B

( ) 2

x $sinx #

x J22sin x sinx # $Jm1 6sinx2 x m26

= % = % = + %' - + = ' ' = % = + %

.

C 0

x $ 1$ 2 $ 2

$ 1 x2 2$ /x 2 $ $2

78 7 7= % 8 8%8 8 88 R % R % R R8 88 8 8 ; ' ; = ; = %9 9 9% 8 8 8 % 8 8 8 8 8 8 8 8: :8:

.

C 0

x J2 1 11J2 2 J6

2 6 6 12J JJ /x 2

2

78 %8 7 78 8= + % % %8 8 88 R + % R % R R8 88 8 8; ' ;9 9 9 %8 8 88 8 8 % 8 8 8 8 8: :8 8:

Mh$ng tfn t\ J nn lo\

ngh-' x J26%= + % .

PgB ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B []. p2 hng trinhI ( )4 4 3

cos x sin x cos x sin 3x #4 4 2

% % + + - - - = ? xTrHch d th t12en 'nh f= hc 5h! ( n/# X``[y

B [^. p2 hng trinhI ( )

sin 2x 3cos x 1 2sinx x 22 2 2

% % % + - - = + @ % ?


41/135


42/135


C "J6 H-nI 22 2sin x # sin x2

- & ' & .

( ) ( )6 6 21 1

2 cos x sin x sin xcosx # 2 1 sin 2x sin2x #2 2

? ' + - = ' - - =

( ) ( )2

sin2x 1

2sin 2x 2sin2x 4 # x $ / $sin2x 2 D 4

= %' - - + = ' ' = + % = -

C Th7 vo SHI2

sin $ 14 2

% + % = & nn h ngh-' hng trinh l ( )x $ / $

4%

= + % .

, g2 th' Hh2o

C "J6 H-nI1 tanx # tanx 1

cosx # sinx 1

7 78 8+ & & -8 8'9 98 8& & 8 8: :

( ) ( )2 sinx

2sin x 1 sinx 1 2sin x cosx 1

4 cosx

% ? ' + + + - = +

( )( ) 2sinx cosx 2sin x sinx 2 cosx sinx' + - + + = +

( )( ) 2sin x cosx 2sin x sin x 1 #' + - + + =

( )( )

( )

( )

2

tanx 1 D x $2sinx cosx # 6sinx 1 D $/J2sin x sin x 1 #

x J216sinx C

2

%= - = - + %+ = ' ' = ' %- + + = = + % = -

.

, g2 th' Hh2o

C "J6 H-nIsin x #

sin2x #cosx #

78 &8 ' &98 &8:

( ) ( ) 1 1

2 sin3x cos3x

sinx cosx

? ' - = +

( ) 3 3 sinx cosx

2 3sinx 4sin x 4cos x 3cosxsinxcosx

+' - - + =

PgB


43/135


44/135


45/135


( )

( )

( )

( ) ( )

cosx 2 D cosx 2 D x $24 $/J12 cosx Ccosx C x J222 3

%= - = - = + % ' ' %== = + %

.

, g2 th' Hh2o

( ) ( )2 6 23 2cos 2x 1 !cos x 2cos x 3 #? ' - - + + =

2 6 23cos 2x 4cos x cos x #' - + =

( )2

2 6 23 2cos x 1 4cos x cos x #' - - + =

6 4 24cos x 12cos x 11cos x 3 #' - + - + =

( )

( )

( )

2 22

2 22

1t C 124t 12t 11t 3 # 2cos x 1 #cos xt 1 C 2

t cos x # sin x #cos x 13t D

2

= 7 8 - + - + = - ==8 8 ' ' = ' '9 8 = K = 8 =8: =

( )cos2x # x $

$/J4 2sin x # x J

% % = = + ' ' = = %

.

L binhI Ty v-c U6Qt h-ncot x tanx- v sin2x O t UB' ch[ng cK ' h-nh thR no #KS vJ nhn t ch6ng h7 c@ng '$-t c6ng h7 Hh$ng #6 tr2 lW nY' S%6 SJI ?cachYng SZng thc lng gac v ' ln h-?. Th-t v-7O t cKI

2 2cosx sinx cos x sin x 2cos2xcot x tanx

sinx cosx sinxcosx sin2x-

- = - = = .

, g2 th' Hh2o

C "J6 H-nI

sinx # sinx 1

cosx # cosx 1

sin2x # cos2x 1

7 78 8& & 8 88 88 8& ' & 9 98 88 8& & 8 88 8: :

( ) 22cos2x 2cos4x

2cos 2x cos2x 1 #sin2x sin2x

? ' = ' - - =

( )

( )

cos2x 1 D2

2x $2 x $ / $1 3 3cos2x2

= % %

' ' = + % ' = + % = - .

? #%n c@ 3@ th$ng 'nhAAAA? PgB


46/135


, g2 th' Hh2o

C "J6 H-nI

sinx # sinx 1

cosx # cosx 1

sin2x # cos2x 1

7 78 8& & 8 88 88 8& ' & 9 98 88 8& & 8 88 8: :

( ) cosx sinx 2 2cos2x 2

4sin2x 4sinxsinx cosx sin2x sin2x sin2x

? ' - + = ' + =

2 2cos2x 2sin 2x 1 # 2cos 2x cos2x 1 #' + - = ' - - =

( )( )

cos2x 1 D2

2x $2 x $ / $1 3 3cos2x2

= % %' ' = + % ' = + % = -

.

, g2 th' Hh2o

C "J6 H-nI cosx # x $2

%& ' & + % .

C #ach g2 ;

( )2

2 2 2 2

2 2

sin x 12 .cos x tan x 2 2tan x. tan x 2

cos x 1 tan x? ' + = ' + =

+

2 2 4 2 4 22tan x tan x tan x 2 2tan x tan x tan x 2 #' + + = + ' + - =

( ) ( )

2

2

tan x 1tanx 1 x $2 / $

tan x 2 D 4

= %' ' = ' = + % = - .

