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10 – 2 s. Lê Vă à
Page - 1 -
Chương
NG
s
1/ 2 2 2cos x sin x 1 2sin x- = - . 2/ 2 22cos x 1 1 2sin x- = - .
3/ 2 23 4sin x 4cos x 1- = - . 4/ sin xcot x cosx tan x sin x cosx+ = + .
5/ 4 4 2 2sin x cos x 1 2sin xcos x+ = - . 6/ 4 4 2 2cos x sin x cos x sin x- = - .
7/ ( )( )24cos x 3 1 2sin x 1 2sin x- = - + . 8/ ( )( )2 2 21 cosx sin x cosx cos x sin x+ - + = .
9/ 4 4 2 2sin x cos x 1 2cos x 2sin x 1- = - = - . 10/ 3 3sin xcosx sin xcos x sin xcosx+ = .
11/ 2 2 2 2tan x sin x tan x sin x- = . 12/ 2 2 2 2cot x cos x cot x cos x- = .
s
1/ 1
tan x cot xsin x cosx
+ = . 2/ 1 cosx sin x
sin x 1 cosx
-=
+.
3/ 1 1
11 tan x 1 cot x
+ =+ +
. 4/ 21 11 1 tan x 0
cosx cosx
æ öæ ö÷ ÷ç ç÷ ÷- + + =ç ç÷ ÷ç ç÷ ÷ç çè øè ø.
5/ 2
2
2
1 sin x1 2tan x
1 sin x
+= +
-. 6/
tan x tan ytan x tan y
cot x cot y
+=
+.
7/ 4
2 4
2 11 cot x
sin x sin x- = - . 8/
cosx 1tan x
1 sin x cosx+ =
+.
9/ ( )( )2 11 cosx 1 cot x
1 cosx- + =
+. 10/
1 cosx 1 cosx 4cot x
1 cosx 1 cosx sin x
+ -- =
- +.
11/ sin x 1 cosx 2
1 cosx sin x sin x
++ =
+. 12/
sin x cosx 1 cosx
sin x cosx 1 1 sin x
+ -=
- + +.
13/ 2
2
sin x 2cosx 1 cosx
1 cosx2 cosx cos x
+ -=
++ -. 14/
sin x cosx 1 2cosx
1 cosx sin x cosx 1
+ -=
- - +.
GOC VA CÔNG THƯC LƯƠNG GIAC 5
A – HÊ THƯC LƯƠNG GIAC CƠ BAN
Hê thưc lương giac cơ ban
. .
. ..
. .
s. Lê Vă à 5. –
Page - 2 - " A e f wer f rr w re e see s f d y "
s
1/ 6 6 2 2sin x cos x 1 3sin xcos x+ = - .
2/ ( )( )6 6 2 2 2 2sin x cos x sin x cos x 1 sin x cos x- = - - .
3/ ( )2
8 8 2 2 4 4sin x cos x 1 2sin x cos x 2sin x cos x+ = - - .
4/ ( )( )8 8 2 2 2 2sin x cos x sin x cos x 1 2sin x cos x- = - - .
s
1/ ( )( )1 sin x cosx tan x 1 cosx 1 tan x+ + + = + + .
2/ ( )( )1 tan x 1 cot x sin x cosx 1 2sin x cosx+ + = + .
3/ ( ) ( ) ( )2
2 21 tan x cos x 1 cot x sin x sin x cosx+ + + = + .
4/ 2 2sin x tan x cos xcot x 2sin xcosx tan x cot x+ + = + .
5/ 2 2 2 2 2sin x tan x 4sin x tan x 3cos x 3+ - + = .
s
1/ 2 2sin cos tan 1
1 2sin cos tan 1
a - a a -=
+ a a a +. 2/
2 26
2 2
tan sintan
cot cos
a - a= a
a - a.
3/ 2 2cos sin
1 sin cos1 tan 1 cot
a a+ = - a a
- a - a. 4/
( )3
tan sin 1
sin cos 1 cos
a - a=
a a + a.
5/ 2 2
2 2
1tan cot 2
sin cos= a + a +
a a. 6/
22
2 2
1 3tantan 1
cos cos
a- a = +
a a.
7/ 2 2 2 2
2 2 2 2
tan tan sin sin
tan tan sin sin
a - b a - b=
a b a b. 8/
( )2
2
1 cos1 cos1 2cot
sin sin
é ù+ aê ú- a
ê ú- = aê úa aê úë û
.
9/ ( )( )2 11 cos 1 cot
1 cos- a + a =
+ a. 10/
2 3
3
sin cos1 tan tan tan
cos
a + a+ a + a + a =
a.
11/ 2
2
sin cos 1 cot
sin cos cos sin 1 cot
a a + a- =
a + a a - a - a. 12/
2 2 4
2 2 2 2
tan 1 cot 1 tan.
1 tan cot tan cot
a + a + a=
+ a a a + a.
13/
2
21 sin 1 sin4 tan
1 sin 1 sin
æ ö÷+ a - aç ÷ç - = a÷ç ÷ç - a + a ÷çè ø. 14/
2
21 cos 1 cos4cot
1 cos 1 cos
æ ö÷+ a - aç ÷ç - = a÷ç ÷ç - a + a ÷çè ø.
s
1/ 4 2 2
1P sin sin cos= a + a a . 2/ 4 4 2
2P sin cos cos= a - a + a .
3/ 2 2 2
3P sin sin cot= a + a a . 4/ 2 2 2
4P cos cos cot= a + a a .
5/ ( )2 2 2
5P 1 sin cot 1 cot= - a a + - a . 6/ 2 2
6P sin tan cos cot 2sin cos= a a + a a + a a .
10 – 2 s. Lê Vă à
Page - 3 -
7/ 2
7
2cos 1P
sin cos
a -=
a + a. 8/
8 2
1 cos 1P
1 cossin
- a= -
+ aa.
9/ 9
cot cotP
tan tan
a + b=
a + b. 10/
10
cosP tan
1 sin
a= a +
+ a.
11/ 11
sin x tan xP sin x cot x
tan x
+= - . 12/
12 2
cosx tan xP cot x cosx
sin x= - .
13/ 2 2
13 2 2
cos x cot xP
sin x tan x
-=
-. 14/
2 22
14 2
1 sin x cos xP cos x
cos x
-= - .
15/ 3 2
15
sin x sin x cos x cosxP
1 2sin x cosx
+ -=
-. 16/
( )2
16
sin x cosx 1P
tan x sin x cosx
+ -=
-.
17/ 2 2 2
17 2 2 2
sin x sin x tan xP
cos x cos x tan x
+=
+. 18/
2 22
18 2
cot x cos xP 1 cos x
cot x
-= + - .
19/ 2 2
2 219 2 2
cos x sin yP cot x cot y
sin x sin y
-= - . 20/
20
1 sin x cosx cot xP
sin x 1 cot x 1 tan x= - -
+ +.
21/ 21
1 cosx 1 sin xP tan x cot x
sin x cosx
- -= + + + . 22/
22
1 1P 1 cot x 1 cot x
sin x sin x
æ öæ ö÷ ÷ç ç÷ ÷= + + + -ç ç÷ ÷ç ç÷ ÷ç çè øè ø.
23/ ( ) ( )2 223
P 1 tan x cos x 1 cot x sin x= + + + . 24/ 24
2 2
1P , 0 x
2sin x cot x cos x
æ öp÷ç ÷= < <ç ÷ç ÷çè ø- -.
s à s
1/ 2A 2cos x 1= - .
2/ 2B 3 4sin x= - .
3/ 2C sin x cosx cos x 1= + - .
4/ 2D sin x sin x cosx 1= + - .
5/ E 1 sin x cosx tan x= + + + .
6/ F tan x cot x sin x cosx= - + + .
7/ ( )2G cosx tan x 1 cosx= - + .
8/ ( ) ( )2H 3 4cos x sin x 2sin x 1= - - + .
9/ 2 2I sin x 3cos x 6 cosx 2 sin x= - + - .
10/ 3 3J cos x sin x sin x cosx= - + + .
11/ 3 2K cos x cos x 2sin x 2= + + - .
12/ 2 3L cos x sin x cosx= + + .
13/ ( )2M 1 cosx cos x sin x 1 cosx= + + - + .
s. Lê Vă à 5. –
Page - 4 - " A e f wer f rr w re e see s f d y "
14/ 3 2N 2cos x 2cos x sin x 1= + + - .
15/ 3 3 2O cos x sin x 2sin x 1= + + - .
16/ ( )( ) 2P 2sin x 1 2cosx 2sin x 1 3 4cos x= - + + - + .
17/ ( )( )Q 2cosx 1 sin x cosx 1= - + - .
18/ 3 3 2R 4sin x 3cos x 3sin x sin x cosx= + - - .
19/ ( ) ( )2S 1 sin x tan x 1 cosx= + - + .
20/ ( ) 2T 2 5sin x 3 1 sin x tan x= - + - .
21/ 2U 2sin x cosx 2sin x 3sin x cosx 1= - + - - .
22/ ( )V tan x 3cot x 4 sin x 3 cosx= - - + .
23/ X 3sin x 2cosx 3tan x 2= + - - .
24/ ( ) ( )Y 2 tan x sin x 3 cot x cosx 5= - + - + .
25/ ( ) ( )Z 3 cot x cosx 5 tan x sin x 2= - - - - .
s à
1/ 4 4 2A cos x sin x 2sin x= - + .
2/ 4 2 2 2B sin x sin x cos x cos x= + + .
3/ 4 2 2 2C cos x sin x cos x sin x= + + .
4/ ( ) ( )4 2 4 2D cos x 2cos x 3 sin x 2sin x 3= - + - .
5/ 6 6 4 4 2E sin x cos x 2sin x cos x sin x= + - - + .
6/ 1 1
F sin x. , 0 x1 cosx 1 cosx 4
æ öp÷ç ÷= + < <ç ÷ç ÷ç+ - è ø.
7/ 4 2 4 2G sin x 4cos x cos x 4sin x= + + + .
8/ 2 2 2 2 2H cos x cot x 5cos x cot x 4sin x= + - + .
9/ ( ) ( )3 3I 1 cot x sin x 1 tan x cos x sin x cosx= + + + - - .
10/ ( )( )4 4 2 2J sin x cos x 1 tan x cot x 2= + - + + .
11/ ( ) ( )8 8 6 6 4K 3 sin x cos x 4 cos x 2sin x 6sin x= - + - + .
12/ ( ) ( )4 2 4 2 2 2L sin x 1 sin x cos x 1 cos x 5sin x cos x 1= + + + + + .
13/ ( ) ( )2
4 4 2 2 8 8M 2 sin x cos x sin x cos x sin x cos x= + + - + .
10 – 2 s. Lê Vă à
Page - 5 -
s à
1/ 2 cot x 1
Atan x 1 cot x 1
+= +
- -.
2/ ( )
22
2 2 2
1 tan x 1B
4tan x 4sin x cos x
-= - .
3/ ( )( )2
2 21 tan xC 1 tan x 1 cot x
tan x
-= - + + .
