Chuyên đề Bồi dưỡng HSG môn Toán lớp 7. Mọi thông tin cần hỗ trợ tư vấn học tập, đăng ký học vui lòng liên hệ tới văn phòng gia sư thủ khoa Hà Nội theo số máy: 0936.128.126. Website: http://giasutoan.giasuthukhoa.edu.vn
Tuyển tập 100 đề thi HSG Toán lớp 7 có đáp án chi tiết - Thầy Thích.
Liên hệ đặt mua tài liệu:
- Điện thoại - Zalo: 0919.281.916
- Email: doanthich@gmail.com
- Website: www.toaniq.com
Chuyên đề Bồi dưỡng HSG môn Toán lớp 7. Mọi thông tin cần hỗ trợ tư vấn học tập, đăng ký học vui lòng liên hệ tới văn phòng gia sư thủ khoa Hà Nội theo số máy: 0936.128.126. Website: http://giasutoan.giasuthukhoa.edu.vn
Tuyển tập 100 đề thi HSG Toán lớp 7 có đáp án chi tiết - Thầy Thích.
Liên hệ đặt mua tài liệu:
- Điện thoại - Zalo: 0919.281.916
- Email: doanthich@gmail.com
- Website: www.toaniq.com
Contents:
1. Geometric Sequence
2. Geometric Means
3. Geometric Series
with activities
Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions
Giáo án bồi dưỡng học sinh giỏi Toán 7Lớp 7 Gia sư
Gia sư lớp 7 tại nhà Hà Nội chất lượng cao. ĐT hỗ trợ: (043).990.6260 - 0936.128.126. Trung tâm gia sư Hà Nội nhận gia sư Toán, Văn, Anh, Lý, Sinh, Tiếng Nhật, Tiếng Pháp lớp 7 tại nhà mọi trình độ từ cơ bản đến nâng cao.
Contents:
1. Geometric Sequence
2. Geometric Means
3. Geometric Series
with activities
Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions
Giáo án bồi dưỡng học sinh giỏi Toán 7Lớp 7 Gia sư
Gia sư lớp 7 tại nhà Hà Nội chất lượng cao. ĐT hỗ trợ: (043).990.6260 - 0936.128.126. Trung tâm gia sư Hà Nội nhận gia sư Toán, Văn, Anh, Lý, Sinh, Tiếng Nhật, Tiếng Pháp lớp 7 tại nhà mọi trình độ từ cơ bản đến nâng cao.
Tuyển tập 21 đề thi vào lớp 10 môn toán năm học 2014 - 2015 (có đáp án).
Mọi thông tin cần hỗ trợ tư vấn, đăng ký học tập ôn luyện thi môn Toán vào lớp 10, vui lòng liên hệ trực tiếp tới văn phòng chúng tôi theo số máy: 0936.128.126.
Tuyển tập các bài Toán Hình học lớp 9 ôn thi vào 10BOIDUONGTOAN.COM
Tuyển tập các bài Toán Hình học lớp 9 ôn thi vào 10. Mọi thông tin cần hỗ trợ tư vấn học tập, đăng ký học, mua tài liệu Toán lớp 9 vui lòng liên hệ: 0976.179.282.
Chuyên đề 6 góc lượng giác và công thức lượng giác
1. 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
. .
. ..
. .
2. 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 .
3. 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= + + - + .
4. s. Lê Vă à 5. –
Page - 4 - " A e f wer f rr w re e see s f d y "
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= + + - + .
5. 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
6. s. Lê Vă à 5. –
Page - 6 - " A e f wer f rr w re e see s f d y "
B – ĐƯỜNG TRÒN LƯỢNG GIÁC – CUNG GÓC LƯỢNG GIÁC
CUNG LIÊN KẾT
7. 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
8. 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 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< < .
9. 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
+
=
+
.
10. s. Lê Vă à 5. –
Page - 10 - " A e f wer f rr w re e see s f d y "
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
= .
11. 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
æ öp÷ç ÷-ç ÷ç ÷çè ø
.
21/
19
cot
4
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 22/
26
cos
3
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 23/
159
tan
4
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 24/
115
sin
6
æ öp÷ç ÷-ç ÷ç ÷çè ø
.
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
æ öp ÷ç ÷+ç ÷ç ÷çè ø
.
9/
3
cot x
2
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 10/
5
cos x
2
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 11/
11
tan x
2
æ öp ÷ç ÷+ç ÷ç ÷çè ø
. 12/
7
sin x
2
æ öp÷ç ÷+ç ÷ç ÷çè ø
.
12. s. Lê Vă à 5. –
Page - 12 - " A e f wer f rr w re e see s f d y "
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
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 22/ 2 9
tan x
2
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 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
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 26/ 2 9
cot x
2
æ öp÷ç ÷-ç ÷ç ÷çè ø
. 27/ 11 11
tan x
2
æ öp ÷ç ÷+ç ÷ç ÷çè ø
. 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
.
13. 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
- -
=
-
.
14. 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 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
- +
= .
15. 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
.
.
. .
:
. .
16. 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 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
æ öp÷ç ÷= +ç ÷ç ÷çè ø
5
cot x 2
2
æ öp ÷ç ÷- =ç ÷ç ÷çè ø
.
8/
7
H sin 2x
4
æ öp÷ç ÷= +ç ÷ç ÷çè ø
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 .
17. 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
+
=
-
.
18. s. Lê Vă à 5. –
Page - 18 - " A e f wer f rr w re e see s f d y "
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
.
19. 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= + - .
20. 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 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
æ ö æ öp p÷ ÷ç ç÷ ÷= + - +ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
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
æ ö æ öp p÷ ÷ç ç÷ ÷= + + + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø
.
4/ 2 2 22 2
D 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 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
= - .
21. 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/ .
22. 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 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
. .
. .
. .
23. 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
+ - -
=
+
.
24. 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 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
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 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+ = .
25. 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
.
26. 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/ 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- = .
27. 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
-
= -
+
.
28. 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 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
-
= + + .
29. 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
30. s. Lê Vă à 5. –
Page - 30 - " A e f wer f rr w re e see s f d y "
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
● .
● .
● .
31. 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= + - .
32. 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 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
.
33. 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
34. 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 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= + .
35. 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= - + + .
36. s. Lê Vă à 5. –
Page - 36 - " A e f wer f rr w re e see s f d y "
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- + + + = .
37. 10 – 2 s. Lê Vă à
Page - 37 -
9/ 8 8 7 1
sin x cos x cos2x cos6x
8 8
- = + .
10/ 8 8 35 7 1
sin x cos x cos4x cos8x
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.
38. 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.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
= - .
39. 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
.
40. 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 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= + - - + .