Chuyên đề 6 góc lượng giác và công thức lượng giác

10 – 2 s. Lê Vă à
Page - 1 -
Chương

NG
s
1/ 2 2 2
cos x sin x 1 2sin x- = - . 2/ 2 2
2cos x 1 1 2sin x- = - .
3/ 2 2
3 4sin x 4cos x 1- = - . 4/ sin xcot x cosx tan x sin x cosx+ = + .
5/ 4 4 2 2
sin x cos x 1 2sin xcos x+ = - . 6/ 4 4 2 2
cos x sin x cos x sin x- = - .
7/ ( )( )2
4cos x 3 1 2sin x 1 2sin x- = - + . 8/ ( )( )2 2 2
1 cosx sin x cosx cos x sin x+ - + = .
9/ 4 4 2 2
sin x cos x 1 2cos x 2sin x 1- = - = - . 10/ 3 3
sin xcosx sin xcos x sin xcosx+ = .
11/ 2 2 2 2
tan x sin x tan xsin x- = . 12/ 2 2 2 2
cot x cos x cot xcos x- = .
s
1/
1
tan x cot x
sin xcosx
+ = . 2/
1 cosx sin x
sin x 1 cosx
-
=
+
.
3/
1 1
1
1 tan x 1 cot x
+ =
+ +
. 4/ 21 1
1 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 x
1 2tan x
1 sin x
+
= +
-
. 6/
tan x tan y
tan x tan y
cot x cot y
+
=
+
.
7/ 4
2 4
2 1
1 cot x
sin x sin x
- = - . 8/
cosx 1
tan x
1 sin x cosx
+ =
+
.
9/ ( )( )2 1
1 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
+ -
=
- - +
.
GÓC VÀ CÔNG THỨC LƯỢNG GIÁC5
A – HỆ THỨC LƯỢNG GIÁC CƠ BẢN
Hệ thức lượng giác cơ bản
 .  .
 .  ..
 .  .

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 2
sin x cos x 1 3sin xcos x+ = - .
2/ ( )( )6 6 2 2 2 2
sin x cos x sin x cos x 1 sin xcos x- = - - .
3/ ( )
2
8 8 2 2 4 4
sin x cos x 1 2sin x cos x 2sin x cos x+ = - - .
4/ ( )( )8 8 2 2 2 2
sin x cos x sin x cos x 1 2sin xcos x- = - - .
s
1/ ( )( )1 sin x cosx tan x 1 cosx 1 tan x+ + + = + + .
2/ ( )( )1 tan x 1 cot x sin xcosx 1 2sin xcosx+ + = + .
3/ ( ) ( ) ( )
22 2
1 tan x cos x 1 cot x sin x sin x cosx+ + + = + .
4/ 2 2
sin xtan x cos xcot x 2sin xcosx tan x cot x+ + = + .
5/ 2 2 2 2 2
sin xtan x 4sin x tan x 3cos x 3+ - + = .
s
1/
2 2
sin cos tan 1
1 2sin cos tan 1
a - a a -
=
+ a a a +
. 2/
2 2
6
2 2
tan sin
tan
cot cos
a - a
= a
a - a
.
3/
2 2
cos sin
1 sin cos
1 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
1
tan cot 2
sin cos
= a + a +
a a
. 6/
2
2
2 2
1 3tan
tan 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 cos
1 2cot
sin sin
é ù
+ aê ú- a
ê ú- = a
ê úa aê úë û
.
9/ ( )( )2 1
1 cos 1 cot
1 cos
- a + a =
+ a
. 10/ 2 3
3
sin cos
1 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 sin
4tan
1 sin 1 sin
æ ö÷+ a - aç ÷ç - = a÷ç ÷ç - a + a ÷çè ø
. 14/
2
21 cos 1 cos
4cot
1 cos 1 cos
æ ö÷+ a - aç ÷ç - = a÷ç ÷ç - a + a ÷çè ø
.
s
1/ 4 2 2
1
P sin sin cos= a + a a . 2/ 4 4 2
2
P sin cos cos= a - a + a .
3/ 2 2 2
3
P sin sin cot= a + a a . 4/ 2 2 2
4
P cos cos cot= a + a a .
5/ ( )2 2 2
5
P 1 sin cot 1 cot= - a a + - a . 6/ 2 2
6
P sin tan cos cot 2sin cos= a a + a a + a a .

10 – 2 s. Lê Vă à
Page - 3 -
7/
2
7
2cos 1
P
sin cos
a -
=
a + a
. 8/ 8 2
1 cos 1
P
1 cossin
- a
= -
+ aa
.
9/ 9
cot cot
P
tan tan
a + b
=
a + b
. 10/ 10
cos
P tan
1 sin
a
= a +
+ a
.
11/ 11
sin x tan x
P sin xcot x
tan x
+
= - . 12/ 12 2
cosx tan x
P cot xcosx
sin x
= - .
13/
2 2
13 2 2
cos x cot x
P
sin x tan x
-
=
-
. 14/
2 2
2
14 2
1 sin xcos x
P cos x
cos x
-
= - .
15/
3 2
15
sin x sin xcos x cosx
P
1 2sin xcosx
+ -
=
-
. 16/
( )
2
16
sin x cosx 1
P
tan x sin xcosx
+ -
=
-
.
17/
2 2 2
17 2 2 2
sin x sin x tan x
P
cos x cos x tan x
+
=
+
. 18/
2 2
2
18 2
cot x cos x
P 1 cos x
cot x
-
= + - .
19/
2 2
2 2
19 2 2
cos x sin y
P cot xcot y
sin xsin y
-
= - . 20/ 20
1 sin x cosx cot x
P
sin x 1 cot x 1 tan x
= - -
+ +
.
21/ 21
1 cosx 1 sin x
P tan x cot x
sin x cosx
- -
= + + + . 22/ 22
1 1
P 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 2
23
P 1 tanx cos x 1 cot x sin x= + + + . 24/ 24 2 2
1
P , 0 x
2sinx cot x cos x
æ öp÷ç ÷= < <ç ÷ç ÷çè ø- -
.
s à s
1/ 2
A 2cos x 1= - .
2/ 2
B 3 4sin x= - .
3/ 2
C sin xcosx cos x 1= + - .
4/ 2
D sin x sin xcosx 1= + - .
5/ E 1 sin x cosx tan x= + + + .
6/ F tan x cot x sin x cosx= - + + .
7/ ( )2
G cosxtan x 1 cosx= - + .
8/ ( ) ( )2
H 3 4cos x sin x 2sin x 1= - - + .
9/ 2 2
I sin x 3cos x 6cosx 2sin x= - + - .
10/ 3 3
J cos x sin x sin x cosx= - + + .
11/ 3 2
K cos x cos x 2sin x 2= + + - .
12/ 2 3
L cos x sin x cosx= + + .
13/ ( )2
M 1 cosx cos x sin x 1 cosx= + + - + .

14/ 3 2
N 2cos x 2cos x sin x 1= + + - .
15/ 3 3 2
O cos x sin x 2sin x 1= + + - .
16/ ( )( ) 2
P 2sin x 1 2cosx 2sin x 1 3 4cos x= - + + - + .
17/ ( )( )Q 2cosx 1 sin x cosx 1= - + - .
18/ 3 3 2
R 4sin x 3cos x 3sin x sin xcosx= + - - .
19/ ( ) ( )2
S 1 sin x tan x 1 cosx= + - + .
20/ ( ) 2
T 2 5sin x 3 1 sin x tan x= - + - .
21/ 2
U 2sin xcosx 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 2
A cos x sin x 2sin x= - + .
2/ 4 2 2 2
B sin x sin xcos x cos x= + + .
3/ 4 2 2 2
C cos x sin xcos x sin x= + + .
4/ ( ) ( )4 2 4 2
D cos x 2cos x 3 sin x 2sin x 3= - + - .
5/ 6 6 4 4 2
E sin x cos x 2sin x cos x sin x= + - - + .
6/
1 1
F sin x. , 0 x
1 cosx 1 cosx 4
æ öp÷ç ÷= + < <ç ÷ç ÷ç+ - è ø
.
7/ 4 2 4 2
G sin x 4cos x cos x 4sin x= + + + .
8/ 2 2 2 2 2
H cos xcot x 5cos x cot x 4sin x= + - + .
9/ ( ) ( )3 3
I 1 cot x sin x 1 tan x cos x sin x cosx= + + + - - .
10/ ( )( )4 4 2 2
J sin x cos x 1 tan x cot x 2= + - + + .
11/ ( ) ( )8 8 6 6 4
K 3 sin x cos x 4 cos x 2sin x 6sin x= - + - + .
12/ ( ) ( )4 2 4 2 2 2
L sin x 1 sin x cos x 1 cos x 5sin x cos x 1= + + + + + .
13/ ( ) ( )
2
4 4 2 2 8 8
M 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
A
tan x 1 cot x 1
+
= +
- -
.
2/
( )
2
2
2 2 2
1 tan x 1
B
4tan x 4sin x cos x
-
= - .
3/ ( )( )
2
2 21 tan x
C 1 tan x 1 cot x
tan x
-
= - + + .
4/
( )
2
2
2 2 2
1 cot x 1
D
cot x sin x cos x
-
= - .
5/
6 2
6 2
1 sin x 3tan x
E
cos x cos x
-
= - .
6/
2 2 2 2
2 2
tan x cos x cot x sin x
F
sin x cos x
- -
= + .
7/
2 2
2
cot x cos x sin xcosx
G
cot xcot x
-
= + .
8/
4 4
6 6
sin x cos x 1
H
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 2
A f x g x h x ,= + +
( ) ( ) ( )4 4 4
B f x g x h x ,= + + ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2
C f x .g x g x .h x h x .f x= + + .
Cho a sin xsin y bcosxcosy 0- = . r
2 2 2 2
1 1
Q
a sin x bcos x a sin y bcos y
= +
+ +
à .
Cho ( )6 6 4 4
Q sin x cos x m sin x cos x= + - + . s Q
à à r Q .
– yê r – ă 1998
Cho
4 4
sin 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


