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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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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
"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$ng Th-t v-7I
PgB ;* ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?
B ^. p2 hng trinhI( )
4 44sin 2x cos 2x cos 4x
tan x tan x4 4
+= ?
% % - + TrHch d th t12en 'nh f= hc 2 (7ng n/# W__]
B _. p2 hng trinhI ( )3 x 1 3x
sin sin 11# 2 2 1# 2
% % - = + TrHch d th t12en 'nh f= hc Thj2 L n/# X``W
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
3x 3x 3x 3 xsin sin sin sin3
1# 2 1# 2 1# 2 1# 2
> % % % % A + = % - + = - = - A A B. Ty SKO t s
S!-t3 x
t1# 2
%= - v s dung c$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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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( ) 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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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( )
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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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( ) ( ) ( ) ( ) ( )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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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( ) ( ) ( )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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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
, 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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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( ) ( ) 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__^
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( ) ( ) 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$ng 'nhAAAA? PgB *;
B Y`. p2 hng trinhI ( )4 2 2 43cos x 4cos xsin x sin x #- + = ?TrHch d th t12en 'nh f= hc k1!c k1a T*8CM W__^ W___ t W
B YW. p2 hng trinhI ( )3 3 2 3 2
cos3xcos x sin3xsin x!
-- = ?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
#hnh vi thRO S7 t Hho lo hn tch S a dung c$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__^
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, 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$ng 'nhAAAA? PgB *9
B YY. p2 hng trinhI ( )34sin3xcos2x 1 6sinx !sin x= + - ?
B YZ. p2 hng trinhI ( )1
cosx cos2x cos3x cos4x cosx2
+ + + + = - ?
B Y[. p2 hng trinhI ( )sin2x 2cosx sinx 1
#tanx 3
+ - -= ?
+TrHch d th t12en 'nh f= hc 5h! ( n/# X`WW
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/o!-c 36 d`n cac ngn c6ng SJ6 H-n v ngn c6ng ngh-' trn c@ng '$-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__^
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
, 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```
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( )
( ) ( )
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```
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( ) ( )
( ) ( )
( ) ( )
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$ng ;gh3* T* 8CM n/# X``]
(Lo\ do sin x #& )
(Nh-n)
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
1 cos2x 1 cos4x 1 cos2x 1 cos4x
cot3x . 11 cos2x 1 cos4x 1 cos2x 1 cos4x
- + - + ' - = - + - + -
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
cot3x 1 cos2x 1 cos4x 1 cos2x 1 cos4x
1 cos2x 1 cos4x 1 cos2x 1 cos4x
>' - + - + - A B= - - - + +
( ) ( ) cot3x 2cos4x 2cos2x 2 cos4x cos2x' - = - +
( )cos3x
4sin3xsinx 4cos3x cosxsin3x
' - = -
( )$
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``\
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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 $ / $t cos2x/ t 1t cos2x/ t 1
7 8 = =- + =88 ' ' ' = ' = % 9 8 = "= " 8 8: .
PgB 9: ? Ell thB FloGBr oF to'orroG rB n thB sBBHs oF tod7AA?
B [W. p2 hng trinhI ( )2cos4x 12sin x 1 #+ - = ?xTrHch d th t12en 'nh Cao )ng 5h! AO BO ( n/# X`WWy
B [X. p2 hng trinhI ( )4 4cos x sin x cos4x #- + = ?
xTrHch d th t12en 'nh Cao )ng 2
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
L binhI Trong v du n7O c{ng tfn t\ h c6ng Hhac nh6 x v 4x nn t S vJ c@ng '$-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
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( ) ( ) 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
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
, 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
% % % + - - = + @ % ?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( )
( )
( )
( ) ( )
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
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
, 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
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( )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$ng l ngh-' ( )? .
? #%n c@ 3@ th$ng 'nhAAAA? PgB
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
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 $/t 2
tan t # 4
7 8 = 8 = %8 8 = -' ' 9 % 8 = - + %8 8 & 8:
ll
.
, g2 th' Hh2o
"J6 H-nI
cos x # 2sin x cos x # sin 2x #4 4 4 2
cos x # 2sin x cos x # cos 2x #4 4 4 2
7 7 8 8% % % % 8 8 - & - - & - &8 8 8 8 8 88 8' '9 9 8 8% % % % 8 8 + & + + & + & 8 8 8 8 8 88 8: :
78888889888888:cos2x # sin2x 1' & ' & .
T cKI tan x tan x tan x tan x cot x tan x 14 4 2 4 4 4 4
> % % % % % % % A - + = - + + = + + = A A B
( ) 4 4 4sin 2x cos 2x cos 4x? ' + = 2 211 sin 4x 1 sin 4x2
' - = - 2sin 4x #' =
? #%n c@ 3@ th$ng 'nhAAAA? PgB
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
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 $ / $41 t
%? ' + = ' - + - = ' = ' = + %
+ .
, g2 th' Hh2oC "J6 H-nI cosx #&
? #%n c@ 3@ th$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- + = + ?
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
, 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
+ - - = ?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( )cosx 1 x $2
$/1cosx x 2
2 3
= - = % + % ' ' % = = + %
ll
.
, g2 th' Hh2oC "J6 H-nI cosx #&
( ) ( )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
+ - + + = ?
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
, g2 th' Hh2oC "J6 H-nI cosx #&
( )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 #+ + + + =
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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
% % + + - + = + - ?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( ) 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?
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
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
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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?
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
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 :;
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
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
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
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__^
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
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``_
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( )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``_
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( )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__^
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
( ) ( ) 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
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( ) 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
-+ = ?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
, 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
+ + = ?
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Phng trinh lng gac v ng dung (Nng co) Ths. L !n"on
( )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
% + + = ?
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Ths. L !n "on Phng trinh lng gac v ng dung (Nng co)
, g2 th' Hh2o
( )
2
2 1 cos