C #ach g2 *

( )2

2 2 2 2 2

2

sin x2sin x 2 2sin xcos x sin x 2cos x #

cos x? ' + = ' + - =

( ) 2 2 2 2 4 22 1 cos x cos x 1 cos x 2cos x # 2cos x cos x 1 #' - + - - = ' + - =( )

( )2

2

2

cos x 1 D$

2cos x 1 # cos2x # x $ / $1 4 2 4cos x2

= - % % %' ' - = ' = ' = + = + % =

.

, g2 th' Hh2oT cKI ( )

2! ! 4 4 4 4sin x cos x sin x cos x 2sin xcos x+ = + -

PgB


47/135


( )2

22 2 2 2 41sin x cos x 2sin xcos x sin 2x

!

>= + - - A

A B

2

2 41 11 sin 2x sin 2x2 !

= - -

2 411 sin 2x sin 2x

!= - +

( ) 2 4 4 21 1

1 sin 2x sin 2x sin 2x !sin 2x #! !

? ' - + = ' - + =

( )( )

2

2

sin 2x 1 D $cos2x # x / $

sin 2x 1 4 2

= N % %' ' = ' = + = .

, g2 th' Hh2o

T cKI ( )2

! ! 4 4 4 4sin x cos x sin x cos x 2sin xcos x+ = + -

( )2

22 2 2 2 41sin x cos x 2sin xcos x sin 2x

!

>= + - - A

A B

2

2 41 11 sin 2x sin 2x

2 !

= - -

2 411 sin 2x sin 2x

!

= - +

( ) ( )2 4 2 4 21 1

16 1 sin 2x sin 2x 1 sin 2x 2sin 2x sin 2x 1 #! 16

? ' - + = - ' + - =

( )( ) ( )

2

2

sin 2x 1 D1 1 $

1 cos4x cos4x # x / $1 2 2 ! 4sin 2x2

= - % %' ' - = ' = ' = + =

.

, g2 th' Hh2o

C ]o ( )cosx 1

xcos # x $2 / $ x2 sin 1

2

78 = -888= ' = % + % ' 98 = 888:

Nn( )

( ) 3

MS sin $ sin $ 1 x

2 2 cos #2MT cos xsinx

7 8 % % 8 ? ' + % = + % = 8 8 ; = 988 ? ' = 88:

Hh$ng l ngh-' ( )? .



48/135


C Nhn h vR ( )? cho ( )x

cos #/ cosx 1 sinx #2

& ' & - ' & O t ScI

( ) 2x x x x

sin cos cos xsin cos2 2 2 2

? ' =

( ) 31

sin3x sin2x cos xsinx2 2

' + =

3 33sinx 4sin x 2sinxcosx cos xsinx' - + =

( ) 2 3sinx 3 4sin x 2cosx cos x #' - + - =

3 2sinx #

cos x 4cos x 2cosx 1 #

=' - - - =

( )

sin x #

cosx 1 x $21 21 x J2 $/J/m

cosx cos1# x m21 21

cosx cos1#

= = = % - -' ' = a + %

= = a = b+ % - + = =

v0

1 21cos

1#

1 21cos1#

78 - -88 a =8898 - +88 =88:

, g2 th' Hh2o

C "J6 H-nIcos2x # sin2x 1

sinx # cosx 1

7 78 8& & 8 8'9 98 8& & 8 8: :

C T cKIcosx sin2x cosxcos2x sinx sin2x cosx

cot x tan2xsinx cos2x sinx cos2x sin xcos2x

++ = + = =

( ) 2cosx

2sinxcosx 4cos xsinxcos2x

? ' =

2

2cos x 2cos xcos2x

' =

2 2cos x 2cos xcos2x #' - =

( ) 2cos x 1 2cos2x #' - =

( )

( ) ( )

2 cosx # C x $cos x # 2 $/11 2cos2x # cos2x C x2 6

% = = + %= ' ' ' %- = = = + %

ll

.

, g2 th' Hh2o

( ) 212x 4x1 cos 1 3 2cos 1 ? ' + + = -

PgB


49/135


3 24x 4x 4x

4cos 3cos 6cos #

' - - + =

( )

( )( )

( )

4x

cos 1 C $

x4x 1 21 2cos C $/ 4 x

4x 1 21 4 2cos D 4

= % = -

' = ' a % = + + =

l

l .

, g2 th' Hh2o

"!-tI t x x t4 4% %

= - ; = +

( ) 3tan t tan t 14

% ? ' = + - 3

tan tan t4tan t 1

1 tan tant4

%+

' = -%

-

3 1 tanttan t 11 tant

tant 1 cost #

78 +8 = -88' 9 -88 & Q &88:

( )

3 4 3 2tan t tan t 2tant tan t tan t tan t 2 #

tant 1 cost # tant 1 cost #

77 88 - = - + =888 8' '9 98 8& Q & & Q &8 88: 8:

( )tan t #

t $tant 1 $/t 2

tan t # 4

7 8 = 8 = %8 8 = -' ' 9 % 8 = - + %8 8 & 8:

ll

.

, g2 th' Hh2o

"J6 H-nI

cos x # 2sin x cos x # sin 2x #4 4 4 2

cos x # 2sin x cos x # cos 2x #4 4 4 2

7 7 8 8% % % % 8 8 - & - - & - &8 8 8 8 8 88 8' '9 9 8 8% % % % 8 8 + & + + & + & 8 8 8 8 8 88 8: :

78888889888888:cos2x # sin2x 1' & ' & .

T cKI tan x tan x tan x tan x cot x tan x 14 4 2 4 4 4 4

> % % % % % % % A - + = - + + = + + = A A B

( ) 4 4 4sin 2x cos 2x cos 4x? ' + = 2 211 sin 4x 1 sin 4x2

' - = - 2sin 4x #' =



50/135


51/135


L binh. Mh g!- hng trinh lng gac d\ng ( )U tanV/cot V/sin2V/cos2V/ tan2V v0 q lh' h6 t thi t S!-t t tan V= . Mh SKI

2

2 2 2

2t 2t 1 ttan2V sin2V cos2V

1 t 1 t 1 t

-= = =

- + +. /n nhnO l[c n7 hng trinh

S cho S vJ hng trinh S\ s v0 3Rn t ' t cK th g2 d` dng ti' t VW . L6 z

rYngO S7 cK th lx x x

.../ / / /x/2x/3x/4x/...4 3 2

, g2 th' Hh2o

C "J6 H-nIsinx # sin2x #cosx #

tanx 1tanx 1

78 &8 788 &88 & '9 98 8 & -8 8:& -88:C "!-t t tanx / t 1= & - .