4/ ( )
22
2 2 2
1 cot x 1D
cot x sin x cos x
-= - .
5/ 6 2
6 2
1 sin x 3tan xE
cos x cos x
-= - .
6/ 2 2 2 2
2 2
tan x cos x cot x sin xF
sin x cos x
- -= + .
7/ 2 2
2
cot x cos x sin x cosxG
cot xcot x
-= + .
8/ 4 4
6 6
sin x cos x 1H
sin x cos
+ -=
+.
Cho ( ) ( )f x 2 1 cosx ,= - ( )g x 2 cosx 3.sin x,= + - ( )h x 2 cosx 3.sin x= + + .
s à ( ) ( ) ( )2 2 2A f x g x h x ,= + +
( ) ( ) ( )4 4 4B f x g x h x ,= + + ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2C f x .g x g x .h x h x .f x= + + .
Cho a sin x sin y bcosx cosy 0- = . r
2 2 2 2
1 1Q
a sin x b cos x a sin y b cos y= +
+ + à .
Cho ( )6 6 4 4Q sin x cos x m sin x cos x= + - + . s Q
à à r Q .
– yê r – ă 1998
Cho 4 4sin x cos x 1
a b a b+ =
+. r
1/
( )
8 8
3 3 3
sin x cos x 1
a b a b
+ =
+
– yê Lê – ă 2001)
2/
( )
10 10
4 4 4
sin x cos x 1
a b a b
+ =
+
– à A s erd – ă 2007
s. Lê Vă à 5. –
Page - 6 - " A e f wer f rr w re e see s f d y "
B – ĐƯƠNG TRON LƯƠNG GIAC – CUNG GOC LƯƠNG GIAC
CUNG LIÊN KÊT
10 – 2 s. Lê Vă à
Page - 7 -
NG
I LIÊN HÊ GI A ĐƠN V Đ VA RA IAN
s s r y 2 3 5 7 13 3 15
, , , , , , , 2, ,3 4 9 5 3 3 4 7 15
p p p 9p p p p p p- .
s s r r d ( )rad : 0 0 0 0 0 0 0 0 090 , 36 , 15 , 72 , 270 , 240 , 540 , 750 , 21- - .
s r s à s r d
0 0 021 30', 75 54', 12 36' 0 0 0, 15 30', 105 45'30'', 27 38'49''- .
s r d r s r s ( )2
radp
à 2,5p .
r à r
00 015 030 045 060 075 090 0120 0150 0180
16
p
8
p
6
p
5
12
p
2
p
2
3
p
3
4
p
5
6
p p
4
3
p
Radian 7
4
p
12
p
5
6
p-
2
p
2
3
p
3
4
p
0135 050- 0810
T NH Đ AI CUNG TRON
r ( )10 cm . dà s à
( ) 0 02 3 7 13 3 3, , , , , 2 , , 45 , rad , 75
3 18 5 3 6 4 2
p p p p p p- p .
dà ( )7 cm dà ( )10 cm .
r 30 t.
e à ( )55 cm . e y ( )/40 km h r
y e y
à à à e dà ( )1,75 m à
( )1,26 m . r 15 à rê r dà
ê
BI U I N NG N CUNG LƯƠNG GIAC
s. Lê Vă à 5. –
Page - 8 - " A e f wer f rr w re e see s f d y "
d s ê r A 0 0 0 0 030 , 45 , 120 , 120 , 330 ,- -
0 0 0 0 07 4 15 2010630 , 750 , 1250 , , , 750 , 1125 , ,
4 3 2 4
p p p p- - - .
s ( ) k2 , k , k , k , k , k2 3 4
p p pp p Î ¢ .
s y rê r k Î ¢ .
1/ x k= p . 2/ x k22
p= + p .
3/ x 4k= p . 4/ x k3
p= + p .
5/ x k26
p= - + p . 6/ x k
4
p= - + p .
7/ k
x3
p= . 8/
k2x
3 3
p p= + .
9/ k3
x4 4
p p= + . 10/
kx
6 3
p p= + .
à s ¼đs AM
6
p= à
¼đk
s AN798
p= k Î ¢ . k Î ¥
1 r . 2 .
s r d 35
3
p à
m
5
p à s yê à
y
G A TR LƯƠNG GIAC C A T CUNG GOC
d r
1/ sin x, cosx, tan x, cot x 3
x2
pp < < .
2/ 3
sin x , cos x , tan x , cot x4 2 2 2
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷+ - - +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø 0 x
2
p< < .
3/ ( )0 0A sin 40 .cos 290= - . 4/ ( )0 0B sin 25 .cos170= - .
5/ ( )0 0 0C sin225 . tan130 .cot 175= - . 6/ ( )0 0 0D cos195 . tan269 .cot 98= - .
7/ ( )0 0E sin 50 .cos 300= - . 8/ 0 21F sin215 . tan
7
p= .
9/ 3 2
G cot .sin5 3
æ öp p÷ç ÷= -ç ÷çè ø. 10/ os
4 4 9H c .sin . tan .cot
5 3 3 5
p p p p= .
r
1/ 1
sin x2
= 0 090 x 180< < . 2/ 4
sin x5
= - 0 0270 x 360< < .
10 – 2 s. Lê Vă à
Page - 9 -
3/ 3
sin x5
= - 3
x2
pp < < . 4/
1cosx
4= 0 x
2
p< < .
5/ 3
cosx5
= 00 x 90< < . 6/ 5
cosx13
= - 0 0180 x 270< < .
7/ 2
cosx5
= x 02
p- < < . 8/
4cosx
5= 0 0270 x 360< < .
9/ 5
sin x13
= x2
p< < p . 10/
1sin x
3= - 0 0180 x 270< < .
11/ tanx 3= 3
x2
pp < < . 12/ tanx 2= - x
2
p< < p .
13/ 1
tan x2
= - x2
p< < p . 14/ cot x 3=
3x
2
pp < < .
15/ 3
tan x4
= 3
x2
pp < < . 16/ tan x 2= - x
2
p< < p .
17/ 2
cot x3
= 0 x2
p< < . 18/ cot x 3= - x
2
p< < p .
19/ cot x 3= - 3
x 22
p< < p . 20/ 0cot15 2 3= +
r s
1/ Cho tanx 2= - . 1 2
5cot x 4 tan x 2sin x cosxA , A
5cot x 4 tan x cosx 3sin x
+ += =
- -.
2/ Cho cot x 2= . 1 2
3sin x cosx sin x 3cosxB , B
sin x cosx sin x 3cosx
- -= =
+ +.
3/ Cho cot x 2= . 1 2 2
2sin x 3cosx 2C , C
3sin x 2cosx cos x sin x cosx
+= =
- -.
4/ Cho tanx 2= . 1
2sin x 3cosxD ,
4sin x 5cosx
+=
-
2 3 3
3sin x 2cosxD ,
5sin x 4cos x
-=
+
3 3
3 2
sin x cos xD ,
sin x sin x cosx
-=
+
4 3 3
sin x 5cosxD ,
sin x 2cos x
+=
-
3 3
5 3
8cos x 2sin x cosxD ,
2cosx sin x
- +=
-
3 3
6 3 2
sin x 2cos x 2sin x 3cosxD
sin x cos x 5sin x
- + +=
- +.
5/ Cho 3
sin x , 0 x5 2
p= < < .
cot x tan xE
cot x tan x
+=
-.
6/ Cho 0 01sin x , 90 x 180
3= < < .
28tan x 3cot x 1F
tan x cot x
+ -=
+.
7/ Cho 2
cosx3
= - . cot x 3 tan x
G2cot x tan x
+=
+.
s. Lê Vă à 5. –
Page - 10 - " A e f wer f rr w re e see s f d y "
8/ Cho 02sin x , 0 x 90
3= < < .
1 2 2
tan x cosx tan x cosxH , H cosx cot x
cot x sin x
-= = - .
9/ Cho 4
cosx , x5 2
p= - < < p .
1 2
cot x tan x sin xI , I cot x
cot x tan x 1 cosx
+= = +
- +.
Cho 5
sin x cosx4
+ = . y r s
1/ A sinx.cosx= . S: 9
A32
= .
2/ B sinx cosx= - . S: 7
B4
= ± .
3/ 3 3C sin x cos x= - . S: 41 7
C128
= ± .
Cho tanx cot x 3- = . y r s
1/ 2 2A tan x cot x= + . S: A 11= .
2/ B tan x cot x= + . S: B 13= ± .
3/ 4 4C tan x cot x= - . S: C 33 13= ± .
Cho sin x cosx m+ = . y e r
1/ A sinxcosx= . 2/ 3 3B sin x cos x= + .
3/ 4 4C sin x cos x= + . 4/ D sin x cosx= - .
5/ 2 2E tan x cot x= + . 6/ 6 6F sin x cos x= + .
sin x, cosx, tan x, cot x . r
1/ sin x cosx 2+ = . 2/ sin x cosx 2- = .
3/ 1
sin x cosx2
+ = . y – ă 2005
4/ 1
sin x cosx5
+ = . S: 4 3 4 3
; ; ;5 5 3 4
- - - .
5/ tan x cot x 4+ = . S: 1 2 3
; ; 2 3; 2 322 2 3
-+ -
-
.
Cho tanx 2cot x 1- = - . y
1/ 2 2A tan x cot x= - . 2/ 3 3B tan x cot x= + .
3/ 4 4C tan x 2cot x= + . 4/ 5 5D tan x 3cot x= - .
r s y
1/ Cho 4 4 33sin x cos x
4+ = . 4 4A sin x 3cos x= + . S: A
7
4= .
10 – 2 s. Lê Vă à
Page - 11 -
2/ Cho 4 4 13sin x cos x
2- = . 4 4B sin x 3cos x= + . S: B 1= .
3/ Cho 4 4 74sin x 3cos x
4+ = . 4 4C 3sin x 4cos x= + . S:
7 57C C
4 28= Ú = .
CUNG GOC LIÊN KÊT
d s d d k2a + p k Î ¢ .
3 5 11 14 21
, , , ,2 4 3 5 6
p p p p p- .
d s d d 0k360a + k Î ¢ .
0 0 0 0 0370 , 512 , 765 , 1000 , 1234- .
d s d d k2a + p k Î ¢ .
8 11 7 18 27 25
, , , , ,3 5 2 7 10 4
p p p p p p- - .
ê d y y r s
1/ 0sin150 . 2/ 0cot 135 . 3/ 0cos225 . 4/ 0tan210 .
5/ 0cot 225 . 6/ 0sin240 . 7/ 0cos315 . 8/ 0tan 300 .
9/ ( )0cot 1380- . 10/ cos11p . 11/ sin13p . 12/ tan10p .
13/ 7
cot6
p. 14/
11cos
3
p. 15/
25cot
4
p. 16/
17sin
3
p.
17/ 29
cos6
p. 18/
45tan
4
p. 19/
16cos
3
æ öp÷ç ÷-ç ÷ç ÷çè ø. 20/
31sin
2
æ öp÷ç ÷-ç ÷ç ÷çè ø.