B – ĐƯỜNG TRÒN LƯỢNG GIÁC – CUNG GÓC LƯỢNG GIÁC
CUNG LIÊN KẾT

10 – 2 s. Lê Vă à
Page - 7 -
NG
I LIÊN HỆ GI A ĐƠN V Đ VÀ 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 0
90 , 36 , 15 , 72 , 270 , 240 , 540 , 750 , 21- - .
s r s à s r d
0 0 0
21 30', 75 54', 12 36' 0 0 0
, 15 30', 105 45'30'', 27 38'49''- .
s r d r s r s ( )2
rad
p
à 2,5p .
r à r
0
0 0
15 0
30 0
45 0
60 0
75 0
90 0
120 0
150 0
180
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
0
135 0
50- 0
810
T NH Đ ÀI CUNG TRÒN
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 LƯỢNG GIÁC

d s ê r A 0 0 0 0 0
30 , 45 , 120 , 120 , 330 ,- -
0 0 0 0 07 4 15 2010
630 , 750 , 1250 , , , 750 , 1125 , ,
4 3 2 4
p p p p
- - - .
s ( )k2 , k , k , k , k , k
2 3 4
p p p
p p Î ¢ .
s y rê r k Î ¢ .
1/ x k= p . 2/ x k2
2
p
= + p .
3/ x 4k= p. 4/ x k
3
p
= + p .
5/ x k2
6
p
= - + p . 6/ x k
4
p
= - + p .
7/
k
x
3
p
= . 8/
k2
x
3 3
p p
= + .
9/
k3
x
4 4
p p
= + . 10/
k
x
6 3
p p
= + .
à s ¼đs AM
6
p
= à ¼đ
k
s AN
798
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 LƯỢNG GIÁC C A T CUNG GÓC
d r
1/ sin x, cosx, tan x, cot x
3
x
2
p
p < < .
2/
3
sin x , cos x , tan x , cot x
4 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 0
A sin 40 .cos 290= - . 4/ ( )0 0
B sin 25 .cos170= - .
5/ ( )0 0 0
C sin225 .tan130 .cot 175= - . 6/ ( )0 0 0
D cos195 .tan269 .cot 98= - .
7/ ( )0 0
E sin50 .cos 300= - . 8/ 0 21
F sin215 .tan
7
p
= .
9/
3 2
G cot .sin
5 3
æ öp p÷ç ÷= -ç ÷çè ø
. 10/ os
4 4 9
H c .sin .tan .cot
5 3 3 5
p p p p
= .
r
1/
1
sin x
2
= 0 0
90 x 180< < . 2/
4
sin x
5
= - 0 0
270 x 360< < .

10 – 2 s. Lê Vă à
Page - 9 -
3/
3
sin x
5
= -
3
x
2
p
p < < . 4/
1
cosx
4
= 0 x
2
p
< < .
5/
3
cosx
5
= 0
0 x 90< < . 6/
5
cosx
13
= - 0 0
180 x 270< < .
7/
2
cosx
5
= x 0
2
p
- < < . 8/
4
cosx
5
= 0 0
270 x 360< < .
9/
5
sin x
13
= x
2
p
< < p . 10/
1
sin x
3
= - 0 0
180 x 270< < .
11/ tanx 3=
3
x
2
p
p < < . 12/ tanx 2= - x
2
p
< < p .
13/
1
tan x
2
= - x
2
p
< < p . 14/ cot x 3=
3
x
2
p
p < < .
15/
3
tan x
4
=
3
x
2
p
p < < . 16/ tan x 2= - x
2
p
< < p .
17/
2
cot x
3
= 0 x
2
p
< < . 18/ cot x 3= - x
2
p
< < p .
19/ cot x 3= -
3
x 2
2
p
< < p . 20/ 0
cot15 2 3= +
r s
1/ Cho tanx 2= - . 1 2
5cot x 4tan x 2sin x cosx
A , A
5cot x 4tan x cosx 3sin x
+ +
= =
- -
.
2/ Cho cot x 2= . 1 2
3sin x cosx sin x 3cosx
B , B
sin x cosx sin x 3cosx
- -
= =
+ +
.
3/ Cho cot x 2= . 1 2 2
2sin x 3cosx 2
C , C
3sin x 2cosx cos x sin x cosx
+
= =
- -
.
4/ Cho tanx 2= . 1
2sin x 3cosx
D ,
4sin x 5cosx
+
=
- 2 3 3
3sin x 2cosx
D ,
5sin x 4cos x
-
=
+
3 3
3 2
sin x cos x
D ,
sin x sin x cosx
-
=
+ 4 3 3
sin x 5cosx
D ,
sin x 2cos x
+
=
-
3 3
5 3
8cos x 2sin x cosx
D ,
2cosx sin x
- +
=
-
3 3
6 3 2
sin x 2cos x 2sin x 3cosx
D
sin x cos x 5sin x
- + +
=
- +
.
5/ Cho
3
sin x , 0 x
5 2
p
= < < .
cot x tan x
E
cot x tan x
+
=
-
.
6/ Cho 0 01
sin x , 90 x 180
3
= < < .
2
8tan x 3cot x 1
F
tan x cot x
+ -
=
+
.
7/ Cho
2
cosx
3
= - .
cot x 3tan x
G
2cot x tan x
+
=
+
.

8/ Cho 02
sin x , 0 x 90
3
= < < . 1 2 2
tan x cosx tan x cosx
H , H cosx cot x
cot x sin x
-
= = - .
9/ Cho
4
cosx , x
5 2
p
= - < < p . 1 2
cot x tan x sin x
I , I cot x
cot x tan x 1 cosx
+
= = +
- +
.
Cho
5
sin x cosx
4
+ = . y r s
1/ A sinx.cosx= . S:
9
A
32
= .
2/ B sinx cosx= - . S:
7
B
4
= ± .
3/ 3 3
C sin x cos x= - . S:
41 7
C
128
= ± .
Cho tanx cot x 3- = . y r s
1/ 2 2
A tan x cot x= + . S: A 11= .
2/ B tan x cot x= + . S: B 13= ± .
3/ 4 4
C tan x cot x= - . S: C 33 13= ± .
Cho sin x cosx m+ = . y e r
1/ A sinxcosx= . 2/ 3 3
B sin x cos x= + .
3/ 4 4
C sin x cos x= + . 4/ D sin x cosx= - .
5/ 2 2
E tan x cot x= + . 6/ 6 6
F 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 cosx
2
+ = . y – ă 2005
4/
1
sin x cosx
5
+ = . S:
4 3 4 3
; ; ;
5 5 3 4
- - - .
5/ tan x cot x 4+ = . S:
1 2 3
; ; 2 3; 2 3
22 2 3
-
+ -
-
.
Cho tanx 2cot x 1- = - . y
1/ 2 2
A tan x cot x= - . 2/ 3 3
B tan x cot x= + .
3/ 4 4
C tan x 2cot x= + . 4/ 5 5
D tan x 3cot x= - .
r s y
1/ Cho 4 4 3
3sin x cos x
4
+ = . 4 4
A sin x 3cos x= + . S: A
7
4
= .