( )2 2

2

2 2 2

1 2t 1 t 1 1 1 tcot x sin2x cos2x sin x 1 cos2x 1

t 2 21 t 1 t 1 t

- -; = = = = - = -+ + +

( )2 2

2 2 2

1 1 t 1 1 1 t 1 2t1 . 1 .

t 1 t 2 21 t 1 t 1 t

- - ? ' - = + - - ++ + +

( )( ) ( ) ( )

( )22

2

t 1 C1 t 1 t 1 t t

2t t 1 # MC

=' - + = - ' - + =

( ) tanx 1 x $ / $4%

' = ; = + % (Nh-n do sin2x 1 #= & )

, g2 th' Hh2oC "J6 H-nI cosx #&

C "!-tI2

2tt tanx sin2x

1 t= ; =

+

( ) ( )3 222t

2t 3 2t 3t 4t 3 # t 1 x $ / $41 t

%? ' + = ' - + - = ' = ' = + %

+ .


? #%n c@ 3@ th$ng 'nhAAAA? PgB +;

B ^W. p2 hng trinhI ( )2cos2x 1

cot x 1 sin x sin2x1 tanx 2

- = + - ?+xTrHch d th t12en 'nh f= hc 5h! A n/# X``Yy

B ^X. p2 hng trinhI ( )sin2x 2tanx 3+ = ?TrHch d th t12en 'nh f= hc k1!c 4a 8 ;$ 5h! A n/# X``W

B ^Y. p2 hng trinhI ( ) ( ) ( )1 tanx 1 sin2x 1 tanx- + = + ?


52/135


53/135


, g2 th' Hh2oC "J6 H-nI sin x #&

( )2

1 14 sinx 2 4 sinx #

sinx sinx

> A ? ' + - + + - = A A B

( )

( )

2 1 3sinx 11 1 sin x 24 sinx 4 sinx 1 #1 sinx sinx sinx 2

sinx 2

+ = ' + + + - = ' + = -

( ) ( )21 2sin x 3sinx 2 # MC' - + =

( )( )

( )

2sinx 2 D x $2

62 2sin x sinx 2 # $/1 sinx x 22 6

% = - = - + % ' + + = ' ' %= - = + %

ll

, g2 th' Hh2o

C "J6 H-nIcosx #

cos3x #

78 &898 &8:

( ) ( )tanx tanx tan3x 2? ' - =

( )

2

2

sin x 3xsinx sinxsin2x 2sin x. 2 2 2

cosx cosxcos3x cosxcos3xcos xcos3x

- - -' = ' = ' =

( ) 2 2 32sin x 2cosxcos3x # 1 cos x cosx 4cos x 3cosx #' + = ' - + - =

( ) 3 2 2 1 $

4cos x 4cos x 1 # cos x cos2x # x / $2 4 2

% %' - + = ' = ' = ' = + .

, g2 th' Hh2o

C "J6 H-nIsinx # cosx 1

sin2x # cos2x 1

7 78 8& & 8 8'9 98 8& & 8 8: :

( ) 2sin 2x sinx sin2x cosx 1 2cos2x? ' + - - =

2 2 2 24sin xcos x 2sin xcosx cosx 1 4cos x #' + - + - =

( ) ( ) 2 2 2 24 1 cos x cos x 2 1 cos x cosx cosx 4cos x 1 #' - + - - - + =

4 3

4cos x 2cos x cosx 1 #' + + - =( )( ) 3 2cosx 1 4cos x 2cos x 2cosx 1 #' + - + - =

? #%n c@ 3@ th$ng 'nhAAAA? PgB +9

B ^]. p2 hng trinhI ( )2tan x tanxtan3x 2- = ?xTrHch d th T12en 'nh f= hc k1!c 4a 8 ;$ n/# W__\y

B ^^. p2 hng trinhI ( )1 1

sin2x sinx 2cot2x2sinx sin2x

+ - - = ?


54/135


( )cosx 1 x $2

$/1cosx x 2

2 3

= - = % + % ' ' % = = + %

ll

.


( ) ( )2 22cosx 2cos x 1 !cos x cosx 1 #? ' - - + - =

3 24cos x !cos x cosx 1 #' - + - =

( )

( )

( ) x $2cosx 1 C

$/1

x 2cosx C 32

= %= ' ' %

= + %=

ll

.

, g2 th' Hh2o

( ) ( )3 24 3sinx 4sin x 1 2sin x sinx #? ' - - + - + =

3 216sin x !sin x sinx 1 #' - + + + =

( )

x $2sin x 1 2x 2 $/J/m1

sinx sinx m24

%= + % = ' ' = a + % = - = a = %- a + %

l .

, g2 th' Hh2o

C "J6 H-nI sinx # cosx 1& & .

C "!-t t sinx t 1= "

( ) ( ) ( ) ( )( )2 2 4 221 1

1 t t t t 1 t t 1 t 1 t 1 t 1t t

? ' - = - ' - = - ' - = - + +

( ) ( ) 3 2t 1 t t 1 # t 1 sinx 1 x $2 / $2% >' - + + = ' = ' = ' = + % A B

.

PgB +< ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B ^_. p2 hng trinhI ( )1

2cos2x !cosx cosx

- + = ?

xTrHch d th T12en 'nh f= hc ;go= Thng n/# X```y

B _`. p2 hng trinhI ( ) ( ) ( )4 sin3x cos2x sinx 1- = - ?xTrHch d th T12en 'nh f= hc L1t n/# X```y

B _W. p2 hng trinhI ( )2 21 1

sinx sin xsinx sin x

- = - ?