21/ 19
cot4
æ öp÷ç ÷-ç ÷ç ÷çè ø. 22/
26cos
3
æ öp÷ç ÷-ç ÷ç ÷çè ø. 23/
159tan
4
æ öp÷ç ÷-ç ÷ç ÷çè ø. 24/
115sin
6
æ öp÷ç ÷-ç ÷ç ÷çè ø.
r s r
1/ ( )0sin x 90- . 2/ ( )0cos 180 x+ . 3/ ( )0sin 270 x- . 4/ ( )0sin x 180- .
5/ ( )0cos x 540+ . 6/ ( )0cot 180 x+ . 7/ ( )0sin x 540- . 8/ ( )0tan 360 x- .
9/ ( )0cos 450 x+ . 10/ ( )2 0sin 270 x+ . 11/ ( )3 0cos 90 x+ . 12/ ( )5 0cot 180 x- .
r s r
1/ ( )cot x - p . 2/ ( )sin xp + . 3/ ( )tan 2 xp - . 4/ ( )cot 3 xp - .
5/ ( )sin x 7- p . 6/ ( )tan x 5- p . 7/ 5
sin x2
æ öp ÷ç ÷+ç ÷ç ÷çè ø. 8/
3cos x
2
æ öp ÷ç ÷+ç ÷ç ÷çè ø.
9/ 3
cot x2
æ öp÷ç ÷-ç ÷ç ÷çè ø. 10/
5cos x
2
æ öp÷ç ÷-ç ÷ç ÷çè ø. 11/
11tan x
2
æ öp ÷ç ÷+ç ÷ç ÷çè ø. 12/
7sin x
2
æ öp÷ç ÷+ç ÷ç ÷çè ø.
s. Lê Vă à 5. –
Page - 12 - " A e f wer f rr w re e see s f d y "
13/ ( )2sin xp + . 14/ ( )9cos xp - . 15/ ( )11cot x 3- p . 16/ ( )4cos x 3+ p .
17/ ( )2cot x 5- p . 18/ ( )6cos x - p . 19/ 8cos x2
æ ö3p÷ç ÷-ç ÷ç ÷çè ø. 20/ 5cos x
2
æ öp ÷ç ÷-ç ÷ç ÷çè ø.
21/ 2012sin x2
æ öp÷ç ÷-ç ÷ç ÷çè ø. 22/ 2 9
tan x2
æ öp÷ç ÷-ç ÷ç ÷çè ø. 23/ 2013 7
cos x2
æ öp÷ç ÷+ç ÷ç ÷çè ø. 24/ 1991 5
sin x2
æ öp ÷ç ÷-ç ÷ç ÷çè ø.
25/ 2015 11cos x
2
æ öp÷ç ÷-ç ÷ç ÷çè ø. 26/ 2 9
cot x2
æ öp÷ç ÷-ç ÷ç ÷çè ø. 27/ 11 11
tan x2
æ öp ÷ç ÷+ç ÷ç ÷çè ø. 28/ ( )cos x 2012- p .
1/ ( )A cos x sin x2
æ öp÷ç ÷= - + - pç ÷ç ÷çè ø.
2/ B cos x sin x cos x sin x2 2 2 2
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= - + - - + - +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
3/ ( )7 3
C 2cosx 3cos x sin x tan x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + p - - - + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
4/ 3
D 2sin x sin(5 x) sin x cos x2 2 2
æ ö æ ö æ öp p p÷ ÷ ÷ç ç ç÷ ÷ ÷= + + p - + + + +ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø.
5/ 7 3
E 2cosx 3cos( x) 5sin x cot x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= - p - + - + -ç ç÷ ÷ç çè ø è ø.
6/ ( ) ( )3
F sin 5 x cos x cot 3 x tan x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= p + + - + p - + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
7/ ( )3 11
G cos 15 x sin x tan x cot x2 2 2
æ ö æ ö æ öp p p÷ ÷ ÷ç ç ç÷ ÷ ÷= p - + - - + -ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø.
8/ ( ) ( )3
H sin x cos x cot 2 x tan x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= p + - - + p - + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
9/ ( ) ( )3 3
I cos 5 x sin x tan x cot 3 x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= p - - + + - + p -ç ç÷ ÷ç çè ø è ø.
10/ ( ) ( ) ( ) ( )0 0 0 0J cos 270 x 2sin x 450 cos x 900 2sin 270 x= - - - + + + - .
11/ ( )2 2 2 23K sin x sin x sin x sin x
4 2 4
æ ö æ ö æ öp p p÷ ÷ ÷ç ç ç÷ ÷ ÷= + + + + + + + pç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø.
12/ 2 2 2 2 2 22 5 7L sin sin sin sin sin sin
3 6 9 9 8 8
p p p p p p= + + + + + .
13/ ( ) ( )2013 2013 2012 2011M cos x cos x .sin x sin x2
æ öp ÷ç ÷= + p + p + - -ç ÷ç ÷çè ø.
10 – 2 s. Lê Vă à
Page - 13 -
14/ ( ) ( ) ( )6 6 4 4 23N sin x cos x 2sin x 2 sin x cos x
2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= p + + - p - + p - - + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
15/
( ) ( )
( )
19tan x .cos 36 x .sin x 5
2O
9sin x .cos x 99
2
æ öp ÷ç ÷- p - - pç ÷ç ÷çè ø=
æ öp ÷ç ÷- - pç ÷ç ÷çè ø
.
16/ ( ) ( )2 285 3P sin x cos 207 x sin 33 x sin x
2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + + p + + p + + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
à r d y
1/ 0 0 0 0A cos0 cos20 cos40 ...... cos180= + + + + .
2/ 0 0 0 0B cos20 cos40 cos60 ...... cos180= + + + + .
3/ 0 0 0 0C cos10 cos40 cos70 ...... cos170= + + + + .
4/ 0 0 0 0D tan20 tan40 tan60 ...... tan180= + + + + .
5/ 0 0 0 0E cot15 cot 30 cot 45 ...... cot165= + + + + .
6/ 0 0 0 0F sin5 sin10 sin15 ...... sin360= + + + + .
7/ 0 0 0 0G cot195 cot 210 cot 225 ...... cot 345= + + + + .
8/ 0 0 0 0H cot15 .cot 35 .cot 55 .cot 75= .
9/ 0 0 0 0I tan10 . tan20 . tan 30 ...... tan 80= .
10/ 0 0 0 0J tan1 . tan2 . tan 3 ...... tan 89= .
11/ 2 0 2 0 2 0 2 0K sin 28 sin 36 sin 54 cos 152= + + + .
12/ 2 0 2 0 2 0 2 0L cos 2 cos 4 cos 6 ...... cos 88= + + + + .
13/ 2 0 2 0 2 0 2 0M sin 10 sin 20 sin 30 ...... sin 90= + + + + .
14/ 2 0 2 0 2 0 2 0N cos 10 cos 20 cos 30 ... cos 180= + + + + .
15/ 0 0 0 0 0O sin20 sin40 sin60 ... sin340 sin360= + + + + + .
à r d y
1/ ( )0 0A cos 315 .sin 765= - .
2/ 0 0 0 0B sin 32 .sin148 sin 302 .sin122= - .
3/ 0 0 0 0C sin810 .cos540 tan135 .cot 585= + .
4/ ( ) ( )0 0 0 0D sin 825 .cos 15 cos75 .sin 555= - + - .
5/ 0 0 0E 2tan540 2cos1170 4sin990= + + .
6/ ( )0 0
0
0 0
sin 234 cos216F . tan 36
sin144 cos126
- -=
-.
s. Lê Vă à 5. –
Page - 14 - " A e f wer f rr w re e see s f d y "
7/ ( )0 0
0
0 0
sin 324 cos666G .cot 36
sin1206 cos36
- +=
+.
8/ ( ) ( ) ( )
( )
0 0 0 0
0 0
sin 328 .sin 958 cos 508 .cos 1022H
cot 572 tan 212
- - -= -
-.
9/ ( )( )
0 0
0
0 0
cos 288 .cot 72I tan18
tan 142 .sin108
-= -
-.
10/ ( ) ( ) ( ) ( )0 0 0 0J 2sin 790 x cos 1260 x tan 630 x . tan 1260 x= + + - + + - .
11/ ( )0 0
0 0 0
2sin2550 .cos 1881K
tan 368 2cos638 cos98
-= +
+.
12/ ( )0 0 0
0 0
0
cos44 tan226 .cos406L cos72 .cot 18
cos316
+= - .
13/ ( )0 0 0 0
0 0
0
tan 46 .sin 44 cot 136 .sin 404M tan 36 . tan 54
cos316
+ -= - .
14/ ( ) ( ) ( )
( )
0 0 0 0
0 0
sin 328 .sin 958 cos 508 .cos 1022N
cos572 tan 212
- - -= -
-.
15/
4 17sin cos
3 6O23 10
sin cos6 3
p p+
=æ öp p÷ç ÷+ -ç ÷ç ÷çè ø
.
16/ ( ) ( )
( )( ) ( )
( )
sin 4,8 .sin 5,7 cos 6,7 .cos 5,8P
cot 5,2 tan 6,2
- p - p - p - p= +
- p - p.
r A à
1/ ( )sin B sin A C= + . 2/ ( )cos A B cosC+ = - .
3/ A B C
sin cos2 2
+= . 4/ ( ) ( )cos B C cos A 2C- = - + .
5/ ( )cos A B C cos2C+ - = - . 6/ ( )sin A 2B C sin B+ + = - .
7/ ( )cot A B C cot 2B- + = - . 8/ 3A B C
cos sin2A2
- + += - .
9/ A B 3C
sin cosC2
+ += . 10/
A B 2C 3Ctan cot
2 2
+ -= .
11/ A 3B C
cos sin B2
+ += - . 12/
A 2B C 3Bcot tan
2 2
- += .
10 – 2 s. Lê Vă à
Page - 15 -
NG
d y . y r
1/ 0 0 0 0A sin12 .cos48 cos12 .sin 48= + .
2/ 0 0 0 0B cos38 .cos22 sin38 .sin22= - .
3/ 0 0 0 0C sin10 .cos55 cos10 .sin55= - .
4/ 0 0 0 0D sin36 .cos6 sin126 .cos84= - .
5/ 0 0 0 0E cos112 .cos23 sin112 .sin23= - .
6/ 0 0 0 0F sin200 .sin 310 cos340 .cos50= + .
7/ 0 0 0 0 0G cos11 .cos21 cos69 .cos79 cos10= + - .
8/ 0 0 0 0 0H cos68 .cos78 cos22 .cos12 sin100= + - .
9/ ( ) ( )0 0 0 0I cos 53 .sin 337 sin 307 .sin113= - - + .
r s
1/ 015 . 2/ 075 . 3/ 0105 . 4/ 0285 .
5/ 19
12
p. 6/
5
12
p. 7/
7
12
p. 8/
13
12
p.
r s
1/ 0
0
1 tan15A
1 tan15
+=
-. 2/
0 0
0 0
tan25 tan20B
1 tan25 tan20
+=
-.