10 – 2 s. Lê Vă à
Page - 11 -
2/ Cho 4 4 1
3sin x cos x
2
- = . 4 4
B sin x 3cos x= + . S: B 1= .
3/ Cho 4 4 7
4sin x 3cos x
4
+ = . 4 4
C 3sin x 4cos x= + . S:
7 57
C C
4 28
= Ú = .
CUNG GÓC LIÊN KẾT
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 0
k360a + k Î ¢ .
0 0 0 0 0
370 , 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/ 0
sin150 . 2/ 0
cot135 . 3/ 0
cos225 . 4/ 0
tan210 .
5/ 0
cot 225 . 6/ 0
sin240 . 7/ 0
cos315 . 8/ 0
tan 300 .
9/ ( )0
cot 1380- . 10/ cos11p . 11/ sin13p. 12/ tan10p.
13/
7
cot
6
p
. 14/
11
cos
3
p
. 15/
25
cot
4
p
. 16/
17
sin
3
p
.
17/
29
cos
6
p
. 18/
45
tan
4
p
. 19/
16
cos
3
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 20/
31
sin
2
.
21/
19
cot
4
. 22/
26
cos
3
. 23/
159
tan
4
. 24/
115
sin
6
.
r s r
1/ ( )0
sin x 90- . 2/ ( )0
cos 180 x+ . 3/ ( )0
sin 270 x- . 4/ ( )0
sin x 180- .
5/ ( )0
cos x 540+ . 6/ ( )0
cot 180 x+ . 7/ ( )0
sin x 540- . 8/ ( )0
tan 360 x- .
9/ ( )0
cos 450 x+ . 10/ ( )2 0
sin 270 x+ . 11/ ( )3 0
cos 90 x+ . 12/ ( )5 0
cot 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 x
2
æ öp ÷ç ÷+ç ÷ç ÷çè ø
. 8/
3
cos x
2
.
9/
3
cot x
2
. 10/
5
cos x
2
. 11/
11
tan x
2
. 12/
7
sin x
2
æ öp÷ç ÷+ç ÷ç ÷çè ø
.

13/ ( )2
sin xp + . 14/ ( )9
cos xp - . 15/ ( )11
cot x 3- p . 16/ ( )4
cos x 3+ p .
17/ ( )2
cot x 5- p . 18/ ( )6
cos x - p . 19/ 8
cos x
2
æ ö3p÷ç ÷-ç ÷ç ÷çè ø
. 20/ 5
cos x
2
æ öp ÷ç ÷-ç ÷ç ÷çè ø
.
21/ 2012
sin x
2
. 22/ 2 9
tan x
2
. 23/ 2013 7
cos x
2
æ öp÷ç ÷+ç ÷ç ÷çè ø
. 24/ 1991 5
sin x
2
æ öp ÷ç ÷-ç ÷ç ÷çè ø
.
25/ 2015 11
cos x
2
. 26/ 2 9
cot x
2
. 27/ 11 11
tan x
2
. 28/ ( )cos x 2012- p .
1/ ( )A cos x sin x
2
æ öp÷ç ÷= - + - pç ÷ç ÷çè ø
.
2/ B cos x sin x cos x sin x
2 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 x
2 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 x
2 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 x
2 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 x
2 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 x
2 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 x
2 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 x
2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= p - - + + - + p -ç ç÷ ÷ç çè ø è ø
.
10/ ( ) ( ) ( ) ( )0 0 0 0
J cos 270 x 2sin x 450 cos x 900 2sin 270 x= - - - + + + - .
11/ ( )2 2 2 23
K 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 7
L sin sin sin sin sin sin
3 6 9 9 8 8
p p p p p p
= + + + + + .
13/ ( ) ( )2013 2013 2012 2011
M cos x cos x .sin x sin x
2
æ öp ÷ç ÷= + p + p + - -ç ÷ç ÷çè ø
.

10 – 2 s. Lê Vă à
Page - 13 -
14/ ( ) ( ) ( )6 6 4 4 23
N 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/
( ) ( )
( )
19
tan x .cos 36 x .sin x 5
2
O
9
sin x .cos x 99
2
æ öp ÷ç ÷- p - - pç ÷ç ÷çè ø
=
æ öp ÷ç ÷- - pç ÷ç ÷çè ø
.
16/ ( ) ( )2 285 3
P 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 0
A cos0 cos20 cos40 ...... cos180= + + + + .
2/ 0 0 0 0
B cos20 cos40 cos60 ...... cos180= + + + + .
3/ 0 0 0 0
C cos10 cos40 cos70 ...... cos170= + + + + .
4/ 0 0 0 0
D tan20 tan40 tan60 ...... tan180= + + + + .
5/ 0 0 0 0
E cot15 cot 30 cot 45 ...... cot165= + + + + .
6/ 0 0 0 0
F sin5 sin10 sin15 ...... sin360= + + + + .
7/ 0 0 0 0
G cot195 cot 210 cot 225 ...... cot 345= + + + + .
8/ 0 0 0 0
H cot15 .cot 35 .cot 55 .cot 75= .
9/ 0 0 0 0
I tan10 .tan20 .tan 30 ......tan 80= .
10/ 0 0 0 0
J tan1 .tan2 .tan 3 ......tan 89= .
11/ 2 0 2 0 2 0 2 0
K sin 28 sin 36 sin 54 cos 152= + + + .
12/ 2 0 2 0 2 0 2 0
L cos 2 cos 4 cos 6 ...... cos 88= + + + + .
13/ 2 0 2 0 2 0 2 0
M sin 10 sin 20 sin 30 ...... sin 90= + + + + .
14/ 2 0 2 0 2 0 2 0
N cos 10 cos 20 cos 30 ... cos 180= + + + + .
15/ 0 0 0 0 0
O sin20 sin40 sin60 ... sin340 sin360= + + + + + .
à r d y
1/ ( )0 0
A cos 315 .sin765= - .
2/ 0 0 0 0
B sin 32 .sin148 sin 302 .sin122= - .
3/ 0 0 0 0
C sin810 .cos540 tan135 .cot 585= + .
4/ ( ) ( )0 0 0 0
D sin 825 .cos 15 cos75 .sin 555= - + - .
5/ 0 0 0
E 2tan540 2cos1170 4sin990= + + .
6/
( )0 0
0
0 0
sin 234 cos216
F .tan 36
sin144 cos126
- -
=
-
.

7/
( )0 0
0
0 0
sin 324 cos666
G .cot 36
sin1206 cos36
- +
=
+
.
8/
( ) ( ) ( )
( )
0 0 0 0
0 0
sin 328 .sin 958 cos 508 .cos 1022
H
cot 572 tan 212
- - -
= -
-
.
9/
( )
( )
0 0
0
0 0
cos 288 .cot 72
I tan18
tan 142 .sin108
-
= -
-
.
10/ ( ) ( ) ( ) ( )0 0 0 0
J 2sin 790 x cos 1260 x tan 630 x .tan 1260 x= + + - + + - .
11/
( )0 0
0 0 0
2sin2550 .cos 1881
K
tan 368 2cos638 cos98
-
= +
+
.
12/
( )0 0 0
0 0
0
cos44 tan226 .cos406
L cos72 .cot18
cos316
+
= - .
13/
( )0 0 0 0
0 0
0
tan 46 .sin 44 cot 136 .sin 404
M tan 36 .tan 54
cos316
+ -
= - .
14/
( ) ( ) ( )
( )
0 0 0 0
0 0
sin 328 .sin 958 cos 508 .cos 1022
N
cos572 tan 212
- - -
= -
-
.
15/
4 17
sin cos
3 6O
23 10
sin cos
6 3
p p
+
=
æ öp p÷ç ÷+ -ç ÷ç ÷çè ø
.
16/
( ) ( )
( )
( ) ( )
( )
sin 4,8 .sin 5,7 cos 6,7 .cos 5,8
P
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 cos
2 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 sin2A
2
- + +
= - .
9/
A B 3C
sin cosC
2
+ +
= . 10/
A B 2C 3C
tan cot
2 2
+ -
= .
11/
A 3B C
cos sin B
2
+ +
= - . 12/
A 2B C 3B
cot tan
2 2
- +
= .

10 – 2 s. Lê Vă à
Page - 15 -

NG
d y . y r
1/ 0 0 0 0
A sin12 .cos48 cos12 .sin 48= + .
2/ 0 0 0 0
B cos38 .cos22 sin38 .sin22= - .
3/ 0 0 0 0
C sin10 .cos55 cos10 .sin55= - .
4/ 0 0 0 0
D sin36 .cos6 sin126 .cos84= - .
5/ 0 0 0 0
E cos112 .cos23 sin112 .sin23= - .
6/ 0 0 0 0
F sin200 .sin 310 cos340 .cos50= + .
7/ 0 0 0 0 0
G cos11 .cos21 cos69 .cos79 cos10= + - .
8/ 0 0 0 0 0
H cos68 .cos78 cos22 .cos12 sin100= + - .
9/ ( ) ( )0 0 0 0
I cos 53 .sin 337 sin 307 .sin113= - - + .
r s
1/ 0
15 . 2/ 0
75 . 3/ 0
105 . 4/ 0
285 .
5/
19
12
p
. 6/
5
12
p
. 7/
7
12
p
. 8/
13
12
p
.
r s
1/
0
0
1 tan15
A
1 tan15
+
=
-
. 2/
0 0
0 0
tan25 tan20
B
1 tan25 tan20
+
=
-
.
3/
0 0 0 0
0 0 0 0
sin10 cos20 sin20 cos10
C
cos17 cos13 sin17 sin13
+
=
-
. 4/
0 0 0
0 0
tan225 cot 81 cot 69
D
cot 261 tan201
-
=
+
.
5/
0 0 0 0
0 0 0 0
sin 73 cos3 sin 87 cos17
E
cos132 cos62 cos42 cos28
-
=
+
. 6/
0 o0 0
0 0
cot 225 cot 79 ot 71
F
cot 259 cot 251
-
=
+
.
C – CÔNG THỨC C NG
  .
  .
 .  .
:
 .  .