B _X. p2 hng trinhI ( )2 21 1

cos x 2 cosx 2 #cosxcos x

+ - + + = ?


55/135



( )2

1 1cosx 2 2 cosx 2 #

cosx cosx

? ' + - - + + =

( )( )

2 1

cosx # 11 1 cosxcosx 2 cosx #1cosx cosx cosx 2 2

cosx

+ = ' + - + = ' + =

( )( ) ( )

2

2

1 cos x 1 # MC

2 cos x 2cosx 1 # cosx 1 x $2 / $

' + =

' - + = ' = ' = %

, g2 th' Hh2o

C "J6 H-nIcosx # sin2x # sin2x #

sinx # cos2x 1

7 78 8& Q & &8 8'9 98 8& & 8 8: :

( ) sinx cosx 1

2 2sin2xcosx sinx sin2x

? ' + = +

2sinxsin2x2 cos2x 2sin 2x 1 #

cosx' + - - =

( ) 2 24sin x cos2x 2 1 cos 2x 1 #' + - - - =

( ) 22 1 cos2x cos2x 3 cos 2x #' - + - + =

22cos 2x cos2x 1 #' - - =

( )( )

cos2x 1 Dx $ / $1 3cos2x

2

= %' ' = + % = -

.

, g2 th' Hh2o

C "J6 H-nIsin x #

sin2x # cos2x 1cosx #

78 &8 ' & ' & 98 &8:.

Cch g W

( ) ( )2 2tan x cot x 2 1 tan x cot x #? ' + + + + =

2 2

2 2

sin x cos x sinx cosx

2 1 #cosx sinxcos x sin x

' + + + + =

4 4 2 2 3 3sin x cos x 2sin xcos x 2sin xcosx 2sinxcos x #' + + + + =

? #%n c@ 3@ th$ng 'nhAAAA? PgB ++

B _Y. p2 hng trinhI ( )1

2tanx cot x 2sin2xsin2x

+ = + ?

B _Z. p2 hng trinhI ( )2 2

tan x cot x 2 1 tan x cot x #+ + + + =


56/135


2 21 11 sin 2x sin 2x sin2x #

2 2' - + + =

( ) ( ) sin2x 1 C x $ / $4

%' = - ' = - + % .

Cch g X (Phng trinh S Ung thBo tn v cot)

C "!-t2 2 2

t tanx cot x t 2 tan x cot x t 2= + K ; + = - .

( ) ( ) ( )

( )

2 2 t # D

t 2 2 1 t # t 2t #t 2 C

=? ' - + + = ' + = ' = -

( ) ( )sinx cosx

tanx cot x 2 2 sin2x 1 C x $ / $cosx sinx 4

%' + = ' + = - ' = - ' = - + % .

, g2 th' Hh2o

( ) 21 11

1 sin 2x sin2x2 !

? ' - = -

24sin 2x !sin2x 3 #' - + =

( )

( )( )

1

sin2x C x $2 12 $/3

sin2x D x2 12

% = = + % ' ' % = = + %

l

l

, g2 th' Hh2o

( ) ( )4 4 24sinx cosx sin x cos x sin 4x? ' - =

( ) ( ) 2 2 2 2 22sin2x sin x cos x sin x cos x sin 4x #' - + - =

2 22sin2xcos2x sin 4x # sin 4x sin4x #' - - = ' + =

( )$4x $ xsin4x # 4 $/

sin4x 1 4x 2 4x2 ! 2

% = % = = ' ' ' % = - % %= - + % = - +

lll

.

, g2 th' Hh2o

( ) ( )2x 2x 2x 2x

4 4 4 42cos cos 4sinx 2 2 1 sinx2 2

% % % %+ + - + - +

? ' + = + -

PgB +: ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B _[. p2 hng trinhI ( )4 4 11

sin x cos x sin2x!

+ = - ?

B _\. p2 hng trinhI ( ) 24sin xcosx 4cos xsinx sin 4x- = ?

B _]. p2 hng trinhI ( ) ( )cos 2x cos 2x 4sinx 2 2 1 sinx4 4

% % + + - + = + - ?


57/135


58/135


( ) 2 23

1 sin 2x sin2x 3sin 2x 4sin2x 4 #4

? ' - = ' + - =

( )sinx 2 D x $2 2$/ sin2 x 2 3sinx sin

3

= - = a + % ' ' a = = % - a + % = = a

ll

.

B9 TVP Qq; Lms;

C1 [W p2 hng trinhI 34cos x 3 2sin2x !cosx+ =

C1 [X p2 hng trinhI 26sin 3x cos12x 14+ =

C1 [Y p2 hng trinhI 3tanx cot x 1 3+ = +

C1 [Z p2 hng trinhI tanx 3cot x 1 3- + =

C1 [[ p2 hng trinhI2 2

1 34

sinxcosxsin xcos x+ =

C1 [\ p2 hng trinhI 21

4tanx 2 #cos x

- + =

C1 [] p2 hng trinhI 21

cotx 3sin x

= +

C1 [^ p2 hng trinhI ( ) 22 2

1 2 2 sinx1 cot x

- + = -+

C1 [_ p2 hng trinhI 24

cosx #

1 tan x

+ - =

+

C1 \` p2 hng trinhI ( )4 4 1

2 sin x cos x cos 2x #2

% + - - =

C1 \W p2 hng trinhI sin4x tanx=

C1 \X p2 hng trinhI4 4 4 sin x sin x sin x

4 4 !

% % + + + - =

C1 \Y p2 hng trinhI tanx cotx 4+ =

C1 \Zp2 hng trinhI ( )

2sinx 3 2 2cosx 2sin x 1

11 sin2x

- - -

=-

C1 \[ p2 hng trinhI 44cos x 3 2sin2x !cosx+ =

C1 \\ p2 hng trinhI1 1 2

cosx sin2x sin4x+ =

C1 \] p2 hng trinhI sin2x 2sin x 14

% + - =

C1 \^ p2 hng trinhI ( ) ( )2 2sin x 1 4 sinx 1 cos 2x sin 2x4 4 % % - = - - + - +

C1 \_ p2 hng trinhI 24x

cos cos x3

=

PgB +4 ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?