3/ 0 0 0 0
0 0 0 0
sin10 cos20 sin20 cos10C
cos17 cos13 sin17 sin13
+=
-. 4/
0 0 0
0 0
tan 225 cot 81 cot 69D
cot 261 tan 201
-=
+.
5/ 0 0 0 0
0 0 0 0
sin 73 cos3 sin 87 cos17E
cos132 cos62 cos42 cos28
-=
+. 6/
0 o0 0
0 0
cot 225 cot 79 ot 71F
cot 259 cot 251
-=
+.
C – CÔNG THƯC C NG
.
.
. .
:
. .
s. Lê Vă à 5. –
Page - 16 - " A e f wer f rr w re e see s f d y "
7/ 2 0 2 0G cos 75 sin 75= - . 8/ 2 0 2 0 2 0H sin 20 sin 100 sin 140= + + .
9/ 2 0 0 2 0I cos 10 cos110 cos 130= + + .
10/ 0 0 0 0 0 0J tan20 . tan 80 tan 80 . tan140 tan140 . tan20= + + .
11/ 0 0 0 0 0 0K tan10 . tan 70 tan 70 . tan130 tan130 . tan190= + + .
12/ 0 0 0 0 0 0L sin160 cos110 sin250 cos340 tan110 tan 340= + + .
13/ ( )( ) ( )( )0 0 0 0 0 0 0 0M cos70 cos50 cos310 cos290 cos40 cos160 cos320 cos380= + + + + - .
r s
1/ A cosx x3
æ öp÷ç ÷= +ç ÷ç ÷çè ø
1sin x
3= à 0 x
2
p< < .
2/ B sin x3
æ öp ÷ç ÷= -ç ÷ç ÷çè ø
12cosx
13= - à
3x
2
pp < < .
3/ ( )0C cos x 30= - tan x 2= à 00 x 90< < .
4/ D tan x3
æ öp÷ç ÷= +ç ÷çè ø
3sin x
5= à x
2
p< < p .
5/ E cos x3
æ öp ÷ç ÷= -ç ÷çè ø
12sin x
13= - à
3x 2
2
p< < p .
6/ F cot x4
æ öp÷ç ÷= -ç ÷ç ÷çè ø
4sin x
5= - à
3x
2
pp < < .
7/ G tan x4
æ öp÷ç ÷= +ç ÷ç ÷çè ø
5cot x 2
2
æ öp ÷ç ÷- =ç ÷ç ÷çè ø.
8/ 7
H sin 2x4
æ öp÷ç ÷= +ç ÷ç ÷çè ø
2cot x
3= .
9/ ( ) ( )I cos a b .cos a b= + - 1
cosa3
= à 1
cosb4
= .
( ) 0 04sin , 0 90
5a = < a < à ( ) 0 08
sin , 90 18017
b = < b < . y r
( )A cos= a + b à ( )B sin= a - b .
8 5
sin , tan b17 12
a = = à ,a b à . y r
( ) ( ) A sin , B cos= a - b = a + b à ( )C tan= a + b .
Cho 0 , ,2 4
p p< a b < a + b = à tan . tan 3 2 2a b = - .
1 y ( ) tan ; tan tana + b a + b . 2 tan ; tana b . Suy ra a à b .
10 – 2 s. Lê Vă à
Page - 17 -
Cho 3
pa - b = . r
1/ ( ) ( )2 2
A cos cos sin sin= a + b + a + b . 2/ ( ) ( )2 2
B cos sin cos sin= a + b + b- a .
1/ A sin x 3cosx= - . 2/ B 3 sin7x cos7x= - .
3/ ( ) 2 2C a sin x b cosx, a b 0= + + ¹ . 4/ D 3 sin x sin x3 6
æ ö æ öp p÷ ÷ç ç÷ ÷= - + +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
5/ E cos7x.cos5x 3 sin2x sin7x.sin5x= - + . 6/ F 3 cos2x sin 2x 2sin 2x6
æ öp÷ç ÷= + + -ç ÷ç ÷çè ø.
7/ G 2sin 2x 4sin x 16
æ öp÷ç ÷= - + +ç ÷ç ÷çè ø. 8/ H sin2x 2 2 cosx 2sin x 3
4
æ öp÷ç ÷= + + + +ç ÷ç ÷çè ø.
1/ A sinxcos5x cosxsin5x= - .
2/ B sin4xcot 2x cos4x= - .
3/ C cos6x tan3x sin6x= - .
4/ ( ) ( ) ( ) ( )D sin x y cos x y sin x y cos x y= + - + - + .
5/ ( ) ( ) ( ) ( )0 0 0 0E cos 40 x cos x 20 sin 40 x sin x 20= - + - - + .
6/ ( ) ( ) ( ) ( )0 0 0 0F sin 14 2x cos 16 2x cos 14 2x sin 16 2x= + - + + - .
7/ ( ) ( ) ( ) ( )0 0 0 0G sin x 10 cos 2x 80 sin x 100 cos 2x 10= + - + + + .
8/ H sin x cos x sin x cos x3 4 4 3
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= - - + - -ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
9/ 3
I cos x cos x cos x cos x3 4 6 4
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= - + + + +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
10/ 9 5 5
J sin x cos x sin x cos x3 4 4 3
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= - - - - +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
11/ K cos x cos x cos x cos x3 4 6 4
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= - - + + +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
12/ 13 13 3
L cos x cos x cos x cos x3 4 6 4
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= - + + + +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
s
1/ tan 3x tan x
A1 tan x tan 3x
-=
+. 2/
tan2x 1B
1 tan2x
+=
-.
s. Lê Vă à 5. –
Page - 18 - " A e f wer f rr w re e see s f d y "
3/ ( )
( )
0
0
tan 2x cot 90 xC
1 cot 90 2x tan x
+ +=
+ -. 4/
2 2
2 2
tan 2x tan xD
1 tan 2x tan x
-=
-.
s
1/ ( ) ( )A sin a b sin a sin b2
æ öp ÷ç ÷= + + - -ç ÷ç ÷çè ø.
2/ ( )B cos a cos b cos a b2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= - - - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
3/ 21C cos a cos a sin a
4 4 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + - +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
4/ 2 2 2 2D sin a sin b cos a cos b= - .
5/ ( )
( ) ( )
2sin a bE tan b
cos a b cos a b
+= -
+ + -.
s
1/ 2 2A cos x 3 sin2x sin x= - - . 2/ 3B 4sin x 3sin x 3 cos3x= - + .
3/ ( ) ( )0 0C sin x 45 cos x 45= + + + . 4/ D tan3x tanx sin2x= - - .
5/ 2E tan2x cot x 8cos x= + - .
s
1/ sin2x 2sinxcosx= .
2/ 2 2cos2x cos x sin x= - .
3/ 2
2tan xtan2x
1 tan x=
-.
4/ 3sin3x 3sin x 4sin x= - .
5/ 3cos3x 4cos x 3cosx= - .
6/ cosx sin x 2 cos x 2 sin x4 4
æ ö æ öp p÷ ÷ç ç÷ ÷+ = - = +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
7/ cosx sin x 2 cos x 2 sin x4 4
æ ö æ öp p÷ ÷ç ç÷ ÷- = + = - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
8/ ( ) ( ) 2 2 2 2sin x y sin x y sin x sin y cos y cos x+ - = - = - .
9/ ( ) ( ) 2 2 2 2cos x y cos x y cos x sin y cos y sin x+ - = - = - .
s
1/ sin x sin x 2 sin x4 4
æ ö æ öp p÷ ÷ç ç÷ ÷+ - - =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
10 – 2 s. Lê Vă à
Page - 19 -
2/ 24sin x sin x 4sin x 33 3
æ ö æ öp p÷ ÷ç ç÷ ÷+ - = -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
3/ ( ) ( ) ( )sin x sin y z sin y sin z x sin zsin x y 0- + - + - = .
4/ ( ) ( ) ( )cosx sin y z cosy sin z x coszsin x y 0- + - + - = .
5/ ( ) ( )tan x y tan x tan y tan x y tan x tan y+ - - = + .
6/ tan 2x tan x tan 2x tan x tan x tan x 16 3 3 6
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷- + - + - - =ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
7/ 2 2
tan x. tan x tan x . tan x tan x . tan x 33 3 3 3
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷+ + + + + + = -ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
8/ ( )3 2cos x .cos x cos x .cos x 1 3
3 4 6 4 4
æ ö æ ö æ ö æ öp p p p÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷- + + + + = -ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø.
9/ ( )( ) ( )( )o o o o o o o ocos70 cos50 cos230 cos290 cos40 cos160 cos320 cos380 0+ + + + + = .
s
1/ ( )( )
cos a b cot a cot b 1
cot a cot b 1cos a b
- +=
-+.
2/ ( ) ( ) ( )sin a b sin b c sin c a
0cosa cosb cosb cosc cosccosa
- - -+ + = .
3/ ( ) ( ) 2 2
2 2
sin a b sin a btan a tan b
cos a cos b
+ -= - .
4/ ( ) ( ) 2 2
2 2
cos a b cos a b1 tan a tan b
cos a cos b
+ -= - .
Ch ng minh các h th c sau, v u ki r c
1/ ( )cos a b 0+ = ( )sin a 2b sin a+ = .
2/ ( )sin 2a b 3sin b+ = ( )tan a b 2tan a+ = . HD: ( )
( )b a b a
2a b a b a
ìï = + -ïïíï + = + +ïïî
.
3/ tana 2tanb= ( ) ( )sin a b 3sin a b+ = - .
4/ 1
tan a. tan b3
= - ( ) ( )cos a b 2cos a b+ = - . HD r .
5/ ( )5sin b sin 2a b= + ( )3
tan a b tan a2
+ = .
6/ ( )sin b sin a cos a b= + ( )2tan a tan a b= + . HD: ( )b a b a= + - .
s. Lê Vă à 5. –
Page - 20 - " A e f wer f rr w re e see s f d y "
7/ ( )cos 2a b 1+ = ( )b
tan a b tan a 2tan2
+ - = .
8/ ( ) ( )cos a b kcos a b+ = - 1 k
tan a tan b1 k
-=
+. HD:
( )( )
a 2b a b b
a a b b
ìï + = + +ïïíï = + -ïïî
.
s
1/ ( )sin a b c
tan a tan b tan c tan a tan b tan ccosa cosb cosc
+ ++ + - = .
2/ ( ) ( )2 2 2cos x cos x y cos y 2cosxcosycos x y 1+ + + - + = .
r yê Lê – . ă 2007
s
1/ 2A sin x cos x cos x3 3
æ ö æ öp p÷ ÷ç ç÷ ÷= + - +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
2/ 2 2 2B cos x cos x cos x3 3
æ ö æ öp p÷ ÷ç ç÷ ÷= + + + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
3/ 2 2 22 2C sin x sin x sin x
3 3
æ ö æ öp p÷ ÷ç ç÷ ÷= + + + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
4/ 2 2 22 2D cos x cos x cos x
3 3
æ ö æ öp p÷ ÷ç ç÷ ÷= + + + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
5/ 3 3a cos x cos3x a sin x sin 3x
F , a constcosx sin x
- += + = .
r tan x tan x tan x tan 3x3 3
æ ö æ öp p÷ ÷ç ç÷ ÷- + =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. r
0 0 0P tan10 tan50 tan110= .