7/ 2 0 2 0
G cos 75 sin 75= - . 8/ 2 0 2 0 2 0
H sin 20 sin 100 sin 140= + + .
9/ 2 0 0 2 0
I cos 10 cos110 cos 130= + + .
10/ 0 0 0 0 0 0
J tan20 .tan 80 tan 80 .tan140 tan140 .tan20= + + .
11/ 0 0 0 0 0 0
K tan10 .tan70 tan70 .tan130 tan130 .tan190= + + .
12/ 0 0 0 0 0 0
L sin160 cos110 sin250 cos340 tan110 tan 340= + + .
13/ ( )( ) ( )( )0 0 0 0 0 0 0 0
M cos70 cos50 cos310 cos290 cos40 cos160 cos320 cos380= + + + + - .
r s
1/ A cosx x
3
æ öp÷ç ÷= +ç ÷ç ÷çè ø
1
sin x
3
= à 0 x
2
p
< < .
2/ B sin x
3
æ öp ÷ç ÷= -ç ÷ç ÷çè ø
12
cosx
13
= - à
3
x
2
p
p < < .
3/ ( )0
C cos x 30= - tan x 2= à 0
0 x 90< < .
4/ D tan x
3
æ öp÷ç ÷= +ç ÷çè ø
3
sin x
5
= à x
2
p
< < p .
5/ E cos x
3
æ öp ÷ç ÷= -ç ÷çè ø
12
sin x
13
= - à
3
x 2
2
p
< < p .
6/ F cot x
4
æ öp÷ç ÷= -ç ÷ç ÷çè ø
4
sin x
5
= - à
3
x
2
p
p < < .
7/ G tan x
4
5
cot x 2
2
æ öp ÷ç ÷- =ç ÷ç ÷çè ø
.
8/
7
H sin 2x
4
2
cot x
3
= .
9/ ( ) ( )I cos a b .cos a b= + -
1
cosa
3
= à
1
cosb
4
= .
( )0 04
sin , 0 90
5
a = < a < à ( )0 08
sin , 90 180
17
b = < b < . y r
( )A cos= a + b à ( )B sin= a - b .
8 5
sin , tan b
17 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
p
a - 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 3sin7x cos7x= - .
3/ ( )2 2
C a sin x bcosx, a b 0= + + ¹ . 4/ D 3 sin x sin x
3 6
æ ö æ öp p÷ ÷ç ç÷ ÷= - + +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
5/ E cos7x.cos5x 3sin2x sin7x.sin5x= - + . 6/ F 3 cos2x sin2x 2sin 2x
6
æ öp÷ç ÷= + + -ç ÷ç ÷çè ø
.
7/ G 2sin 2x 4sin x 1
6
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 sin4xcot2x cos4x= - .
3/ C cos6xtan3x sin6x= - .
4/ ( ) ( ) ( ) ( )D sin x y cos x y sin x y cos x y= + - + - + .
5/ ( ) ( ) ( ) ( )0 0 0 0
E cos 40 x cos x 20 sin 40 x sin x 20= - + - - + .
6/ ( ) ( ) ( ) ( )0 0 0 0
F sin 14 2x cos 16 2x cos 14 2x sin 16 2x= + - + + - .
7/ ( ) ( ) ( ) ( )0 0 0 0
G sin x 10 cos 2x 80 sin x 100 cos 2x 10= + - + + + .
8/ H sin x cos x sin x cos x
3 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 x
3 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 x
3 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 x
3 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 x
3 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
A
1 tan x tan 3x
-
=
+
. 2/
tan2x 1
B
1 tan2x
+
=
-
.

3/
( )
( )
0
0
tan2x cot 90 x
C
1 cot 90 2x tan x
+ +
=
+ -
. 4/
2 2
2 2
tan 2x tan x
D
1 tan 2x tan x
-
=
-
.
s
1/ ( ) ( )A sin a b sin a sin b
2
æ öp ÷ç ÷= + + - -ç ÷ç ÷çè ø
.
2/ ( )B cos a cos b cos a b
2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= - - - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
3/ 21
C 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 2
D sin a sin b cos a cos b= - .
5/
( )
( ) ( )
2sin a b
E tan b
cos a b cos a b
+
= -
+ + -
.
s
1/ 2 2
A cos x 3sin2x sin x= - - . 2/ 3
B 4sin x 3sin x 3cos3x= - + .
3/ ( ) ( )0 0
C sin x 45 cos x 45= + + + . 4/ D tan3x tanx sin2x= - - .
5/ 2
E tan2x cot x 8cos x= + - .
s
1/ sin2x 2sinxcosx= .
2/ 2 2
cos2x cos x sin x= - .
3/
2
2tan x
tan2x
1 tan x
=
-
.
4/ 3
sin3x 3sin x 4sin x= - .
5/ 3
cos3x 4cos x 3cosx= - .
6/ cosx sin x 2 cos x 2 sin x
4 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 x
4 4
æ ö æ öp p÷ ÷ç ç÷ ÷- = + = - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
8/ ( ) ( ) 2 2 2 2
sin x y sin x y sin x sin y cos y cos x+ - = - = - .
9/ ( ) ( ) 2 2 2 2
cos x y cos x y cos x sin y cos y sin x+ - = - = - .
s
1/ sin x sin x 2 sin x
4 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/ 2
4sin x sin x 4sin x 3
3 3
æ ö æ öp p÷ ÷ç ç÷ ÷+ - = -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
3/ ( ) ( ) ( )sin xsin y z sin ysin z x sin zsin x y 0- + - + - = .
4/ ( ) ( ) ( )cosxsin y z cosysin z x coszsin x y 0- + - + - = .
5/ ( ) ( )tan x y tan x tan y tan x y tan x tan y+ - - = + .
6/ tan2x tan x tan2x tan x tan x tan x 1
6 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 3
3 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 2
cos 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 o
cos70 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
0
cosa cosb cosbcosc cosccosa
- - -
+ + = .
3/
( ) ( ) 2 2
2 2
sin a b sin a b
tan a tan b
cos a cos b
+ -
= - .
4/
( ) ( ) 2 2
2 2
cos a b cos a b
1 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 2tana+ = . HD:
( )
( )
b a b a
2a b a b a
ìï = + -ïïí
ï + = + +ïïî
.
3/ tana 2tanb= ( ) ( )sin a b 3sin a b+ = - .
4/
1
tana.tan b
3
= - ( ) ( )cos a b 2cos a b+ = - . HD r .
5/ ( )5sin b sin 2a b= + ( ) 3
tan a b tana
2
+ = .
6/ ( )sin b sin a cos a b= + ( )2tana tan a b= + . HD: ( )b a b a= + - .

7/ ( )cos 2a b 1+ = ( ) b
tan a b tana 2tan
2
+ - = .
8/ ( ) ( )cos a b kcos a b+ = -
1 k
tan a tan b
1 k
-
=
+
. HD:
( )
( )
a 2b a b b
a a b b
ìï + = + +ïïí
ï = + -ïïî
.
s
1/
( )sin a b c
tana tan b tan c tana tan b tan c
cosa cosbcosc
+ +
+ + - = .
2/ ( ) ( )2 2 2
cos x cos x y cos y 2cosxcosycos x y 1+ + + - + = .
r yê Lê – . ă 2007
s
1/ 2
A sin x cos x cos x
3 3
.
2/ 2 2 2
B cos x cos x cos x
3 3
æ ö æ öp p÷ ÷ç ç÷ ÷= + + + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
3/ 2 2 22 2
C sin x sin x sin x
3 3
.
4/ 2 2 22 2
D cos x cos x cos x
3 3
.
5/
3 3
a cos x cos3x a sin x sin 3x
F , a const
cosx sin x
- +
= + = .
r 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
. r
0 0 0
P 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/ ( )0sinC
tan A tan B, A,B 90
cosA.cosB
= + ¹ .
5/ ( )0
tan 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 sin
2 2 2 2 2
= - .
8/
A B C B C
cos sin cos cos sin
2 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 1
2 2 2 2 2 2
+ + = .
10/
A B C A B C
cot cot cot cot .cot .cot
2 2 2 2 2 2
+ + = .
11/ ( )0cosC cosB
cot 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 sin
2 2 2 2 2 2 2 2 2 2 2 2
= + + .
13/ 2 2 2A B C A B C
sin 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/ tanA tanB tanC 3 3, A,B,C+ + ³ " .
2/ 2 2 2
tan A tan B tan C 9, A,B,C+ + ³ " .
3/ 6 6 6
tan A tan B tan C 81, A,B,C+ + ³ " .
4/ 2 2 2A B C
tan tan tan 1
2 2 2
+ + ³ .
5/
A B C
tan tan tan 3
2 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 2
a b c ab bc ca+ + ³ + + à
A B B C C A
tan .tan tan .tan tan .tan 1
2 2 2 2 2 2
+ + = .
5/ r
2
A B C
tan tan tan
2 2 2
æ ö÷ç ÷+ +ç ÷çè ø
à s d 3/ .