59/135


C1 ]` p2 hng trinhIx

tan cosx sin2x #2

+ =

C1 ]W p2 hng trinhI 1 3tanx 2sin2x+ =

C1 ]X p2 hng trinhI 23x 4x

2cos 1 3cos

+ =

C1 ]Y p2 hng trinhI cot x tanx 2tan2x= +

C1 ]Z p2 hng trinhI 23x

2cos 1 3cos2x2

+ =

C1 ][ p2 hng trinhI 23cos4x 2cos 3x 1- =

C1 ]\ p2 hng trinhIx

cosx tan 12

+ =

C1 ]] p2 hng trinhI 23tan2x 4tan3x tan 3x tan2x- =

C1 ]^ p2 hng trinhI 2 3

cosx cos4x cos2xcos3x cos 4x

2

+ + =

C1 ]_ p2 hng trinhI ( ) ( )1 tan x 1 sin2x 1 tan x- + = +

C1 ^` p2 hng trinhI 6 6 213

sin x cos x cos 2x!

+ =

C1 ^W p2 hng trinhI 6 6 1

sin x cos x cos2x16

+ = +

C1 ^X p2 hng trinhI ( )2sinx 2 3tan x 1 sinx- = -

C1 ^Y p2 hng trinhI ( )44 1sin x sin x 1

!

+ - =

C1 ^Z p2 hng trinhI 2!x 2x

cos cos3 3

=

C1 ^[ p2 hng trinhI 2x

cos2x 3cosx 4cos2

- =

C1 ^\ p2 hng trinhI 2cosxcosx cos4xcos2x 4 3sin x= + -C1 ^] p2 hng trinhI 2cosx cos2x 1 cos2x cos3x= + +

C1 ^^ p2 hng trinhI ( )sin3x cos2x 2 sin2xcosx 1+ = -

C1 ^_p2 hng trinhI

( )

4 42 cos 2x sin 2x cos!x cos4x #- + - =

C1 _` p2 hng trinhI ( ) 2x 1 1 2cos 2 2x cos 1#cos x sin x

2 2 2 2 2

% % % - + - - - = -

C1 _W p2 hng trinhI

4cos x cos 2x3 3 2

% % - + + =

C1 _X p2 hng trinhI 4 4

sin x cos x 2sinx2

+ = -

C1 _Y p2 hng trinhI 6 6 1

sin x cos x sin2x

4

+ =

C1 _Z p2 hng trinhI2 23sin 2x !sin x 11 3cos2x

#sin2x

+ - -=

? #%n c@ 3@ th$ng 'nhAAAA? PgB +8


60/135


C1 _[ p2 hng trinhI( ) 2cos2x 3 2 2sinx 3 2sin x

1 #sin2x 1

+ + -- =

+

C1 _\ p2 hng trinhIsinx 1

11 cos2x

+=

+

C1 _]p2 hng trinhI cos2x 3cot2x sin4x

2cot2x cos2x

+ +

=-

C1 _^ p2 hng trinhI

4 4

4

x xsin cos

2 2 cos x

tan x tan x4 4

+=

% % - +

C1 __ p2 hng trinhI2

2

1 1cos x cosx

cosxcos x+ = +

C1 W`` p2 hng trinhI ( ) ( )cos2x 2 2 cosx sinx cosx+ = - -

PgB :> ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?


61/135


C P8;4 TQR;8 BVC ;8zT T8+ 9; D C xPT C{ f9|;y

]\ngI ( ) X Y( )asinx bcosx c / a/b Z #+ = ? [

Phng *h* WI

"J6 H-n S hng trinh cK ngh-'I 2 2 2a b c+ K

#h * vR hng trinh cho 2 2a b #+ & O t ScI

( )2 2 2 2 2 2

a b csinx cosx

a b a b a b? ' + =

+ + +.

"!-t ( )

2 2 2 2

a bsin cos / #2

a b a b >a = a = a % A B

+ +. Phng trinh tr thnhI

2 2 2 2

c c

sin sinx cos cosx cos(x )a b a ba + a = ' - a =+ + S 3Rt cach g2.

Phng *h* XI

M' tr UB' x xcos # $ x $2

2 2 2%

= ' = + % ' = % + % cK h2 l ngh-' h7 Hh$ng NR6

h2 thi gh nh-n ngh-' n7.

0 x xcos # $ x $2

2 2 2%

& ' & + % ' & % + % O t S!-tI

2

2 2

x 2t 1 tt tan sinx / cosx2 1 t 1 t

-= ; = =+ + . Th7 vo hng trinhO t ScI

( ) ( )2(b c)t 2at c b #? ' + - + - = ? ? .

i x $2 b c #& % + % ' + & nn ( )? ? cK ngh-' HhI

2 2 2 2 2 2\ a (c b ) # a b cE = - - K ' + K .

p2 hng trinh ( )? ? O ng v0 ' ngh-' t t cK hng trinhI xtan t x

2= ;

B9 TVP >P (U;4

B W`W. p2 hng trinhI ( )2 6

cosx 3sinx 2 / x

% %- = - ? @

B W`X. p2 hng trinhI ( )2

x xsin cos 3cosx 2

2 2

+ + = ?

B W`Y. p2 hng trinhI ( )1

tan x sin2x cos2x 2 2cosx #

cosx

- - + - = ?

B W`Z. p2 hng trinhI( )

( ) ( ) ( )

1 2sinx cosx3

1 2sinx 1 sinx

-= ?

+ -

? #%n c@ 3@ th$ng 'nhAAAA? PgB :;


62/135


63/135


B WYX. p2 hng trinhI ( ) 2 x2 3 cosx 2sin

2 41

2cosx 1

% - - - =

-

( ) ?

B WYY. p2 hng trinhI ( ) ( )2

3cosx 4sin x 6 2 3 3cosx 4sin x 6- - + = - - - ( ) ?

B WYZ. p2 hng trinhI ( )sinx sin2x 3cosx cos2x

- = ?-

B WY[. p2 hng trinhI ( )sin3x 3cos3x 2sin2x- = ?