A A,B,C à . ng minh
1/ sinC sin A.cosB sin B.cosA= + .
2/ sin A sin BcosC sinCcosB= + .
3/ cosA sinBsinC cosBcosC= - .
4/ ( ) 0sinCtan A tan B, A,B 90
cosA.cosB= + ¹ .
5/ ( ) 0tan A tan B tanC tan A. tan B. tanC, A,B,C 90+ + = ¹ .
6/ cot A.cot B cot B.cot C cot C.cot A 1+ + = .
7/ A B C B C
sin cos cos sin sin2 2 2 2 2
= - .
8/ A B C B C
cos sin cos cos sin2 2 2 2 2
= - .
10 – 2 s. Lê Vă à
Page - 21 -
9/ A B B C C A
tan . tan tan . tan tan . tan 12 2 2 2 2 2
+ + = .
10/ A B C A B C
cot cot cot cot .cot .cot2 2 2 2 2 2
+ + = .
11/ ( ) 0cosC cosBcot B cot C , A 90
sin B.cosA sinC.cosA+ = + ¹ .
12/ A B C A B C A B C A B C
cos .cos .cos sin sin cos sin cos sin cos sin sin2 2 2 2 2 2 2 2 2 2 2 2
= + + .
13/ 2 2 2A B C A B Csin sin sin 1 2sin sin sin
2 2 2 2 2 2+ + = + .
14/ sin AcosBcosC sin BcosCcosA sinCcosAcosB sin Asin BsinC+ + = .
A A,B,C à . ng minh
1/ tan A tan B tanC 3 3, A,B,C+ + ³ " .
2/ 2 2 2tan A tan B tan C 9, A,B,C+ + ³ " .
3/ 6 6 6tan A tan B tan C 81, A,B,C+ + ³ " .
4/ 2 2 2A B Ctan tan tan 1
2 2 2+ + ³ .
5/ A B C
tan tan tan 32 2 2+ + ³ .
HD:
1/ , 2/ , 3/ , s d tan A tan B tanC tan A.tan B.tanC+ + = à
Cauchy.
4 / s d 2 2 2a b c ab bc ca+ + ³ + + à
A B B C C Atan . tan tan . tan tan . tan 1
2 2 2 2 2 2+ + = .
5 / r
2A B C
tan tan tan2 2 2
æ ö÷ç ÷+ +ç ÷çè ø à s d 3 / .
s. Lê Vă à 5. –
Page - 22 - " A e f wer f rr w re e see s f d y "
NG
R T G N VA T NH GIA TR C A BI U THƯC
r s d y
1/ 2 2A cos sin8 8
p p= - . 2/ 0 3 0B 3cos10 4cos 10= - .
3/ ( )0 2 0C sin20 1 4cos 20= - . 4/ 3 0 0D 4sin 40 3sin130= + .
5/ 3 0 0E 4sin 50 3cos140= + . 6/ 0
2 0
tan15F
1 tan 15=
-.
7/ 2
7tan
8G
1 tan8
p
=p
-
. 8/ 0 0
1 3H
sin10 cos10= - .
9/ 0
0
1I 4sin 70
sin10= - . 10/ 2 2 5
J tan tan12 12
p p= + .
11/ 2 0 2 0K tan 36 tan 72= . 12/ 2 0 2 0 0L cos 70 sin 40 sin100= + .
13/ 2 0 2 0 0M cos 20 2sin 55 2 sin65= + - . 14/ 0 0 0 0N sin 6 .sin 42 .sin 66 .sin 78= .
D – CÔNG THƯC NHÂN
Công thưc nhân đôi
.
.
.
.
Công thưc ha bâc Công thưc nhân ba ơ r
. .
. .
. .
10 – 2 s. Lê Vă à
Page - 23 -
r s d y
1/ 0 0A cos36 cos72= . 2/ 2
B cos cos5 5
p p= .
3/ C sin .cos cos8 8 4
p p p= . 4/ D sin .cos .cos
16 16 8
p p p= .
5/ 0 0 0E sin10 sin 50 sin 70= . 6/ 2 4
F cos cos cos7 7 7
p p p= .
7/ 4 5
G cos cos cos7 7 7
p p p= . 8/ 0 0 0 0H sin6 cos12 cos24 cos48= .
9/ 0 0 0 0I sin 6 sin 42 sin 66 sin 78= . 10/ 0 0 0 0 0J sin 5 .sin15 .sin25 .... sin 75 .sin 85= .
11/ 0 0 0 0 0K cos10 .cos20 .cos30 ...cos70 .cos80= .
12/ 0 0 0 0L 8tan18 cos18 cos36 cos72= .
13/ 0 0 0 0M cos20 cos40 cos60 cos80= .
14/ 2 3 4
N cos cos cos cos9 9 9 9
p p p p= .
15/ O 96 3 sin cos cos cos cos48 48 24 12 6
p p p p p= .
16/ 2 4 8 16 32
P cos .cos .cos .cos .cos31 31 31 31 31
p p p p p= .
17/ 2 4 8 16
Q cos cos cos cos cos33 33 33 33 33
p p p p p= .
18/ 2 3 4 5 6 7
R cos .cos .cos .cos .cos .cos .cos15 15 15 15 15 15 15
p p p p p p p= .
r s d y
1/ 0 0
0 0 0 0
cos80 cos20A
cos35 cos15 sin 35 sin15
+=
-.
2/ 0
0 0
4 0 4 0
sin 60B 3sin15 cos15
sin 15 cos 15= -
-.
3/ 0 0 0 0C tan 9 tan27 tan 63 tan 81= - - + .
4/ D 4tan 2tan tan cot8 16 32 32
p p p p= + + - .
à ă 2000
5/ ( )2 0 0 0 2 0
2 0 2 0
cos 696 tan 260 tan 530 cos 156E
tan 252 cot 342
+ - -=
+.
s. Lê Vă à 5. –
Page - 24 - " A e f wer f rr w re e see s f d y "
1/ ( )2
A sin x cosx= + . 2/ 2 2B 1 4sin xcos x= - .
3/ C sinxcosxcos2x= . 4/ 4 4D cos 2x sin 2x= - .
5/ 2 2E cos x sin x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + - +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. 6/ F sinxcosxcos2x= .
7/ G 4sin x sin x sin 2x2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. 8/ 2 2x x
H sin sin8 2 8 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
9/ ( )2 0I sin 8x 2cos 45 4x= + + . 10/ ( )J sin2x cos2x 2cosx sin x cosx 1= + - + + .
r s d y
1/ 22cos x 1
Asin x cosx
-=
+. 2/
21 2sin 2xB
cos2x sin2x
-=
-.
3/ ( )( )
2C
1 tan x 1 cot x=
- +. 4/ ( )2D 1 tan x cot x= - .
5/ sin2x cos2x
Esin x cosx
= - . 6/ cot x tan x
Fcos2x
-= .
7/
31 sin x
2G
1 sin x2
æ öp ÷ç ÷+ -ç ÷ç ÷çè ø=
æ öp ÷ç ÷+ +ç ÷ç ÷çè ø
. 8/ 2 2
2 2
sin 2x 4sin xH
sin 2x 4sin x 4
-=
+ -.
9/ sin 4x cos2x
I .1 cos4x 1 cos2x
=+ +
. 10/ sin 3x cos3x
Jsin x cosx
-=
+.
11/
1 cos x2x
K tan .4 2
sin x2
æ öp ÷ç ÷+ +ç ÷çæ ö ÷çp è ø÷ç ÷= +ç ÷ç æ ö÷ç pè ø ÷ç ÷+ç ÷ç ÷çè ø
. 12/ 1
L cot 2x tan x2sin2x
= + + .
13/ 1 cos2x sin2x
M .cot x1 cos2x sin2x
- +=
+ +. 14/
( )22 sin 2x 2cos x 1N
cosx sin x cos3x sin 3x
+ -=
- - +.
15/ O 1 sin x 1 sin x, 0 x4
æ öp÷ç ÷= + + - < <ç ÷ç ÷çè ø. 16/
1 1 1P cot 8x
sin2x sin 4x sin 8x= + + + .
17/ O 1 sin2x 1 sin2x, x4 4
æ öp p÷ç ÷= - + + - < <ç ÷ç ÷çè ø.
Tính giá tr c a bi u th ng giác, khi bi t
1/ 3
sin2x, cos2x khi sin x , x5 2
p= < < p .
2/ sin2x, cos2x khi sin x cosx 2+ = .
10 – 2 s. Lê Vă à
Page - 25 -
3/ 5 3
cos2x, sin2x, tan2x khi cosx , x13 2
p= - p < < .
4/ cos2x, sin2x, tan2x khi tan x 2= .
5/ 4 3
sin x, cosx khi sin2x , x5 2 2
p p= - < < .
6/ 7
cos2x, sin2x, tan2x khi tan x8
= .
7/ sin2x, cos2x, tan2x khi tan x 2 3, 0 x2
p= - < < .
Tính giá tr c a bi u th ng giác, khi bi t
1/ 5 5A sin xcosx cos xsin x= - x16
p= .
2/ 4 3 3 4B sin x sin x cosx cos x sin x cos x= + - + x48
p= .
3/ C cos4x= tan x cot x 3+ = .
4/ ( )0D cos 270 4x= + ( )0cot 45 x 2+ = .
5/ 4 4E sin x cos x= + sin x cosx m+ = m Î ¡ .
6/ tan x cot x
Ftan x cot x
-=
+
3sin x cosx , 0 x
4 4
æ öp÷ç ÷= < <ç ÷ç ÷çè ø.
r 0 5 1
sin184
-= .
r d y
1/ 0cos15 . 2/ 0cos18 . 3/ 0sin36 . 4/ 0cos36 .
x
tan2
1/ 4
cosx , 0 x5 2
p= < < . 2/
24 3tan x , x
7 2
p= p < < .
3/ 7
sin x cosx , 0 x2 6
p+ = < < .
Cho 1
tan x , 0 x2 2
æ öp÷ç ÷= < <ç ÷ç ÷çè ø à
1tan y , 0 y
3 2
æ öp÷ç ÷= < <ç ÷ç ÷çè ø.
x y+ . 2/ ( )sin x 2y- ( )cos 2x y+ .
1/ 0 x
2tan x 2 1
ìï pï < <ïïíïï = -ïïî
. 2/ 0 x
26 2
cosx4
ìï pï < <ïïïíï +ï =ïïïî
.
s. Lê Vă à 5. –
Page - 26 - " A e f wer f rr w re e see s f d y "
e cos2x s
1/ 2A 1 cos x= + . 2/ 2 2B sin xcos x= .
3/ 2
2
1 sin xC
cos x
+= . 4/
2
2
1 cot xD
1 cot x
+=
-.