NG
R T G N VÀ T NH GIÁ TR C A BI U THỨC
r s d y
1/ 2 2
A cos sin
8 8
p p
= - . 2/ 0 3 0
B 3cos10 4cos 10= - .
3/ ( )0 2 0
C sin20 1 4cos 20= - . 4/ 3 0 0
D 4sin 40 3sin130= + .
5/ 3 0 0
E 4sin 50 3cos140= + . 6/
0
2 0
tan15
F
1 tan 15
=
-
.
7/
2
7
tan
8G
1 tan
8
p
=
p
-
. 8/ 0 0
1 3
H
sin10 cos10
= - .
9/ 0
0
1
I 4sin70
sin10
= - . 10/ 2 2 5
J tan tan
12 12
p p
= + .
11/ 2 0 2 0
K tan 36 tan 72= . 12/ 2 0 2 0 0
L cos 70 sin 40 sin100= + .
13/ 2 0 2 0 0
M cos 20 2sin 55 2sin65= + - . 14/ 0 0 0 0
N sin 6 .sin 42 .sin 66 .sin 78= .
D – CÔNG THỨC NHÂN
Công thức nhân đôi
 .
 .
 .
 .
Công thức hạ bậc Công thức nhân ba ở r
 .  .
 .  .
 .  .

10 – 2 s. Lê Vă à
Page - 23 -
r s d y
1/ 0 0
A cos36 cos72= . 2/
2
B cos cos
5 5
p p
= .
3/ C sin .cos cos
8 8 4
p p p
= . 4/ D sin .cos .cos
16 16 8
p p p
= .
5/ 0 0 0
E sin10 sin50 sin70= . 6/
2 4
F cos cos cos
7 7 7
p p p
= .
7/
4 5
G cos cos cos
7 7 7
p p p
= . 8/ 0 0 0 0
H sin6 cos12 cos24 cos48= .
9/ 0 0 0 0
I sin6 sin 42 sin66 sin78= . 10/ 0 0 0 0 0
J sin5 .sin15 .sin25 .... sin75 .sin 85= .
11/ 0 0 0 0 0
K cos10 .cos20 .cos30 ...cos70 .cos80= .
12/ 0 0 0 0
L 8tan18 cos18 cos36 cos72= .
13/ 0 0 0 0
M cos20 cos40 cos60 cos80= .
14/
2 3 4
N cos cos cos cos
9 9 9 9
p p p p
= .
15/ O 96 3 sin cos cos cos cos
48 48 24 12 6
p p p p p
= .
16/
2 4 8 16 32
P cos .cos .cos .cos .cos
31 31 31 31 31
p p p p p
= .
17/
2 4 8 16
Q cos cos cos cos cos
33 33 33 33 33
p p p p p
= .
18/
2 3 4 5 6 7
R cos .cos .cos .cos .cos .cos .cos
15 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 cos20
A
cos35 cos15 sin 35 sin15
+
=
-
.
2/
0
0 0
4 0 4 0
sin60
B 3sin15 cos15
sin 15 cos 15
= -
-
.
3/ 0 0 0 0
C tan9 tan27 tan63 tan 81= - - + .
4/ D 4tan 2tan tan cot
8 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 156
E
tan 252 cot 342
+ - -
=
+
.

1/ ( )
2
A sin x cosx= + . 2/ 2 2
B 1 4sin xcos x= - .
3/ C sinxcosxcos2x= . 4/ 4 4
D cos 2x sin 2x= - .
5/ 2 2
E cos x sin x
2 2
. 6/ F sinxcosxcos2x= .
7/ G 4sin x sin x sin 2x
2 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
. 8/ 2 2x x
H sin sin
8 2 8 2
æ ö æ öp p÷ ÷ç ç÷ ÷= + - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
9/ ( )2 0
I sin 8x 2cos 45 4x= + + . 10/ ( )J sin2x cos2x 2cosx sinx cosx 1= + - + + .
r s d y
1/
2
2cos x 1
A
sin x cosx
-
=
+
. 2/
2
1 2sin 2x
B
cos2x sin2x
-
=
-
.
3/
( )( )
2
C
1 tan x 1 cot x
=
- +
. 4/ ( )2
D 1 tan x cot x= - .
5/
sin2x cos2x
E
sin x cosx
= - . 6/
cot x tan x
F
cos2x
-
= .
7/
3
1 sin x
2
G
1 sin x
2
æ öp ÷ç ÷+ -ç ÷ç ÷çè ø
=
æ öp ÷ç ÷+ +ç ÷ç ÷çè ø
. 8/
2 2
2 2
sin 2x 4sin x
H
sin 2x 4sin x 4
-
=
+ -
.
9/
sin 4x cos2x
I .
1 cos4x 1 cos2x
=
+ +
. 10/
sin 3x cos3x
J
sin x cosx
-
=
+
.
11/
1 cos x
2x
K tan .
4 2
sin x
2
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 x
2sin2x
= + + .
13/
1 cos2x sin2x
M .cot x
1 cos2x sin2x
- +
=
+ +
. 14/
( )2
2 sin2x 2cos x 1
N
cosx sin x cos3x sin 3x
+ -
=
- - +
.
15/ O 1 sin x 1 sin x, 0 x
4
æ öp÷ç ÷= + + - < <ç ÷ç ÷çè ø
. 16/
1 1 1
P cot 8x
sin2x sin 4x sin 8x
= + + + .
17/ O 1 sin2x 1 sin2x, x
4 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 , x
5 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 , x
13 2
p
= - p < < .
4/ cos2x, sin2x, tan2x khi tan x 2= .
5/
4 3
sin x, cosx khi sin2x , x
5 2 2
p p
= - < < .
6/
7
cos2x, sin2x, tan2x khi tan x
8
= .
7/ sin2x, cos2x, tan2x khi tan x 2 3, 0 x
2
p
= - < < .
Tính giá tr c a bi u th ng giác, khi bi t
1/ 5 5
A sin xcosx cos xsin x= - x
16
p
= .
2/ 4 3 3 4
B sin x sin xcosx cos xsin x cos x= + - + x
48
p
= .
3/ C cos4x= tan x cot x 3+ = .
4/ ( )0
D cos 270 4x= + ( )0
cot 45 x 2+ = .
5/ 4 4
E sin x cos x= + sin x cosx m+ = m Î ¡ .
6/
tan x cot x
F
tan x cot x
-
=
+
3
sin x cosx , 0 x
4 4
æ öp÷ç ÷= < <ç ÷ç ÷çè ø
.
r 0 5 1
sin18
4
-
= .
r d y
1/ 0
cos15 . 2/ 0
cos18 . 3/ 0
sin36 . 4/ 0
cos36 .
x
tan
2
1/
4
cosx , 0 x
5 2
p
= < < . 2/
24 3
tan x , x
7 2
p
= p < < .
3/
7
sin x cosx , 0 x
2 6
p
+ = < < .
Cho
1
tan x , 0 x
2 2
æ öp÷ç ÷= < <ç ÷ç ÷çè ø
à
1
tan y , 0 y
3 2
æ öp÷ç ÷= < <ç ÷ç ÷çè ø
.
x y+ . 2/ ( )sin x 2y- ( )cos 2x y+ .
1/
0 x
2
tan x 2 1
ìï pï < <ïïí
ïï = -ïïî
. 2/
0 x
2
6 2
cosx
4
ìï pï < <ïïïí
ï +ï =ïïïî
.