B WY\. p2 hng trinhI ( )2cosx sin2x

32cos x sinx 1

-= ?

- -

B WY]. p2 hng trinhI ( )1

tanx 3cosx

- = ?

B WY^. p2 hng trinhI ( )33sinx 3cos1x 1 4sin x- = + ?

B WY_. p2 hng trinhI ( )3 3 cos xcos3x sin x sin3x!

- = ?

B WZ`. p2 hng trinhI ( )1#cosx 3cot x 4= + ?

B WZW. p2 hng trinhI ( ) ( )cos3x sinx 3 cosx sin3x- = - ?

B WZX. p2 hng trinhI ( )3

4sin2x 3cos2x cos 3x #2

% - - + = ?

B WZY. p2 hng trinhI ( ) ( )2 2x 3

4sin 3cos2x 1 2cos x / x #2 4

% - = + - ? @ %

B WZZ. p2 hng trinhI ( )cos2x 3sin2x 3sinx cosx 4 #- - - + = ?

B WZ[. p2 hng trinhI ( ) ( ) ( )3

sin x cosx 2 sin2x 1 sinx cosx 2+ - + + + = ?

B WZ\. p2 hng trinhI ( )2 23cos x 2sinxcosx 3sin x 1 #+ - - = ?

B WZ]. p2 hng trinhI ( ) ( )2

1 2sinx cosx 1 sinx cosx+ = + + ?

B WZ^. p2 hng trinhI ( )34sin x 1 3sinx 3cos3x- = - ?

B WZ_. p2 hng trinhI ( ) ( )6 6! sin x cos x 3 3cos2x 11 3 3sin4x sin2x+ - = - - ?

B W[`. p2 hng trinhI ( ) ( )3sin2x 2cosx 1 3cosx 2 cos2x cos3x+ + + = + ?

? #%n c@ 3@ th$ng 'nhAAAA? PgB :9


64/135


8;4 (; 49S9 P8;4 TQR;8 BVC ;8zT T8+ 9; D C

xPT C{ f9|;y

, g2 th' Hh2o

( ) 3 1 2

sinx cosx2 2 2

? ' - = sinxcos cosxsin sin6 6 4% % %

' - =

sin x sin6 4

% % ' - = ( )

$2xx $2

!4 6 4 $/3 11 2x 2 x6 4 !4

% %% % = +- = + % ' ' % % % % - = + % = +

lll

3x

2 $2 6 !41/1 $ 2/ $ 22 6 !4 Eo x #/4 2/4 1 x

2 11 2 6 12$/ 2 !4 x

!4

%= % % % % R R = R + R % % % ' ' R R ' = ' = % % % % R + R = % =

l ll

l l

-7 ngh-' c%n ti' c hng trinh lI 3 x x x!4 12 !4% % %= = = .

, g2 th' Hh2o

( ) 1 sin x 3cosx 2? ' + + = 1 3 1sinx cosx2 2 2' + = sinxcos cosxsin sin

3 3 6% % %' + =

sin x sin3 6

% % ' + = ( )

x $2 x $23 6 6 $/

x 2 x 2

3 6 2

% % % + = + % = - + % ' ' % % % + = + % = + %

l

l l

.

, g2 th' Hh2o

PgB :< ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B W`W. p2 hng trinhI ( ) 2 6cosx 3sinx 2 / x

% %- = - ? @ TrHch d th t12en 'nh f= hc nh tF k1!c (n n/# W__] W__^

B W`X. p2 hng trinhI ( )2

x xsin cos 3cosx 2

2 2

+ = = ? TrHch d th t12en 'nh f= hc 5h! ( n/# X``]

B W`Y. p2 hng trinhI ( )1

tan x sin2x cos2x 2 2cosx #cosx

- - + - = ?

TrHch d th t12en 'nh f= hc L1t n/# W__^


65/135


C "J6 HnI ( )cosx # sinx 1& ' & ? ?

( ) sinx 2sinx cosx cosx cosx cos2x 2cos2x #? ' - - + =

( ) ( ) 2sinx 1 2cos x cos2x cosx 2 #' - - - = ( )sinx cos2x cos2x cosx 2 #' + - =

( ) cos2x sinx cosx 2 #' + - =

( )( )

2x $cos2x # 2 x $ / $sinx cosx 2 4 2sin x 2 D

4

% = + % = % % ' ' ' = + %+ = + =

C Th7 vo ( )? ? O t Sc h ngh' hng trinh lI ( )x $ / $4 2% %

= + .

L binh. " v0hng trinh 3-c nhQt v0 sn v cos d\ngI asinx bcosx c+ = thi cac 3\n hcsnh cK th g2 '$-t cach d` dng 3Yng cach ch h vR cho 2 2 2a b c+ K . Nhng nR6

g!- d\ngI

( )( )

( )

( )asinx bcosx a\sin$x 1

asinx bcosx a\cos$x 2 $ 1

asinx bcosx a\sin$x b\cos$x 3

+ = + = &

+ = +

thi h0ng g2 k67Rt

nh thR no # 3inh t_nhO t UB' vR tra c ( ) ( )1 / 2 l hng trinh 3-c nhQt thBo sn v

cos thi cach l' c{ng tng tX nh d\ngI asinx bcosx c+ = . #^n S v0 d\ng ( )3 O tco h vR c hng trinh l h hng trinh 3-c nhQt S v0 sn v cosO hn nhncach g2 c{ng tng tX. Nhng l6 z rYngO t h2 ch67n vR so cho vR tra l c@ng '$-tc6ngO vR h2 l c@ng '$-t c6ng. /7 ch' ngh-' h0ng s67 ngh_ SK k6 lW g2 s6.

