TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG
BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC
PHẦN V: BIẾN ĐỔI LƯỢNG GIÁC
A – HỆ THỨC LƯỢNG GIÁC CƠ BẢN
tan
sin
.
cos
cot
tan .cot 1 .
sin 2 cos2 1 .
1 tan2
cos
.
sin
1
.
cos2
1 cot2
1
.
sin2
BÀI TẬP ÁP DỤNG
Bài 1.
Chứng minh các đẳng thức sau
1/
cos2 x sin2 x 1 2 sin2 x .
2/
2 cos2 x 1 1 2 sin2 x .
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3/
3 4 sin2 x 4 cos2 x 1 .
4/
sin x cot x cos x tan x sin x cos x .
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5/
sin 4 x cos 4 x 1 2 sin 2 x cos2 x .
6/
cos4 x sin4 x cos2 x sin2 x .
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7/
4 cos2 x 3 1 2 sin x 1 2 sin x . 8/
1 cos xsin
2
x cos x cos2 x sin2 x .
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9/
sin4 x cos4 x 1 2cos2 x 2sin2 x 1 . 10/ sin 3 x cos x sin x cos 3 x sin x cos x .
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11/ tan2 x sin2 x tan2 x sin2 x .
12/ cot2 x cos2 x cot2 x cos2 x .
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Bài 2.
Chứng minh các đẳng thức sau
1/
tan x cot x
1
.
sin x cos x
2/
1 cos x
sin x
.
sin x
1 cos x
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3/
1
1
1.
1 tan x 1 cot x
4/
1 1 1 1 tan2 x 0 .
cos x
cos x
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5/
1 sin2 x
1 2 tan2 x .
2
1 sin x
6/
tan x tan y
tan x tan y
.
cot x cot y
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7/
1 cot4 x
2
1
.
2
sin x sin 4 x
8/
tan x
cos x
1
.
1 sin x
cos x
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9/
1
1 cos x1 cot x 1 cos x .
2
10/
1 cos x 1 cos x
4 cot x
.
1 cos x 1 cos x
sin x
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11/
sin x
1 cos x
2
.
1 cos x
sin x
sin x
12/
sin x cos x 1
cos x
.
sin x cos x 1 1 sin x
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sin2 x 2 cos x 1
cos x
13/
.
2
1 cos x
2 cos x cos x
14/
sin x cos x 1
2 cos x
.
1 cos x
sin x cos x 1
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Bài 3.
Chứng minh các đẳng thức sau
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1/
BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC
sin 6 x cos6 x 1 3 sin2 x cos2 x .
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2/
sin 6 x cos6 x sin2 x cos2 x 1 sin2 x cos2 x .
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3/
2
sin 8 x cos 8 x 1 2 sin2 x cos 2 x 2 sin 4 x cos 4 x .
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4/
sin 8 x cos8 x sin2 x cos2 x 1 2 sin2 x cos2 x .
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Bài 4.
Chứng minh các đẳng thức sau
1/
1 sin x cos x tan x 1 cos x1 tan x .
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2/
1 tan x1 cot x sin x cos x 1 2 sin x cos x .
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BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC
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3/
1 tan x cos
2
x 1 cot x sin 2 x sin x cos x .
2
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4/
sin 2 x tan x cos2 x cot x 2 sin x cos x tan x cot x .
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5/
sin 2 x tan 2 x 4 sin2 x tan 2 x 3 cos2 x 3 .
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Bài 5.
Chứng minh các đẳng thức sau
1/
sin2 cos2
tan 1
.
1 2 sin cos
tan 1
2/
tan2 sin2
tan 6 .
2
2
cot cos
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3/
cos2
sin2
1 sin cos .
1 tan 1 cot
4/
tan sin
1
.
3
sin
cos 1 cos
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5/
1
tan2 cot2 2 .
2
2
sin cos
6/
1
3 tan2
tan2
1.
cos2
cos2
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7/
tan2 tan2 sin2 sin 2
.
tan2 tan2
sin2 sin2
8/
2
1 cos 1 cos
1 2 cot .
sin sin2
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9/
1
1 cos 1 cot 1 cos .
2
10/ 1 tan tan2 tan3
sin cos
.
cos3
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11/
sin
cos
1 cot2
.
sin cos cos sin 1 cot2
12/
tan2 1 cot2
1 tan4
.
.
1 tan2 cot2
tan2 cot2
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TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG
2
1 sin
1 sin
4 tan2 .
13/
1 sin
1 sin
BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC
2
1 cos
1 cos
4 cot2 .
14/
1 cos
1 cos
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Bài 6.
Rút gọn các biểu thức sau
1/ P1 sin 4 sin2 cos2 .
2/ P2 sin 4 cos4 cos2 .
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3/ P3 sin2 sin2 cot2 .
4/ P4 cos2 cos2 cot2 .
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5/ P5 1 sin2 cot2 1 cot2 .
6/ P sin2 tan cos2 cot 2sin cos .