e cos2x s
1/ 2
A 1 cos x= + . 2/ 2 2
B sin xcos x= .
3/
2
2
1 sin x
C
cos x
+
= . 4/
2
2
1 cot x
D
1 cot x
+
=
-
.
5/ 6 6
E sin x cos x= + . 6/ 6 2 6 2
F sin xcos x cos xsin x= + .
e
x
t tan
2
= s
1/ A sinx= . 2/ B cosx= .
3/
sin x
C
3 2cosx
=
-
. 4/ D tan x cot x= + .
5/
1 tan x
E
1 cot x
-
=
+
. 6/
tan x sin x
F
tan x sin x
+
=
-
.
CHỨNG INH Đ NG THỨC
s
1/ 4 4
cos x sin x cos2x- = . 2/ ( )2
sin 4x 4sin x cosx 1 2sin x= - .
3/ 2 2
cos 2x sin x cosxcos3x- = . 4/ 4 2
cos4x 8cos x 8cos x 1= - + .
5/ 4
8sin x 3 4cos2x cos4x= - + . 6/ 4 4 3 1
sin cos x cos4x
4 4
+ = + .
7/ 4 4 2 2
sin cos x 6cos xsin x cos4x+ - = . 8/ 6 6 5 3
sin x cos x cos4x
8 8
+ = + .
9/ 6 6 15 1
sin x cos x cos2x cos6x
16 16
- = + . 10/ ( )6 6 2x x 1
sin cos cosx sin x 4
2 2 4
- = - .
11/ 8 8 7 1
cos x sin x cos2x cos6x
8 8
- = + . 13/ ( )3 3 4 4
cos xcos3x sin xsin3x 3 sin x cos x 2- = + - .
r a b
4
p
+ = ( )( )1 tana 1 tan b 2+ + = .
s
1/
1 cos2x
tan x
sin2x
-
= . 2/
2
tan x cot x
sin2x
+ = .
3/ cot x tanx 2cot2x- = . 4/
cosx x
cot
1 sin x 4 2
æ öp ÷ç ÷= -ç ÷ç ÷ç- è ø
.
5/ 21 cosx x
.tan 1
1 cosx 2
+
=
-
. 6/
1
cot 2x cot x
sin2x
+ =
7/
x 2
tan cot
4 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 2tan2x
4 4
æ ö æ öp p÷ ÷ç ç÷ ÷+ - - =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
. 10/ 3 3 1
cos xsin x sin xcosx sin 4x
2
- = .
11/ 3 3 3
cos3xsin x sin 3xcos 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 3tan3x
3 3
æ ö æ öp p÷ ÷ç ç÷ ÷+ + + + =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
. 14/ cot x tanx 2tan2x 4tan4x 8cot8x- - - = .
s
1/
1 sin x x
cot
cosx 4 2
æ ö+ p ÷ç ÷= -ç ÷ç ÷çè ø
. 2/
2
1 1 2sin x
tan2x
cos2x 1 sin2x
-
+ =
-
.
3/
( )
2sin2x sin 4x
tan2x cosx
2 cosx cos3x
+
=
+
. 4/ 2 26 2cos4x
tan x cot x
1 cos4x
+
= +
-
.
5/
2
1 2sin x 1 tan x
1 sin2x 1 tan x
- -
=
+ +
. 6/
1 x
1 tan tan x
cosx 2
æ ö÷ç ÷+ =ç ÷ç ÷çè ø
7/
1 sin2x
tan x
cos2x 4
æ ö+ p ÷ç ÷= +ç ÷ç ÷çè ø
. 8/ 22sin x sin2x x
tan
2sin x sin2x 2
-
=
+
.
9/
2
2
1 2sin x
1
2tan x cos x
4 4
-
=
æ ö æ öp p÷ ÷ç ç÷ ÷+ +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
. 10/
x
sin x sin x2 tan
x 2
1 cosx cos
2
+
=
+ +
.
11/
1 cosx cos2x
cot x
sin2x sin x
- +
=
-
. 12/
1 cos4x 1
sin 4x
cot x tan x 2
+
=
-
.
13/
( )( )
sin2x cosx x
tan
21 cos2x 1 cosx
=
+ +
. 14/
( )
4 4 2
2sin x cos x cos x x
cos
21 1 cosx
- +
=
-
.
15/ 2 2 6 2cos4x
cot x tan x
1 cos4x
+
+ =
-
. 16/
2
x x 4
cot tan
2 2 1 2tan x cot 2x
æ ö÷ç ÷- =ç ÷ç ÷ç -è ø
.
17/
2
cot 2x 1
cos8xcot 4x sin 8x
2cot 2x
-
- = . 18/
2
1 1 2sin x
tan2x
cos2x 1 sin2x
-
+ =
-
.
19/
1 sin2x cos2x
tan 4x
cos4x sin2x cos2x
-
- =
+
.
20/
cosx sin x cosx sin x
2tan2x
cosx sin x cosx sin x
+ -
- =
- +
.
21/
x x
cos sin 12 2 tan x
x x cosx
cos sin
2 2
-
= -
+
.

22/ 2
1 cosx cos2x cos3x
2cosx
2cos x cosx 1
+ + +
=
+ -
.
23/
1 x x
2 cot 2x cot tan
sin2x 2 2
æ ö÷ç ÷+ = -ç ÷ç ÷çè ø
.
24/
4 4
sin x sin2x cos x
cos2x
tan2x 1
+ -
=
-
.
25/
2 2
4
2 2
sin 2x 4sin x
tan x
sin 2x 4sin x 4
-
=
+ -
.
26/
2 2
2 2
sin 3x cos 3x
8cos2x
sin x cos x
- = .
27/
2 3
cos x cos3x sin x sin 3x
3
cosx sin x
- +
+ = .
28/ 2
1 1 sin2x
1 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 a
A cos cos cos ... cos
2 a2 2 2 2 .sin
2
= = .
2/
n
2 n 1
B cos .cos ... cos
2n 1 2n 1 2n 1 2
p p p
= =
+ + +
.
3/
2 4 2n 1
C cos .cos ... cos
2n 1 2n 1 2n 1 2
p p p
= = -
+ + +
.
4/
1 1 1 1 1 1 x
D cosx cos , 0 x
2 2 2 2 2 2 8 2
æ öp÷ç ÷= + + + = < <ç ÷ç ÷çè ø
.
s y à s
1/ 2sin2x 2sin x x
A tan
sin2x 2sin x 2
-
= +
+
.
2/
3 3
cos x cos3x sin x sin 3x
B
cosx sin x
- +
= + .
3/ 4 1
C 4sin x 2cos2x cos4x
2
= + - .
4/ 4 2 2 4
D 3cos2x 5sin x 4sin xcos x cos x= + + - .
5/
2
tan x 1
E cot x cos4xcot 2x sin 4x
2
-
= + + .

10 – 2 s. Lê Vă à
Page - 29 -
6/ 4 4 4 4 3
F 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
.
BIẾN Đ I THÀNH 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/ ( )( ) 2
C 2sin x 1 2cos2x 2sin x 3 4sin x 1= - + + - + .
4/ ( )( ) 2
D 2cosx 1 2cosx 2sin x 1 4sin x 3= - + + + - .
5/ ( )( ) 2
E 2sin x 1 3cos4x 2sin x 4 4cos x 3= + + - + - .
6/ F 1 sin x cosx sin2x cos2x= + + + + .
7/ ( )2
G 1 cosx cos x sin x 1 cosx= + + - + .
8/ ( ) ( )2 2
H 1 sin x cosx sin2x 1 cos x sin x 1= + - + + - .
à s
1/ 3 3
A sin xcosx cos xsinx= - .
2/ 3 3
B sin x cos x sin x cosx= + - + .
3/ 3 3
C sin x cos x cos2x= + - .
4/ 2 3
D cos x sin x cosx= + + .
5/ 3 2
E cos x cos x 2sin x 2= + + - .
6/ 3 3 2
F 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 THỨC BIẾN Đ I


NG
CÔNG THỨC BIẾN Đ I T NG THÀNH T CH
r
1/ 0cos2x cos4x
A khi x 20
sin 4x sin2x
-
= =
-
.
2/
cosx.cos13x
B khi x
cos3x cos5x 17
p
= =
+
.
3/
cosx.cos10x
C khi x
cos2x cos4x 13
p
= =
+
.
4/
tan2x sin2x 2
D khi tan x
tan2x sin2x 15
-
= =
+
.
 Công thức biến đổi tổng thành tích
● . ● .
● . ● .
● . ● .
● . ● .

● .
● .
 Công thức biến đổi tích thành tổng
● .
● .
● .

10 – 2 s. Lê Vă à
Page - 31 -
5/
sin x sin2x sin 3x 1
E khi sin x , x
cosx cos2x cos3x 3 4 2
+ + p p
= = < <
+ +
.
1/
1 2cosx
A
1 2cosx
-
=
+
. 2/
3 2cos3x
B
3 2cos3x
-
=
+
.
3/
1 2cos2x
C
3 2sin2x
-
=
+
. 4/
2 2sin2x
D
2 2sin2x
-
=
+
.
5/
sin5x sin 3x
E
2cos4x
-
= . 6/
cos4x cos2x
F
sin 4x sin2x
-
=
+
.
7/
( )sin x y
G
sin x sin y
+
=
+
. 8/
sin x sin y
H
cosx cosy
+
=
+
.
9/
cosx sin x
I
cosx sin x
+
=
-
. 10/
2 2
2 2
sin 4x sin 2x
J
cos x cos 2x
-
=
-
.
11/
2
sin 4x
K
2cosx cos3x cos5x
=
+ +
. 12/
sin2x
L
tan x cot 2x
=
+
.
13/
tan 3x tan 5x
M
cot 3x cot 5x
+
=
+
. 14/
tan2x cot 2x
N
1 tan2x.tan 4x
+
=
+
.
15/
1 sin2x cos2x
O
1 sin2x cos2x
+ +
=
+ -
. 16/
1 sin 4x cos4x
P
1 cos4x sin 4x
+ -
=
+ +
.
17/
sin2x 2sin 3x sin 4x
Q
cos3x 2cos4x cos5x
+ +
=
+ +
. 18/
sin x sin 4x sin 7x
R
cosx cos4x cos7x
+ +
=
+ +
.
19/
cos2x sin 4x cos6x
S
cos2x sin 4x cos6x
- -
=
+ -
. 20/ 2
1 cosx cos2x cos3x
T
2cos x cosx 1
+ + +
=
+ -
.
21/
( )2
2 sin2x 2cos x 1
U
cosx sin x cos3x sin 3x
+ -
=
- - +
. 22/
( )
( )
( )
( )
sin x y sin x cos x y cosx
V
sin x y sin x cos x y cosx
+ - + +
= -
+ + - -
.
23/
cos7x cos8x cos9x cos10x
X
sin 7x sin 8x sin 9x sin10x
- - +
=
- - +
. 24/
sin 4x sin 5x sin 6x
Y
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 2sin2x 1= - .
9/ I 3 2cos2x= + . 10/ ( ) ( )J sin a b sin a b= + - - .
11/ 2 2
K cos x cos y= - . 12/ L 1 sin x cos2x= + - .