, g2 th' Hh2o

C "J6 HnI ( )sin x 1 cosx #

1 1sin x sin x

2 2

7 78 8& &8 88 88 8' ? ?9 98 8& - & -8 88 88 8: :

( ) ( )2cosx sin2x 3 1 sinx 2sinx 2sin x? ' - = - + -

cosx sin2x 3cos2x 3sinx' - = +

3 1 1 3

cos2x sin2x cosx sin x2 2 2 2

' + = -

cos2xcos sin sin2x cosxcos sin sinx

6 6 3 3

% % % %' + = -

? #%n c@ 3@ th$ng 'nhAAAA? PgB :+

B W`Z. p2 hng trinhI ( )( ) ( ) ( )1 2sinx cosx 3

1 2sinx 1 sinx- = ?

+ -

TrHch d th t12en 'nh f= hc 5h! A n/# X``_


66/135


( )x $2

2cos 2x cos x $/26 3 x

1! 3

%= + % % % ' - = + ' % % = - +

ll

.

C Th7 vo ( )? ? O t Sc ngh' c hng trinh lI ( )2

x /1! 3

% %= - +

ll .

, g2 th' Hh2o

C "J6 HnI ( )cosx #

sin2x 1sin x #

78 &8 ' & ? ?98 &8:

( ) 2!sin xcosx 3sinx cosx? ' = +( ) 4cosx 1 cos2x 3sinx cosx' - = +

3cosx 4cos2xcosx 3sinx' - =

( ) 3cosx 2 cos3x cosx 3sinx' - + =

cosx 3sinx 2cos3x' - =

1 3

cos3x cosx sin x2 2

' = -

( )x $

6cos3x cos x $/3 x

12 2

%= + % % ' = + ' % % = - +

ll

C Th7 vo ( )? ? O t Sc h ngh' hng trinh lI ( )x $ x / $/6 12 2% % %

= + % = - + l

l .

, g2 th' Hh2o

( ) 3sinx sin3x

sinx sin2x cosx 3 cos3x 2cos4x2

-? ' + + = +

2sinx 2sin2x cosx 2 3cos3x 4cos4x 3sinx sin3x' + + = + -

( )1

2sinx 2. sin3x sinx 2 3cos3x 4cos4x 3sinx sin3x2

' + + + = + -

2sin3x 2 3cos3x 4cos4x' + =

3 1

cos3x sin3x cos4x2 2

' + =

PgB :: ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B W`[. p2 hng trinhI ( )3 1

!sinxcosx sinx

= + ?

B W`\. p2 hng trinhI ( ) ( )3sinx cosxsin2x 3cos3x 2 cos4x sin x+ + = + ?

TrHch d th t12en 'nh f= hc 5h! B n/# X``_


67/135


( )x $2

6cos 3x cos4x $/26 x

42

%= - + % % ' - = ' % % = +

ll

.

, g2 th' Hh2o

( ) ( )33sin3x 4sin 3x 3cosx 1? ' - - =

sinx 3cosx 1' - =

1

sinx cos cosx sin3 3 2

% %' - =

sin x sin3 6

% % ' - =

( )$2

x $2 x3 6 1! $/

2x 2 x

3 6 4

% % % % - = + % = + ' ' % % % % - = % - + % = +

ll

l

.

, g2 th' Hh2o

( ) 3cosx sinx sinx sinx #? ' - - - =

3 1

cosx sinx sinx2 2

' - =

sin x sinx3

% ' - =

( )$x x $2 x

3 12 2 $/2

x x 2 x3 3

% % % - = + % = + ' ' % % % - = % - + % = +

ll

l

.

, g2 th' Hh2o

( ) 2sinx 6cosx 6sinx cosx 2cos x ? ' + - + =

( ) ( ) 2 sinx 1 6cosx 1 sinx 2cos x #' - + - + =

? #%n c@ 3@ th$ng 'nhAAAA? PgB :=

B W`]. p2 hng trinhI ( )os 33sin3x 3c x 1 4sin 3x- = + ?

B W`^. p2 hng trinhI ( )3cosx 2sin3xcoss2x sinx #- - = ?TrHch d th t12en 'nh f= hc 5h! ( n/# X``_

B W`_. p2 hng trinhI ( )sinx 6cosx 3sin2x cos2x !+ - + = ?TrHch d th t12en 'nh f= hc ;go= Thng n/# W__] W__^


68/135


( ) ( ) 2sin x 1 6cosx 2cos x #' - - + =

( ) ( ) ( ) ( ) 1 sin x 6cosx 2 1 sin x 1 sin x #' - - + - + =

( ) ( ) 1 sin x 6cosx 2 2sin x #' - - + + =

( )

( )

sinx 1 1

6cosx 2sinx 2

=' + =

C p2 ( ) ( )1 x $2 / $2

%' = + %

C p2 6cosx 2sinx + = ; Phng trinh v$ ngh' do 2 2 26 2 + R .

C 7 h ngh' c hng trinh lI ( )x $2 / $2

%= + % .

, g2 th' Hh2o

( ) 22sinx cosx 4cos x 2 1 sinx 4cosx? ' + - = + -

( ) ( ) ( ) 24cos x 1 sinx 2cosx 1 2 2cosx 1 #' - + - + - =

( ) ( ) 2cosx 1 2cosx 1 sin x 2 #' - + + + =

( ) ( )

2cosx 1 #cosx cos x $2 / $

2cosx sin x 3 MC 3 3

- = % %' ' = ' = + % + = - .

, g2 th' Hh2o

( ) 2sin2x 2cosx cos2x 4 sinx #? ' - - + - =

( ) 22cos x 2sinx 1 1 2sin x 4 sinx 6sinx #' - - + + - - =

( ) ( ) ( ) 2cosx 2sinx 1 sinx 2sinx 1 3 2sinx 1 #' - + - - - =( ) ( ) 2sin x 1 2cosx sin x 3 #' - + - =

( )( )

1 x $2sin x sin 6 $/2 6 2cosx sin x 3 # MC x 26

% % = + % = = ' ' %+ - = = + %

l

l

.

, g2 th' Hh2o

PgB :4 ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B WW`. p2 hng trinhI ( )sin2x 2cos2x 1 sinx 4cosx+ = + - ?