6
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7/
P7
2 cos2 1
.
sin cos
8/
P8
1 cos
2
sin
1
.
1 cos
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9/
P9
cot cot
.
tan tan
10/ P10 tan
cos
.
1 sin
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11/ P11
sin x tan x
sin x cot x .
tan x
12/ P12
cos x tan x
sin2 x
cot x cos x .
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13/ P13
cos2 x cot2 x
2
2
sin x tan x
14/ P14
.
1 sin2 x cos2 x
2
cos x
cos2 x .
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sin3 x sin x cos2 x cos x
15/ P15
.
1 2 sin x cos x
2
16/ P16
sin x cos x
1
tan x sin x cos x
.
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17/ P17
sin2 x sin2 x tan2 x
2
2
2
cos x cos x tan x
.
18/ P18
cot2 x cos2 x
2
cot x
1 cos2 x .
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TÀI LIỆU HỌC TẬP TOÁN 10 – CHẤT LƯỢNG
19/ P19
cos2 x sin2 y
2
2
sin x sin y
cot2 x cot2 y .
BÀI GIẢNG: BIẾN ĐỔI LƯỢNG GIÁC
20/ P20
1
sin x
cos x cot x
.
sin x 1 cot x
1 tan x
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21/ P21
1 cos x 1 sin x
1
1 cotx 1 .
tan x cotx . 22/ P22 1 cotx
sin x
cos x
sin x
sin x
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23/ P 1 tanx cos2 x 1 cotx sin2 x .
23
24/ P
24
, 0 x .
2
2
2
sinx cot x cos x
1
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Bài 7.
Biến đổi các biểu thức sau thành tích số
1/
A 2 cos2 x 1 .
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2/
B 3 4 sin2 x .
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3/
C sin x cos x cos2 x 1 .
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4/
D sin2 x sin x cos x 1 .
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5/
BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC
E 1 sin x cos x tan x .
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6/
F tan x cot x sin x cos x .
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7/
G cos x tan2 x 1 cos x .
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8/
H 3 4 cos2 x sin x 2 sin x 1 .
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9/
I sin2 x 3 cos2 x 6 cos x 2 sin x .
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10/ J cos3 x sin3 x sin x cos x .
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11/ K cos3 x cos2 x 2 sin x 2 .
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12/ L cos2 x sin3 x cos x .
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13/ M 1 cos x cos2 x sin x 1 cos x .
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14/ N 2 cos3 x 2 cos2 x sin x 1 .
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15/ O cos3 x sin3 x 2 sin2 x 1 .
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16/ P 2 sin x 12 cos x 2 sin x 1 3 4 cos2 x .
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17/ Q 2 cos x 1sin x cos x 1 .
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18/ R 4 sin3 x 3 cos3 x 3 sin x sin2 x cos x .
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19/ S 1 sin x tan2 x 1 cos x .
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20/ T 2 5 sin x 3 1 sin x tan2 x .
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21/ U 2 sin x cos x 2 sin2 x 3 sin x cos x 1 .
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22/ V tan x 3 cot x 4 sin x 3 cos x .
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23/ X 3 sin x 2 cos x 3 tan x 2 .
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24/ Y 2 tan x sin x 3 cot x cos x 5 .
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25/ Z 3 cot x cos x 5 tan x sin x 2 .
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Bài 8.
Chứng minh các biểu thức sau không phụ thuộc vào biến x
1/
A cos4 x sin4 x 2 sin2 x .
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2/
B sin4 x sin2 x cos2 x cos2 x .
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3/
BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC
C cos4 x sin2 x cos2 x sin2 x .
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4/
D cos4 x 2 cos2 x 3 sin 4 x 2 sin2 x 3 .
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5/
E sin6 x cos6 x 2 sin4 x cos4 x sin2 x .
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6/
F sin x.
1
1
,
1 cos x 1 cos x
0 x
.
4
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7/
G sin 4 x 4 cos2 x cos 4 x 4 sin2 x .
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8/
H cos2 x cot2 x 5 cos2 x cot2 x 4 sin2 x .
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9/
I 1 cot x sin 3 x 1 tan x cos3 x sin x cos x .
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10/ J sin4 x cos 4 x 1 tan2 x cot2 x 2 .
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11/ K 3 sin8 x cos 8 x 4 cos6 x 2 sin 6 x 6 sin 4 x .
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12/ L sin4 x 1 sin2 x cos 4 x 1 cos2 x 5 sin 2 x cos2 x 1 .
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2
13/ M 2 sin4 x cos4 x sin2 x cos2 x sin 8 x cos8 x .
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Bài 9.
Chứng minh các biểu thức sau không phụ thuộc vào biến x
1/
A
2
cot x 1
.
tan x 1 cot x 1
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2/
B
1 tan2 x
2
4 tan x
2
1
2
4 sin x cos2 x
.
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3/
C
BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC
1 tan2 x
1 tan2 x 1 cot2 x .
tan x
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1 cot x
D
2
2
4/
cot2 x
1
sin2 x cos2 x
.