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 0
X cos46 cos22 2cos78= - - .
23/
2 3
Y cos cos cos
7 7 7
p p p
= - + . 24/
2 4 6
Z cos cos cos
7 7 7
p p p
= + + .
25/ 0 0 0
W sin70 sin20 sin50= - + . 26/ ( )cos5x cos7x cos 6xD = + - p + .
27/ cos 5x sin x cos3x
2
æ öp ÷ç ÷F = + + -ç ÷ç ÷çè ø
. 28/
x y
sin 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 0
cos 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 0
sin 35 cos40 sin55 cos20+ + + . 8/ 0 0 0 0
sin57 sin59 sin93 sin61- - + .
9/ 0 0 0 0
sin 47 sin61 sin11 sin25+ - - . 10/ cos5x 3cos7x 3cos9x cos11x+ + + .
11/ 2 3x
1 sin x cos5x sin7x 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 2
cos x cos 2x cos 3x 1+ + - . 2/ 2 2 2 3
sin x sin 2x sin 3x
2
+ + - .
3/ 2 2 2
sin 3x sin 2x sin x- - . 4/ 2 2 2
sin x cos 2x cos 3x- - .
5/ 2 2 2
sin x 2sin 2x sin 3x- + . 6/ ( )2 2
sin 4x cos 6x sin 10,5 10x- - p + .
7/ 2 2 2 2
sin 3x cos 4x sin 5x cos 6x- - + . 8/ 2 2 2 2 3
cos x cos 2x cos 3x cos 4x
2
+ + + - .
9/ 2 2 2 2
cos x cos 2x cos 3x cos 4x 2+ + + - . 10/ 2 25 9x
cos3x sin7x 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 cos5x
2
æ öp ÷ç ÷+ + p + -ç ÷ç ÷çè ø
. 12/
cos3xcos 5x
2
2sin2x cos 3x
2 cos5x
æ öp ÷ç ÷-ç ÷çæ ö ÷çp è ø÷ç ÷- + -ç ÷ç ÷çè ø
.
1/ 0 0 6
sin 75 cos75
2
+ = . 2/ 0 0 0
cos12 cos48 sin18- = .
3/ 0 0 0
sin65 sin55 3cos5+ = . 4/ 0 0
tan267 tan93 0+ = .
5/ 0 0 0
cos85 cos35 cos25 0+ + = . 6/ 0 0 0 0
tan9 tan27 tan63 tan 81 4- - + = .
7/ 0 0 0 0 1
cos24 cos48 cos84 cos12
2
+ - - = .
8/ 0 0 0 0 0 0
tan9 tan15 cot15 cot 9 tan27 cot 27+ + + = + .
9/ 2 31 1
cosx cos3x cos5x 8sin xcos x
2 2
- - = .
10/ ( )sin x 1 2cos2x 2cos4x 2cos6x sin7x+ + + = .
11/ 4 x
1 4cosx 6sin2x 4sin x 16sin2xsin
2
+ + - = .
12/ ( ) ( )2 0 0
8sin xsin x 60 sin x 60 cos4x cos2x+ - = - .
13/ ( ) ( ) ( )2 0 0
sin x cosx cos4x 4sin2xsin x 15 cos x 15- - = + - .
Cho a b c+ = .
a b c
sina sin b sin c 4cos cos sin
2 2 2
+ + = .
r
1/ 6 6
A sin cos
24 24
p p
= + .
2/ 2 2
B tan cot
12 12
p p
= + .
3/
3 5 7 9
C cos cos cos cos cos
11 11 11 11 11
p p p p p
= + + + + .
4/
2 4 6 8 10
D cos cos cos cos cos
11 11 11 11 11
p p p p p
= + + + + .
5/ 0 2 3 4 5 6
E 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 0
F sin 40 cos10 cos40 sin10 cos140= + - + + .
r A à à A,B,C. r

1/
A B C
sin A sin B sinC 4cos cos cos
2 2 2
+ + = .
2/
A B C
sin A sin B sinC 4sin sin cos
2 2 2
+ - = .
3/
A B C
cosA cosB cosC 1 4sin sin sin
2 2 2
+ + = + .
4/ sin2A sin2B sin2C 4sin Asin BsinC+ + = .
5/ 1 cos2A cos2B cos2C 4cosAcosBcosC+ + + = - .
6/ ( )2 2 2
sin A sin B sin C 2 1 cosAcosBcosC+ + = + .
7/ 2 2 2
cos A cos B cos C 1 2cosAcosBcosC+ + = - .
CÔNG THỨC BIẾN Đ I T CH THÀNH T NG
à s
1/
2
A sin sin
5 5
p p
= . 2/ B sin5xcos3x= .
3/
3
C sin cos
4 6
p p
= . 4/
7
D sin cos
12 12
p p
= .
5/ ( ) ( )E sin x y cos x y= + - . 6/ ( ) ( )0 0
F sin x 30 cos x 30= + - .
7/ G 2sinxsin2xsin3x= . 8/ H 8cosxsin2xsin3x= .
9/ I sin x sin x cos2x
6 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 0
A cos75 cos15= .
2/
5
B sin sin
12 12
p p
= .
3/
11 5
C sin cos
12 12
p p
= .
4/ 0 0 0
D sin20 .sin40 .sin80= .
5/ 0x 5x
E sin .sin , khi x 60
4 4
= = .
6/ 0 0 0 0
F sin20 sin40 sin60 sin80= .
7/ 4 5 7 11
G 2 .sin .sin .sin .sin
24 24 24 24
p p p p
= .
8/ 0 0 0 0 0 0
H sin5 .sin15 .sin25 .......sin65 sin75 .sin 85= .
9/ 2 0 0 0
I sin 10 cos70 cos50= + .

10 – 2 s. Lê Vă à
Page - 35 -
10/ 0 0 0 0 01
J cos10 cos50 cos5 cos25 cos10
2
= - + .
11/ 2 0 2 0 0 0
K sin 50 sin 10 sin50 sin10= + + .
12/ 2 0 2 0 0 0
L cos 73 cos 47 cos73 .cos47= + + .
13/ 2 0 2 0 0 0
M sin 50 sin 70 cos50 cos70= + - . A A ă 2001
14/ 0 0 0 0 01
N cos10 cos50 cos5 cos25 sin10
2
= - + .
15/ ( )
0
0 0
0
sin55
O 2 cos22 cos44
sin11
= + - .
16/
0
0 0
4 0 4 0
sin60
P 3sin15 sin75
sin 15 sin 75
= +
-
.
17/
0
0 0
0 0
1 cos40
Q 2 2 sin10 2sin 35
2cos5 sin 5
æ ö÷ç ÷ç= - - - ÷ç ÷ç ÷è ø
.
18/ 0
0
1
R 4sin70
sin10
= - .
19/ 0 0
1 3
S
sin10 cos10
= - .
20/
2 3
T cos cos cos
7 7 7
p p p
= - + .
21/
2 4 5 2 4 6 8
U cos cos cos cos cos cos cos
7 7 7 5 5 5 5
p p p p p p p
= + + + + + + + .
22/
2 4 6 8
V cos cos cos cos
9 9 9 9
p p p p
= + + + .
( )A 2sin x cosx cos3x cos5x= + + . s y r r
3 5
B cos cos cos
7 7 7
p p p
= + + .
s
1/ A cos11xcos3x cos17xcos9x= - .
2/ B sin18xcos13x sin9xcos4x= - .
3/ C sin xsin 3x sin 4xsin 8x= + .
4/ D sin2xsin6x cosxcos3x= - .
5/ E cos3xcos6x cos4xcos7x= - .
6/ ( ) ( )0 0
F sin xsin 60 x sin 60 x= - + .
7/ ( ) ( )0 0
G 8cosxcos 60 x cos 60 x 1= - + + .