B WWW. p2 hng trinhI ( )2sin2x cos2x sinx 2cosx 4- = + - ?TrHch d th t12en 'nh f= hc k1!c 4a T*8CM n/# X``W

B WWX. p2 hng trinhI ( )sin2x cos2x 3sinx cosx 2- = + - ?

TrHch d th t12en 'nh f= hc ;$ng ;gh3* 9 5h! B n/# X``W


69/135


( ) 22sinxcosx 1 2sin x 3sinx cosx 2 #? ' - + - - + =

( ) ( ) ( ) 22sin x cosx cosx 2sin x sin x 2sin x 1 #' - + - - - =

( ) ( ) ( ) cosx 2sin x 1 sin x 2sin x 1 2sin x 1 #' - + - - - =

( ) ( ) 2sin x 1 cosx sin x 1 #' - + - =

1sinx sin2sin x 1 # 2 6

cosx sinx 1 cos x cos4 4

% = = - = ' ' % %+ = - =

( )

x $2 x J2

6 6 $/J/m/nx m2 x n2

2

% %= + % = + %' % = + % = %

.

, g2 th' Hh2o

( ) ( )3 22cos x 2cos x sin x 1 #? ' + + - =

( ) ( ) 22cos x cosx 1 sin x 1 #' + + - =

( ) ( ) ( ) ( ) 2 1 sin x 1 sin x cosx 1 1 sin x #' - + + - - =

( ) ( ) ( ) 1 sin x 2 1 sin x 1 cosx 1 # >' - + + - = A B( ) ( ) 1 sinx 1 2cosx 2sinx 2sinxcosx #' - + + + =

( ) ( ) ( )2

1 sin x sin x cosx 2 sin x cosx # >

' - + + + = A A B

( ) 21 sinx 2sin x 2 2 sin x #4 4

> % % A ' - + + + = A A B

2

1 sinx #

2sin x 2 2sin x #4 4

- =

' % % + + + =

( )

( )

sin x 1x $2

2sin x # $/4 x

4sin x 2 D

4

= % = + % % ' + = ' % = - + % % + = -

l

l

.

? #%n c@ 3@ th$ng 'nhAAAA? PgB :8

B WWY. p2 hng trinhI ( )32cos x cos2x sinx #+ + = ?

B WWZ. p2 hng trinhI ( )21 cos2x

1 cot2xsin 2x

-+ = ?


70/135


, g2 th' Hh2o

C "J6 HnI ( )sin2x # cos2x 1& ' & ? ?

( )( ) ( )

sin2x cos2x 1 cos2xsin2x 1 cos2x 1 cos2x

+ -? ' =

- +

( ) ( ) sin2x cos2x 1 cos2x sin2x' + + =

2sin2x sin2x cos2x cos2x cos 2x sin2x #' + + + - =

( ) cos2x sin2x cos2x 1 #' + + =

cos2x #

x $ x J x m3cos 2x cos 4 2 2 4

4 4

= % % % % ' ' = + = + % = - + %% % - =

C Th7 vo ( )? ? ; h ngh' hng trinh l ( )x $ / $4 2

% %= + .

, g2 th' Hh2o

( ) 3 cos4x

4 3sin4x 24

+? ' + =

3 1

sin 4x cos4x 12 2' + = -

( ) cos 4x 1 4x $2 x $ / $3 3 3 2

% % % % ' - = - ' - = % + % ' = + .

, g2 th' Hh2o

( ) ( ) ( ) 11 sin2x cos2x 1 sin2xcos2x sin4x2

? ' + + - =

( )1 1

1 sin 4x sin2x cos2x 1 sin 4x #2 2

' - + + - =

( )1

1 sin4x sin2x cos2x 1 #2

' - + + =

( )1 sin4x 2 D1 sin4x #2

sin2x cos2x 1sin2x cos2x 1 #

=- = ' ' + = -+ + =

PgB => ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?

B WW[. p2 hng trinhI ( )21 cos2x

1 cot2xsin 2x

-+ = ?

B WW\. p2 hng trinhI ( )3 3 1

1 sin 2x cos 2x sin4x2

+ + = ?


71/135


( )x $3 42cos 2x 1 cos 2x cos $/

4 4 4 x2

%= - + % % % % ' - = - ' - = ' % = + %

l

l

.

, g2 th' Hh2o

C "J6 H-nIsin x #

sin2x # cos2x 1cosx #

78 &8 ' & ' & 98 &8:.

( ) ( )sinx 3cosx

4 sinx 3cosxcosx sinx? ' - = +

( ) 2 2sin x 3cos x 4sinxcosx sinx 3cosx #' - - + =

( ) ( ) ( ) sinx 3cosx sinx 3cosx 2sin2x sinx 3cosx #' - + - + =

( ) ( ) sinx 3cosx sinx 3cos2x 2sin2x #' + - - =

( )tanx 3sin x 3cosx # x $

3 $/1 3 4sin x sin2xsinx cos2x sin2x x32 2 3

%= -+ = = - + % ' ' ' % % %- = - = = +

ll

.

, g2 th' Hh2o

( ) 3 3sin x sinx cos x cosx #? ' - + + =

( )2 3

sinx sin x 1 cos x cosx #' - + + =

2 3sinxcos x cos x cosx #' - + + =

( ) 2cosx sinxcosx cos x 1 #' - + + =

1 1 cos2x

cosx sin2x 1 #2 2

+ ' - + + =

sin2x cos2x 3

cosx #2

- + + ' =

( ) ( )

o

cosx # x $ / $sin2x cos2x 3 MC Po 1 1 2

= %' ' = + % - + = - + R

.

? #%n c@ 3@ th$ng 'nhAAAA? PgB =;

Bai 117. p2 hng trinhI ( ) ( )tanx 3cotx 4 sinx 3cosx- = + ?

B WW^. p2 hng trinhI ( )3 3sin x cos x sinx cosx+ = - ?TrHch d th t12en 'nh f= hc Thj2 n n/# W__\

B WW_. p2 hng trinhI ( )4 4 1

cos x sin x4 4

% + + = ?


72/135


, g2 th' Hh2o

( )

2

2 1 cos

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Phuong Trinh Luong Giac Va Ung Dung1