5/ 6 6E sin x cos x= + . 6/ 6 2 6 2F sin x cos x cos x sin x= + .
e x
t tan2
= s
1/ A sinx= . 2/ B cosx= .
3/ sin x
C3 2cosx
=-
. 4/ D tan x cot x= + .
5/ 1 tan x
E1 cot x
-=
+. 6/
tan x sin xF
tan x sin x
+=
-.
CHƯNG INH Đ NG THƯC
s
1/ 4 4cos x sin x cos2x- = . 2/ ( )2sin 4x 4sin x cosx 1 2sin x= - .
3/ 2 2cos 2x sin x cosxcos3x- = . 4/ 4 2cos4x 8cos x 8cos x 1= - + .
5/ 48sin x 3 4cos2x cos4x= - + . 6/ 4 4 3 1sin cos x cos4x
4 4+ = + .
7/ 4 4 2 2sin cos x 6cos x sin x cos4x+ - = . 8/ 6 6 5 3sin x cos x cos4x
8 8+ = + .
9/ 6 6 15 1sin x cos x cos2x cos6x
16 16- = + . 10/ ( )6 6 2x x 1
sin cos cosx sin x 42 2 4
- = - .
11/ 8 8 7 1cos x sin x cos2x cos6x
8 8- = + . 13/ ( )3 3 4 4cos x cos3x sin x sin 3x 3 sin x cos x 2- = + - .
r a b4
p+ = ( )( )1 tan a 1 tan b 2+ + = .
s
1/ 1 cos2x
tan xsin2x
-= . 2/
2tan x cot x
sin2x+ = .
3/ cot x tanx 2cot 2x- = . 4/ cosx x
cot1 sin x 4 2
æ öp ÷ç ÷= -ç ÷ç ÷ç- è ø.
5/ 21 cosx x. tan 1
1 cosx 2
+=
-. 6/
1cot 2x cot x
sin2x+ =
7/ x 2
tan cot4 2 4 2 cosx
æ ö æ öp p p÷ ÷ç ç÷ ÷+ + + =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. 8/ ( )tan2x tan x cos2x tan x- = .
10 – 2 s. Lê Vă à
Page - 27 -
9/ tan x tan x 2tan2x4 4
æ ö æ öp p÷ ÷ç ç÷ ÷+ - - =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. 10/ 3 3 1
cos x sin x sin x cosx sin 4x2
- = .
11/ 3 3 3cos3x sin x sin 3x cos x sin 4x
4+ = . 12/ tan x tan x . tan x tan 3x
3 3
æ ö æ öp p÷ ÷ç ç÷ ÷- + =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
13/ 2
tan x tan x tan x 3tan 3x3 3
æ ö æ öp p÷ ÷ç ç÷ ÷+ + + + =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. 14/ cot x tanx 2tan2x 4tan4x 8cot 8x- - - = .
s
1/ 1 sin x x
cotcosx 4 2
æ ö+ p ÷ç ÷= -ç ÷ç ÷çè ø. 2/
21 1 2sin xtan2x
cos2x 1 sin2x
-+ =
-.
3/ ( )
2sin 2x sin 4xtan 2x cosx
2 cosx cos3x
+=
+. 4/ 2 26 2cos4x
tan x cot x1 cos4x
+= +
-.
5/ 21 2sin x 1 tan x
1 sin 2x 1 tan x
- -=
+ +. 6/
1 x1 tan tan x
cosx 2
æ ö÷ç ÷+ =ç ÷ç ÷çè ø
7/ 1 sin 2x
tan xcos2x 4
æ ö+ p ÷ç ÷= +ç ÷ç ÷çè ø. 8/ 22sin x sin2x x
tan2sin x sin2x 2
-=
+.
9/ 2
2
1 2sin x1
2tan x cos x4 4
-=
æ ö æ öp p÷ ÷ç ç÷ ÷+ +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
. 10/
xsin x sin
x2 tanx 2
1 cosx cos2
+=
+ +
.
11/ 1 cosx cos2x
cot xsin2x sin x
- +=
-. 12/
1 cos4x 1sin 4x
cot x tan x 2
+=
-.
13/ ( )( )
sin 2x cosx xtan
21 cos2x 1 cosx=
+ +. 14/
( )
4 4 22sin x cos x cos x x
cos21 1 cosx
- +=
-.
15/ 2 2 6 2cos4xcot x tan x
1 cos4x
++ =
-. 16/
2x x 4
cot tan2 2 1 2tan x cot 2x
æ ö÷ç ÷- =ç ÷ç ÷ç -è ø.
17/ 2cot 2x 1
cos8x cot 4x sin 8x2cot 2x
-- = . 18/
21 1 2sin xtan2x
cos2x 1 sin2x
-+ =
-.
19/ 1 sin2x cos2x
tan 4xcos4x sin2x cos2x
-- =
+.
20/ cosx sin x cosx sin x
2tan2xcosx sin x cosx sin x
+ -- =
- +.
21/
x xcos sin
12 2 tan xx x cosx
cos sin2 2
-= -
+
.
s. Lê Vă à 5. –
Page - 28 - " A e f wer f rr w re e see s f d y "
22/ 2
1 cosx cos2x cos3x2cosx
2cos x cosx 1
+ + +=
+ -.
23/ 1 x x
2 cot 2x cot tansin 2x 2 2
æ ö÷ç ÷+ = -ç ÷ç ÷çè ø.
24/ 4 4sin x sin2x cos x
cos2xtan2x 1
+ -=
-.
25/ 2 2
4
2 2
sin 2x 4sin xtan x
sin 2x 4sin x 4
-=
+ -.
26/ 2 2
2 2
sin 3x cos 3x8cos2x
sin x cos x- = .
27/ 2 3cos x cos3x sin x sin 3x
3cosx sin x
- ++ = .
28/ 2
1 1 sin 2x1 tan x 1 tan x
cosx cosx cos x
æ öæ ö÷ ÷ç ç÷ ÷+ + + - =ç ç÷ ÷ç ç÷ ÷ç çè øè ø.
s
1/ 2 3 n
n
n
a a a a sin aA cos cos cos ... cos
2 a2 2 2 2 .sin2
= = .
2/ n
2 n 1B cos .cos ... cos
2n 1 2n 1 2n 1 2
p p p= =
+ + +.
3/ 2 4 2n 1
C cos .cos ... cos2n 1 2n 1 2n 1 2
p p p= = -
+ + +.
4/ 1 1 1 1 1 1 x
D cosx cos , 0 x2 2 2 2 2 2 8 2
æ öp÷ç ÷= + + + = < <ç ÷ç ÷çè ø.
s y à s
1/ 2sin2x 2sin x xA tan
sin2x 2sin x 2
-= +
+.
2/ 3 3cos x cos3x sin x sin 3x
Bcosx sin x
- += + .
3/ 4 1C 4sin x 2cos2x cos4x
2= + - .
4/ 4 2 2 4D 3cos2x 5sin x 4sin x cos x cos x= + + - .
5/ 2tan x 1
E cot x cos4x cot 2x sin 4x2
-= + + .
10 – 2 s. Lê Vă à
Page - 29 -
6/ 4 4 4 4 3F sin x sin x sin x sin x
4 2 4
æ ö æ ö æ öp p p÷ ÷ ÷ç ç ç÷ ÷ ÷= + + + + + +ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø.
BIÊN Đ I THANH T CH
à s
1/ ( )( )A 2sin x 1 2cosx sin x sin2x cosx= - + - + .
2/ ( )( )B 2cosx 1 2sin x cosx sin2x sin x= - + - + .
3/ ( )( ) 2C 2sin x 1 2cos2x 2sin x 3 4sin x 1= - + + - + .
4/ ( )( ) 2D 2cosx 1 2cosx 2sin x 1 4sin x 3= - + + + - .
5/ ( )( ) 2E 2sin x 1 3cos4x 2sin x 4 4cos x 3= + + - + - .
6/ F 1 sin x cosx sin2x cos2x= + + + + .
7/ ( )2G 1 cosx cos x sin x 1 cosx= + + - + .
8/ ( ) ( )2 2H 1 sin x cosx sin2x 1 cos x sin x 1= + - + + - .
à s
1/ 3 3A sin xcosx cos xsin x= - .
2/ 3 3B sin x cos x sin x cosx= + - + .
3/ 3 3C sin x cos x cos2x= + - .
4/ 2 3D cos x sin x cosx= + + .
5/ 3 2E cos x cos x 2sin x 2= + + - .
6/ 3 3 2F cos x sin x 2sin x 1= + + - .
à s
1/ A sin2x cos2x cosx sin x= + + - .
2/ B sin2x cos2x 3sin x cosx 2= + + - - .
3/ C cos2x sin2x 3sin x cosx 2= - + + - .
4/ ( )D sin2x cos2x 2 cosx sin x 3= + + - - .
5/ E 9sin x 6cosx 3sin2x cos2x= + - + .
6/ F sin2x 2cos2x 4cosx sin x 1= + + - - .
7/ ( )( )G 2cosx 1 sin x cosx 1= - + - .
8/ H 2sin2x cos2x 7sin x 2cosx 4= - - - + .
E – CÔNG THƯC BIÊN Đ I
s. Lê Vă à 5. –
Page - 30 - " A e f wer f rr w re e see s f d y "
NG
CÔNG THƯC BIÊN Đ I T NG THANH T CH
r
1/ 0cos2x cos4xA khi x 20
sin 4x sin2x
-= =
-.
2/ cosx.cos13x
B khi xcos3x cos5x 17
p= =
+.
3/ cosx.cos10x
C khi xcos2x cos4x 13
p= =
+.
4/ tan2x sin2x 2
D khi tan xtan2x sin2x 15
-= =
+.
Công thưc biên đôi tông thanh tich
● . ● .
● . ● .
● . ● .
● . ● .
● .
● .
Công thưc biên đôi tich thanh tông
● .
● .
● .
10 – 2 s. Lê Vă à
Page - 31 -
5/ sin x sin2x sin 3x 1
E khi sin x , xcosx cos2x cos3x 3 4 2
+ + p p= = < <
+ +.
1/ 1 2cosx
A1 2cosx
-=
+. 2/
3 2cos3xB
3 2cos3x
-=
+.
3/ 1 2cos2x
C3 2sin2x
-=
+. 4/
2 2sin2xD
2 2sin2x
-=
+.
5/ sin 5x sin 3x
E2cos4x
-= . 6/
cos4x cos2xF
sin 4x sin2x
-=
+.
7/ ( )sin x y
Gsin x sin y
+=
+. 8/
sin x sin yH
cosx cosy
+=
+.
9/ cosx sin x
Icosx sin x
+=
-. 10/
2 2
2 2
sin 4x sin 2xJ
cos x cos 2x
-=
-.
11/ 2sin 4x
K2cosx cos3x cos5x
=+ +
. 12/ sin 2x
Ltan x cot 2x
=+
.
13/ tan 3x tan 5x
Mcot 3x cot 5x
+=
+. 14/
tan2x cot 2xN
1 tan2x. tan 4x
+=
+.