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5/
E
1 sin6 x
cos6 x
3 tan2 x
cos2 x
.
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6/
F
tan2 x cos2 x
sin2 x
cot2 x sin2 x
cos2 x
.
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7/
G
cot2 x cos2 x
2
cot x
sin x cos x
.
cot x
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8/
H
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sin4 x cos4 x 1
sin6 x cos6 x
.
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B – ĐƯỜNG TRÒN LƯỢNG GIÁC – CUNG GÓC LƯỢNG GIÁC
CUNG LIÊN KẾT
1. Đường tròn lượng giác và dấu góc lượng giác
sinx
Cung
1
I
II III IV
+
+
–
–
+
–
–
+
+
–
+
–
+
phần tư
π
–
+
–
2
Giá trị LG
(II)
(I)
π
0 1
2π
O
-1
cosx
(IV)
(III)
3π
2
(Nhất cả – Nhì sin – Tam tan – Tứ cos)
-1
2.Cung liên kết
Cung đối nhau
Cung bù nhau
cos(a) cos a
sin( a) sin a
sin a cos a
2
sin(a) sin a
cos( a) cos a
cos a sin a
2
tan(a) tan a
tan( a) tan a
tan a cot a
2
cot(a) cot a
cot( a) cot a
cot a tan a
2
Cung hơn kém
Cung phụ nhau
Cung hơn kém
sin( a) sin a
cos a sin a
2
tan( a) tan a
tan a cot a
2
cot( a) cot a
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sin a cos a
2
cos( a) cos a
2
cot a tan a
2
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BÀI TẬP ÁP DỤNG
MỐI LIÊN HỆ GIỮA ĐƠN VỊ ĐỘ VÀ RADIAN
Bài 10.
Đổi số đo các cung sau ra độ, phút, giây:
2 3 5 7 13 3
15
,
,
,
,
,
, , 2,
, .
3
4
9
5
3
3
4
7 15
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Bài 11.
Đổi số đo các góc sau ra radian rad :
900, 360, 150 , 720 , 2700, 2400, 5400, 7500 , 210 .
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Bài 12.
Đổi số đo độ của các cung tròn sau thành số đo radian (chính xác đến phần nghìn):
21030 ', 75054 ', 12036 ' , 15030 ', 105045 ' 30 '', 27 038 ' 49 '' .
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Bài 13.
Điền các giá trị thích hợp vào ô trống
00
150
300
450
600
750
900
1200
1500
1800
16
8
6
5
12
2
2
3
3
4
5
6
4
3
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TÀI LIỆU HỌC TẬP TOÁN 10- CHẤT LƯỢNG
Radian
7
4
12
BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC
5
6
Độ
2
2
3
3
4
1350
500
8100
TÍNH ĐỘ DÀI CUNG TRÒN
Bài 14.
Một đường tròn có bán kính 10 cm . Tính độ dài các cung có số đo là
2 3 7 13
3
3
,
,
,
,
, 2,
, 450, rad, 75 0 .
3 18 5
3
6
4
2
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Bài 15.
Kim giờ của đồng hồ dài 7 cm , kim phút dài 10 cm . Tính quãng đường kim phút,
kim giờ đi được trong 30 phút.
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Bài 16.
Bánh xe có đường kính (tính cả lốp) là 55 cm . Nếu xe chạy với vận tốc 40 km /h
thì trong một giây bánh xe quay được bao nhiều vòng ?
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Bài 17.
Kim phút và kim giờ của đồng hồ lớn nhà Bưu Điện Hà Nội theo thứ tự dài
1, 75 m và 1, 26 m . Hỏi trong 15 phút, mũi kim phút và mũi kim giờ vạch trên
cung tròn có độ dài bằng bao nhiêu mét ?
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BIỂU DIỄN NGỌN CUNG LƯỢNG GIÁC
Bài 18.
Biểu diễn các góc sau lên đường tròn lượng giác gốc A:
300 , 450, 1200, 1200, 3300,
6300, 7500 , 12500,
7
4
15 2010
,
, 750 0, 11250 ,
,
.
4
3
2
4
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Bài 19.
Xác định điểm cuối của cung có số đo: k2, k, k , k , k , k .
2
3
4
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Bài 20.
Xác định điểm ngọn của các họ nghiệm sau đây trên đường tròn lượng giác với
k .
1/ x k .
2/ x
k2 .
2
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3/ x 4k .
BÀI GIẢNG: BIẾN ĐỔI LƯỢN GIÁC
4/ x
k .
3
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5/ x
k2 .
6
6/ x
k .
4
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7/ x
k
.
3
8/ x
k2
.
3
3
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9/ x
k3
.
4
4
10/ x
k
.
6
3
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Bài 21.
k
Cho hai điểm M và N sao cho sđ AM
và sđ AN
với k . Tìm k để:
6
798
1/ M trùng với N.
2/ M đối xứng với N qua tâm O.
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