8/
1
H cosxcos2xsin 3x sin12x
4
= - .
9/ I 4sin2xsin5xsin7x sin 4x= - .
10/
1
J sin2xsin6xcos4x cos12x
4
= + .
11/
1
K sin xsin2xsin 3x sin 4x
4
= - .
12/ L 4cosx sin x sin x cos2x
6 6
æ ö æ öp p÷ ÷ç ç÷ ÷= + - -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
13/ M sin 4xsin10x sin11xsin 3x sin7xsin x= - - .
14/
x 3x
N cos cos sin xsin 3x sin2xsin 3x
2 2
= - - .
15/
7x 3x x 5x
O sin cos sin cos sin2xcos7x
2 2 2 2
= + + .
16/ P cos2x cos4x cos6x 4cosxcos2xcos3x 2= + + - - .
17/ ( )Q sin x 1 2cos2x 2cos4x 2cos6x= + + + .
18/ ( )sin5x
R 2 cos2x cos4x
sin x
= - + .
19/ 2 2
S sin x sin x sin cos 2x
4 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 1
cos12 cos18 4cos15 cos21 cos24
2
+
+ - = - .
à A ă 2001
2/ 0 0 0 0
tan9 tan63 tan 81 tan27 4- + - = .
3/ 0 0 0 0 08 3
tan 30 tan 40 tan 50 tan 60 cos20
3
+ + + = .
4/ ( ) ( )0 0 1 1
cos 2x 60 cos 2x 60 cos4x
2 4
+ - = - .
5/ ( ) ( ) ( )2 0 0
sin x cosx cos4x 4sin2xsin x 15 cos x 15+ - = + - .
6/ 2 2 3
sin 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/ ( ) ( ) ( )sina sin b c sin bsin c a sin csin a b 0- + - + - = .
8/ ( ) ( )2 2 2
cos x 2cosa cosxcos a x cos a x sin a- + + + = .

10 – 2 s. Lê Vă à
Page - 37 -
9/ 8 8 7 1
8 8
- = + .
10/ 8 8 35 7 1
64 16 64
+ = + + .
11/
3 2
tan 4tan 11
11 11
p p
+ = .
12/ 0 0 0
tan20 tan40 tan80 3= .
13/ 3 0 2 0
8sin 18 8sin 18 1+ = . V à ă 2000
14/ 4 4 4 43 5 6 3
sin sin sin sin
16 16 16 16 2
p p p p
+ + + = .
15/ 0 0 0 0 0 0
tan10 tan25 tan25 tan55 tan55 tan10 1+ + = .
16/ 0 0 0 0
tan15 tan25 tan35 tan85 1= .
17/ 0 0 0
tan20 tan40 tan80 3 3- + = .
18/ 0 o 0 0
tan10 tan50 tan60 tan70 2 3- + + = .
19/ 0 0 0 0 0
tan20 tan 40 tan 80 tan 60 8sin 40+ + + = .
20/ 6 0 4 0 2 0
tan 20 33tan 20 27tan 20 3 0- + - = .
A r
1/ B C
3
p
- = à
1
sin B.sinC
2
= . S: B , C , A
2 6 3
p p p
= = = .
2/
2
B C
3
p
+ = à
1 3
sin B.cosC
4
+
= . S:
5
A , 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 c
cot
2 b
+
= .
Ch u ki n c à A cân:
1/ ( ) A B
a tan A b tan B a b tan
2
+
+ = + . 2/ 2
2tan B tanC tan B.tanC+ = .
3/ ( )sin A sin B 1
tan A tan B
cosA cosB 2
+
= +
+
. 4/
C 2sin A.sin B
cot
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 sinC
2
+ + £ . HD: C ng sin
3
p
vào VT.
2/
3
cosA cosB cosC
2
+ + £ . HD: C ng cos
3
p
vào VT.

3/ tan A tan B tanC 3 3+ + ³ .
4/
1
cosA.cosB.cosC
8
£ .
HD
1
cosA.cosB.cosC
8
- d .
Cho cos cosa cosbj = a,b k
2
p
¹ + p . 2a a b
tan tan tan
2 2 2
j + j -
= .
Cho ( )sin 2a b 5sin b+ = .
( )2tan a b
3
tana
+
=
a k
2
a b l
2
ìï pï ¹ + pïïïí
ï pï + ¹ + pïïïî
.
Cho ( )tan a b 3tana+ = . ( )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
cos
aB bA
+
= a - b
+
.
r
x a
tan
2 b
= A a sin x bcosx= + à à .
Cho a,b,c à à A,B,C.
γ, ,a b ở γ
a b c
cos , cos , cos
b c c a a b
a = b = =
+ + +
.
1/
γ2 2 2
tan tan tan 1
2 2 2
a b
+ + = . 2/
γ A B C
tan tan tan tan tan tan
2 2 2 2 2 2
a b
= .
1 1 1 tan 8x tan x
...
cosxcos2x cos2xcos3x cos7xcos8x sin x
-
+ + + = .
r 8 0 8 0 8 0
M sin 20 sin 40 sin 80= + + .
s 11 ă 200
r
0 0 0 0
0
2sin2 4sin 4 ... 178sin178 180sin180
N
cot1
+ + + +
= .
s
1/ ( )A cos cos3 cos5 ... cos 2n 1= a + a + a + + - a . S:
sin2n
A
2sin
a
=
a
.
2/
( )n 12 3
B sin sin sin ... sin
n n n n
- pp p p
= + + + + . S: B cot
2n
p
= .
3/
( )2n 13 5
C cos cos cos ... cos
n 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
= + + + a
5
p
= . S: D 1 5= - .
5/
n 1
1 1 1 1
E 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 1
tan2 x
E
x
tan
2
-
= .
6/
n
1 1 1 1
F ...
sin2x sin 4x sin 8x sin2 x
= + + + + .
7/ in3 3 2 3 n 1 3
n
x x x x
G sin 3s 3 sin ...... 3 sin
3 3 3 3
-
= + + + + .
yê 4 4 2 23 3 2
M tan tan 2 tan tan
4 4 7 7
æ öp p p p÷ç ÷= + + +ç ÷ç ÷çè ø
.
( ) ( )3 1
sin x 3sin x sin 3x
4
= - * . Khi thay
n
a
x
3
= à ( )* . y
3 3 n 1 3
n 2 n
a a a
S sin 3sin ... 3 sin
3 3 3
-
= + + + .
S: n
n n
1 a
S 3 sin sina
4 3
æ ö÷ç ÷= -ç ÷ç ÷è ø
.
sin2a
cosa
2sina
= . s y r r n 2 n
x x x
P cos cos ... cos
2 2 2
= .
S: n
n
n
sin x
P
x
2 sin
2
= .
1 x
cot cot x
sin x 2
= - .
( )n 1
n 1
1 1 1
S ... , 2 k
sin sin2 sin2
-
-
= + + + a ¹ p
a a a
.
S: n 1
S cot cot 2
2
-a
= - a .
2
tan x.tan2x tan2x 2tan x= - .
2 2 n 1 2
n 2 n n 1
a a a a a
S tan .tana 2tan .tan ... 2 tan .tan
2 22 2 2
-
-
= + + + .
S: n
n n
a
S tana 2 tan
2
= - .
Tính 2
sin 2x, bi t: 2 2 2 2
1 1 1 1
7
tan x cot x sin x cos x
+ + + = .
S:
8
9
.

sina m= à 0 0
270 x 450< < .
1/ ( )a 1
cos 1 m 1 m
2 2
= - + + - . 2/ ( )a 1
sin 1 m 1 m
2 2
= - + - - .
Cho 0
2
p
< a < . tan cot 2a + a ³ .
s
1/ 0 0
sin75 sin15 1+ > . 2/ 0 0
cos36 tan 36> .
s
1/ sin 3xcosx cos3xsin x cos2x 2- - £ .
2/ 3 sin 3x cos2xcosx sin2xsin x 2+ - £ .
3/ ( ) ( )0 0 1
sin xsin x 60 sin x 60
4
+ - £ .
4/ cosxcos3x sin2xsin4x 1- £ .
5/ 2
cos2xcosx sin xsin 3x sin xcos3x 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 , x
cosx 2 2
æ öp p÷ç ÷= + - < <ç ÷ç ÷çè ø
.
3/ y sin xcosxcos2xcos4x= .
4/ 4 4
y sin x cos x, 0 x
2
æ öp÷ç ÷= + £ £ç ÷ç ÷çè ø
.
5/ 2
y sin x 2 sin x= + - .
6/ 4 4 2 2
y cos x sin x cos xsin x= + + .
7/ 4 2 5
y sin 4x cos x cos2x
4
= - + .
8/ 6 6 4 4
y 2sin x 2cos x sin x cos x cos2x= + - - + .

Chuyên đề 6 góc lượng giác và công thức lượng giác

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