15/ 1 sin 2x cos2x
O1 sin 2x cos2x
+ +=
+ -. 16/
1 sin 4x cos4xP
1 cos4x sin 4x
+ -=
+ +.
17/ sin2x 2sin 3x sin 4x
Qcos3x 2cos4x cos5x
+ +=
+ +. 18/
sin x sin 4x sin 7xR
cosx cos4x cos7x
+ +=
+ +.
19/ cos2x sin 4x cos6x
Scos2x sin 4x cos6x
- -=
+ -. 20/
2
1 cosx cos2x cos3xT
2cos x cosx 1
+ + +=
+ -.
21/ ( )22 sin 2x 2cos x 1
Ucosx sin x cos3x sin 3x
+ -=
- - +. 22/
( )( )
( )( )
sin x y sin x cos x y cosxV
sin x y sin x cos x y cosx
+ - + += -
+ + - -.
23/ cos7x cos8x cos9x cos10x
Xsin 7x sin 8x sin 9x sin10x
- - +=
- - +. 24/
sin 4x sin 5x sin 6xY
cos4x cos5x cos6x
+ +=
+ +.
à s y
1/ A cos3x cosx= + . 2/ B sin 3x sin2x= + .
3/ C cos4x cosx= - . 4/ D sin5x sinx= - .
5/ E 1 sin2x= + . 6/ F 1 sinx= - .
7/ G 1 2cosx= + . 8/ H 2 sin2x 1= - .
9/ I 3 2cos2x= + . 10/ ( ) ( )J sin a b sin a b= + - - .
11/ 2 2K cos x cos y= - . 12/ L 1 sin x cos2x= + - .
s. Lê Vă à 5. –
Page - 32 - " A e f wer f rr w re e see s f d y "
13/ M 1 sin x cosx= + + . 14/ N cosx sin2x cos3x= + - .
15/ O sin 3x sin x sin2x= - + . 16/ Q cosx cos2x sin 3x= - - .
17/ R sin x sin2x sin 3x= + + . 18/ S cosx cos2x cos3x= + + .
19/ T 2sin2x cos5x cos9x= + - . 20/ U sin 3x 2sin2x sin x= - + .
21/ V cosx cos3x 2cos5x= + + . 22/ 0 0 0X cos46 cos22 2cos78= - - .
23/ 2 3
Y cos cos cos7 7 7
p p p= - + . 24/
2 4 6Z cos cos cos
7 7 7
p p p= + + .
25/ 0 0 0W sin 70 sin20 sin 50= - + . 26/ ( )cos5x cos7x cos 6xD = + - p + .
27/ cos 5x sin x cos3x2
æ öp ÷ç ÷F = + + -ç ÷ç ÷çè ø. 28/
x ysin x sin y sin
2
+G= + + .
29/ ( )sin x sin y sin x yL = + + + . 30/ ( )cosx cosy sin x yP = + + + .
31/ ( )cosx cosy cos x y 1Q = + + + + . 32/ ( ) ( )0 0cos 60 x cos 60 x cos3xS = + + - + .
33/ 1 cos2x cos4x cos6x¡ = + + + . 34/ sin2x sin 4x sin6xW= + + .
35/ sin5x sin6x sin7x sin 8xX= + + + . 36/ cos5x cos8x cos9x cos12xY = + + + .
à s y
1/ 1 cosx cos2x cos3x+ + + . 2/ sin x sin 3x sin7x sin5x- + - .
3/ sin x sin2x sin5x sin 8x- + + . 4/ cos7x sin 3x sin2x cos3x+ + - .
5/ cos9x cos7x cos3x cosx- + - . 6/ cos10x cos8x cos6x 1- - + .
7/ 0 0 0 0sin 35 cos40 sin 55 cos20+ + + . 8/ 0 0 0 0sin 57 sin 59 sin 93 sin 61- - + .
9/ 0 0 0 0sin 47 sin 61 sin11 sin25+ - - . 10/ cos5x 3cos7x 3cos9x cos11x+ + + .
11/ 2 3x1 sin x cos5x sin 7x 2cos
2+ - - - . 12/ ( )sin 3x sin x sin2x 2 1 cosx cosx+ - + - .
13/ sin x sin2x sin 3x 1 cosx cos2x+ + - - - .
14/ 1 sin x cos3x cosx sin2x cos2x+ + - - - .
15/ sin x sin2x sin 3x cosx cos2x cos3x+ + - - - .
à s y
1/ 2 2 2cos x cos 2x cos 3x 1+ + - . 2/ 2 2 2 3sin x sin 2x sin 3x
2+ + - .
3/ 2 2 2sin 3x sin 2x sin x- - . 4/ 2 2 2sin x cos 2x cos 3x- - .
5/ 2 2 2sin x 2sin 2x sin 3x- + . 6/ ( )2 2sin 4x cos 6x sin 10,5 10x- - p + .
7/ 2 2 2 2sin 3x cos 4x sin 5x cos 6x- - + . 8/ 2 2 2 2 3cos x cos 2x cos 3x cos 4x
2+ + + - .
9/ 2 2 2 2cos x cos 2x cos 3x cos 4x 2+ + + - . 10/ 2 25 9xcos3x sin 7x 2sin 2cos
4 2 2
æ öp p÷ç ÷+ - + +ç ÷ç ÷çè ø.
10 – 2 s. Lê Vă à
Page - 33 -
11/ ( )sin 2x cot 3x sin 2x 2 cos5x2
æ öp ÷ç ÷+ + p + -ç ÷ç ÷çè ø. 12/
cos3x cos 5x2
2sin 2x cos 3x2 cos5x
æ öp ÷ç ÷-ç ÷çæ ö ÷çp è ø÷ç ÷- + -ç ÷ç ÷çè ø.
1/ 0 0 6sin 75 cos75
2+ = . 2/ 0 0 0cos12 cos48 sin18- = .
3/ 0 0 0sin65 sin55 3cos5+ = . 4/ 0 0tan267 tan 93 0+ = .
5/ 0 0 0cos85 cos35 cos25 0+ + = . 6/ 0 0 0 0tan 9 tan27 tan 63 tan 81 4- - + = .
7/ 0 0 0 0 1cos24 cos48 cos84 cos12
2+ - - = .
8/ 0 0 0 0 0 0tan 9 tan15 cot 15 cot 9 tan27 cot 27+ + + = + .
9/ 2 31 1cosx cos3x cos5x 8sin x cos x
2 2- - = .
10/ ( )sin x 1 2cos2x 2cos4x 2cos6x sin 7x+ + + = .
11/ 4 x1 4cosx 6sin2x 4sin x 16sin2x sin
2+ + - = .
12/ ( ) ( )2 0 08sin x sin x 60 sin x 60 cos4x cos2x+ - = - .
13/ ( ) ( ) ( )2 0 0sin x cosx cos4x 4sin2x sin x 15 cos x 15- - = + - .
Cho a b c+ = . a b c
sin a sin b sin c 4cos cos sin2 2 2
+ + = .
r
1/ 6 6A sin cos24 24
p p= + .
2/ 2 2B tan cot12 12
p p= + .
3/ 3 5 7 9
C cos cos cos cos cos11 11 11 11 11
p p p p p= + + + + .
4/ 2 4 6 8 10
D cos cos cos cos cos11 11 11 11 11
p p p p p= + + + + .
5/ 0 2 3 4 5 6E cos0 cos cos cos cos cos cos
7 7 7 7 7 7
p p p p p p= + + + + + + .
6/ ( ) ( )2 2
0 0 0 0 0F sin 40 cos10 cos40 sin10 cos140= + - + + .
r A à à A,B,C . r
s. Lê Vă à 5. –
Page - 34 - " A e f wer f rr w re e see s f d y "
1/ A B C
sin A sin B sinC 4cos cos cos2 2 2
+ + = .
2/ A B C
sin A sin B sinC 4sin sin cos2 2 2
+ - = .
3/ A B C
cosA cosB cosC 1 4sin sin sin2 2 2
+ + = + .
4/ sin2A sin2B sin2C 4sin Asin BsinC+ + = .
5/ 1 cos2A cos2B cos2C 4cosAcosBcosC+ + + = - .
6/ ( )2 2 2sin A sin B sin C 2 1 cosAcosBcosC+ + = + .
7/ 2 2 2cos A cos B cos C 1 2cosAcosBcosC+ + = - .
CÔNG THƯC BIÊN Đ I T CH THANH T NG
à s
1/ 2
A sin sin5 5
p p= . 2/ B sin5xcos3x= .
3/ 3
C sin cos4 6
p p= . 4/
7D sin cos
12 12
p p= .
5/ ( ) ( )E sin x y cos x y= + - . 6/ ( ) ( )0 0F sin x 30 cos x 30= + - .
7/ G 2sinxsin2xsin3x= . 8/ H 8cosxsin2xsin3x= .
9/ I sin x sin x cos2x6 6
æ ö æ öp p÷ ÷ç ç÷ ÷= + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø. 10/ ( ) ( ) ( )J 4cos a b cos b c cos c a= - - - .
r
1/ 0 0A cos75 cos15= .
2/ 5
B sin sin12 12
p p= .
3/ 11 5
C sin cos12 12
p p= .
4/ 0 0 0D sin20 .sin40 .sin80= .
5/ 0x 5xE sin .sin , khi x 60
4 4= = .
6/ 0 0 0 0F sin20 sin40 sin60 sin80= .
7/ 4 5 7 11G 2 .sin .sin .sin .sin
24 24 24 24
p p p p= .
8/ 0 0 0 0 0 0H sin 5 .sin15 .sin25 .......sin 65 sin 75 .sin 85= .
9/ 2 0 0 0I sin 10 cos70 cos50= + .
10 – 2 s. Lê Vă à
Page - 35 -
10/ 0 0 0 0 01J cos10 cos50 cos5 cos25 cos10
2= - + .
11/ 2 0 2 0 0 0K sin 50 sin 10 sin 50 sin10= + + .
12/ 2 0 2 0 0 0L cos 73 cos 47 cos73 .cos47= + + .
13/ 2 0 2 0 0 0M sin 50 sin 70 cos50 cos70= + - . A A ă 2001
14/ 0 0 0 0 01N cos10 cos50 cos5 cos25 sin10
2= - + .
15/ ( )0
0 0
0
sin 55O 2 cos22 cos44
sin11= + - .
16/ 0
0 0
4 0 4 0
sin 60P 3sin15 sin 75
sin 15 sin 75= +
-.
17/ 0
0 0
0 0
1 cos40Q 2 2 sin10 2sin 35
2cos5 sin 5
æ ö÷ç ÷ç= - - - ÷ç ÷ç ÷è ø.
18/ 0
0
1R 4sin 70
sin10= - .
19/ 0 0
1 3S
sin10 cos10= - .
20/ 2 3
T cos cos cos7 7 7
p p p= - + .
21/ 2 4 5 2 4 6 8
U cos cos cos cos cos cos cos7 7 7 5 5 5 5
p p p p p p p= + + + + + + + .
22/ 2 4 6 8
V cos cos cos cos9 9 9 9
p p p p= + + + .
( )A 2sin x cosx cos3x cos5x= + + . s y r r
3 5B cos cos cos
7 7 7
p p p= + + .
s
1/ A cos11xcos3x cos17xcos9x= - .
2/ B sin18xcos13x sin9xcos4x= - .
3/ C sin x sin 3x sin 4x sin 8x= + .
4/ D sin2xsin6x cosxcos3x= - .
5/ E cos3xcos6x cos4xcos7x= - .
6/ ( ) ( )0 0F sin x sin 60 x sin 60 x= - + .
7/ ( ) ( )0 0G 8cosx cos 60 x cos 60 x 1= - + + .
s. Lê Vă à 5. –
Page - 36 - " A e f wer f rr w re e see s f d y "
8/ 1
H cosx cos2x sin 3x sin12x4
= - .
9/ I 4sin2x sin5x sin7x sin 4x= - .
10/ 1
J sin2x sin 6x cos4x cos12x4
= + .
11/ 1
K sin x sin2x sin 3x sin 4x4
= - .
12/ L 4cosx sin x sin x cos2x6 6
æ ö æ öp p÷ ÷ç ç÷ ÷= + - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
13/ M sin 4x sin10x sin11x sin 3x sin7x sin x= - - .
14/ x 3x
N cos cos sin x sin 3x sin2x sin 3x2 2
= - - .
15/ 7x 3x x 5x
O sin cos sin cos sin2x cos7x2 2 2 2
= + + .
16/ P cos2x cos4x cos6x 4cosxcos2xcos3x 2= + + - - .
17/ ( )Q sin x 1 2cos2x 2cos4x 2cos6x= + + + .
18/ ( )sin 5x
R 2 cos2x cos4xsin x
= - + .
19/ 2 2S sin x sin x sin cos 2x4 4 12 12
æ ö æ ö æ öp p p p÷ ÷ ÷ç ç ç÷ ÷ ÷= + - - - +ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çè ø è ø è ø.
s
1/ 0 0 0 0 0 3 1cos12 cos18 4cos15 cos21 cos24
2
++ - = - .
à A ă 2001
2/ 0 0 0 0tan 9 tan 63 tan 81 tan27 4- + - = .
3/ 0 0 0 0 08 3tan 30 tan 40 tan 50 tan 60 cos20
3+ + + = .
4/ ( ) ( )0 0 1 1cos 2x 60 cos 2x 60 cos4x
2 4+ - = - .
5/ ( ) ( ) ( )2 0 0sin x cosx cos4x 4sin2x sin x 15 cos x 15+ - = + - .
6/ 2 2 3sin x sin x sin x sin x
3 3 4
æ ö æ öp p÷ ÷ç ç÷ ÷+ - + - =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø.
7/ ( ) ( ) ( )sin a sin b c sin b sin c a sin csin a b 0- + - + - = .
8/ ( ) ( )2 2 2cos x 2cosa cosxcos a x cos a x sin a- + + + = .
10 – 2 s. Lê Vă à
Page - 37 -
9/ 8 8 7 1sin x cos x cos2x cos6x
8 8- = + .
10/ 8 8 35 7 1sin x cos x cos4x cos8x
64 16 64+ = + + .
11/ 3 2
tan 4tan 1111 11
p p+ = .
12/ 0 0 0tan20 tan40 tan80 3= .
13/ 3 0 2 08sin 18 8sin 18 1+ = . V à ă 2000
14/ 4 4 4 43 5 6 3sin sin sin sin
16 16 16 16 2
p p p p+ + + = .
15/ 0 0 0 0 0 0tan10 tan25 tan25 tan 55 tan 55 tan10 1+ + = .
16/ 0 0 0 0tan15 tan25 tan35 tan85 1= .
17/ 0 0 0tan20 tan 40 tan 80 3 3- + = .
18/ 0 o 0 0tan10 tan50 tan60 tan70 2 3- + + = .
19/ 0 0 0 0 0tan20 tan 40 tan 80 tan 60 8sin 40+ + + = .
20/ 6 0 4 0 2 0tan 20 33tan 20 27 tan 20 3 0- + - = .
A r
1/ B C3
p- = à
1sin B.sinC
2= . S: B , C , A
2 6 3
p p p= = = .
2/ 2
B C3
p+ = à
1 3sin B.cosC
4
+= . S:
5A , B , C
3 12 4
p p p= = = .
Ch u ki n c à A
1/ cos2A cos2B cos2C 1+ + = - . 2/ tan2A tan2B tan2C 0+ + = .
3/ b c a
cosB cosC sin B.sinC+ = . 4/
B a ccot
2 b
+= .
Ch u ki n c à A cân:
1/ ( )A B
a tan A b tan B a b tan2
++ = + . 2/ 22tan B tanC tan B. tanC+ = .
3/ ( )sin A sin B 1
tan A tan BcosA cosB 2
+= +
+. 4/
C 2sin A.sin Bcot
2 sinC= .
Ch ng minh b ng th c, t s y r u ki n c à A u:
1/ 3 3
sin A sin B sinC2
+ + £ . HD: C ng sin3
p vào VT.
2/ 3
cosA cosB cosC2
+ + £ . HD: C ng cos3
p vào VT.
s. Lê Vă à 5. –
Page - 38 - " A e f wer f rr w re e see s f d y "
3/ tan A tan B tanC 3 3+ + ³ .
4/ 1
cosA.cosB.cosC8
£ .
HD 1
cosA.cosB.cosC8
- d .
Cho cos cosa cosbj = a,b k2
p¹ + p . 2a a b
tan tan tan2 2 2
j + j -= .
Cho ( )sin 2a b 5sin b+ = . ( )2tan a b
3tan a
+=
a k2
a b l2
ìï pï ¹ + pïïïíï pï + ¹ + pïïïî
.
Cho ( ) tan a b 3tan a+ = . ( )sin 2a 2b sin2a 2sin2b+ + = .
Cho ( )( )
( )( )
sin x cos xa A
;b Bsin x cos x
- a - a= =
- b - b à aB bA 0+ ¹ .
( )aA bB
cosaB bA
+= a - b
+.
r x a
tan2 b
= A a sin x bcosx= + à à .
Cho a,b, c à à A,B,C .
γ, ,a b ơ γa b c
cos , cos , cosb c c a a b
a = b = =+ + +
.
1/ γ2 2 2tan tan tan 1
2 2 2
a b+ + = . 2/
γ A B Ctan tan tan tan tan tan
2 2 2 2 2 2
a b= .
1 1 1 tan 8x tan x
...cosx cos2x cos2x cos3x cos7xcos8x sin x
-+ + + = .
r 8 0 8 0 8 0M sin 20 sin 40 sin 80= + + .
s 11 ă 200
r 0 0 0 0
0
2sin2 4sin 4 ... 178sin178 180sin180N
cot 1
+ + + += .
s
1/ ( )A cos cos3 cos5 ... cos 2n 1= a + a + a + + - a . S: sin2n
A2sin
a=
a.
2/ ( )n 12 3
B sin sin sin ... sinn n n n
- pp p p= + + + + . S: B cot
2n
p= .
3/ ( )2n 13 5
C cos cos cos ... cosn n n n
- pp p p= + + + . S: C cos
n
p= - .
10 – 2 s. Lê Vă à
Page - 39 -
4/ 1 1 1
D ... ,cosa.cos2a cos2a.cos3a cos4a.cos5a
= + + + a5
p= . S: D 1 5= - .
5/ n 1
1 1 1 1E 1 1 1 ... 1
cosx cos2x cos3x cos2 x-
æ öæ öæ ö æ ö÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷= + + + +ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç ÷è øè øè ø è ø. S:
n 1tan 2 xE
xtan
2
-
= .
6/ n
1 1 1 1F ...
sin2x sin 4x sin 8x sin2 x= + + + + .
7/ in3 3 2 3 n 1 3
n
x x x xG sin 3s 3 sin ...... 3 sin
3 3 3 3
-= + + + + .
yê 4 4 2 23 3 2M tan tan 2 tan tan
4 4 7 7
æ öp p p p÷ç ÷= + + +ç ÷ç ÷çè ø.
( ) ( ) 3 1sin x 3sin x sin 3x
4= - * . Khi thay
n
ax
3= à ( )* . y
3 3 n 1 3n 2 n
a a aS sin 3sin ... 3 sin
3 3 3
-= + + + .
S: nn n
1 aS 3 sin sin a
4 3
æ ö÷ç ÷= -ç ÷ç ÷è ø.
sin2a
cosa2sin a
= . s y r r n 2 n
x x xP cos cos ... cos
2 2 2= .
S: n
n
n
sin xP
x2 sin
2
= .
1 x
cot cot xsin x 2
= - .
( ) n 1
n 1
1 1 1S ... , 2 k
sin sin2 sin2
-
-= + + + a ¹ p
a a a.
S: n 1S cot cot 22
-a= - a .
2tan x. tan2x tan2x 2tan x= - .
2 2 n 1 2n 2 n n 1
a a a a aS tan . tan a 2tan . tan ... 2 tan . tan
2 22 2 2
-
-= + + + .
S: nn n
aS tan a 2 tan
2= - .
Tính 2sin 2x, bi t:
2 2 2 2
1 1 1 17
tan x cot x sin x cos x+ + + = .
S: 8
9.
s. Lê Vă à 5. –
Page - 40 - " A e f wer f rr w re e see s f d y "
sina m= à 0 0270 x 450< < .
1/ ( )a 1cos 1 m 1 m
2 2= - + + - . 2/ ( )a 1
sin 1 m 1 m2 2
= - + - - .
Cho 02
p< a < . tan cot 2a + a ³ .
s
1/ 0 0sin 75 sin15 1+ > . 2/ 0 0cos36 tan 36> .
s
1/ sin 3x cosx cos3x sin x cos2x 2- - £ .
2/ 3 sin 3x cos2x cosx sin2x sin x 2+ - £ .
3/ ( ) ( )0 0 1sin x sin x 60 sin x 60
4+ - £ .
4/ cosxcos3x sin2xsin4x 1- £ .
5/ 2cos2x cosx sin x sin 3x sin x cos3x 1+ - £ .
6/ ( )2sin x cosx cos3x cos5x cos7x cos9x 1+ + + + £ .
r à r s
1/ y 3 sin x cosx 2= + + .
2/ 1
y cosx , xcosx 2 2
æ öp p÷ç ÷= + - < <ç ÷ç ÷çè ø.
3/ y sin x cosx cos2x cos4x= .
4/ 4 4y sin x cos x, 0 x2
æ öp÷ç ÷= + £ £ç ÷ç ÷çè ø.
5/ 2y sin x 2 sin x= + - .
6/ 4 4 2 2y cos x sin x cos x sin x= + + .
7/ 4 2 5y sin 4x cos x cos2x
4= - + .
8/ 6 6 4 4y 2sin x 2cos x sin x cos x cos2x= + - - + .
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