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Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
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Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
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Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
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Tuấn
2
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
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D/Y_+U$\+
»
CD
G&?+*DU$\+OA"Oh&'
$\U`+#b@N/U$\+&'DU$\+$%GU'B
#

#$#%&#'
()*"k{+0@ND0
( 2 2 ( 2 4
T T4T T  T T
2 8 4! 3 4< (
π π π π π
()+"k@+0@ND0+28
!
T4(
!
2!
~
T4!
!
T48
!
T((
!
2!
~
T((8
!
Gv: Hồ Thanh
Tuấn
44
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
(),"YaD-V48*#)*9'+$Qa
DD0@N


4<
π
(8
!
;!
!
92
()@")$QaU$\+"+F*Y-+m

¼
AM
D0@N
k
π

(
k
π

(
 
8
k k Z
π
∈
9
 
2 (
k k Z

π π
+ ∈
$-K$##L#
B!.##C
*)$QaU$\+@g#N
¼
AM
D0
¼
AM
H
α
0+
α
H
OK
H
M
y
T N0
α
H
M
OH x=

α
H
0+
N0
α

α
N0
α
!≠
T N
α
H
N0
0+
α
α

0+ !
α
≠

*/ 3
 Vôùi moïi
α
ta coù :
1 sin 1 hay sin 1
α α
− ≤ ≤ ≤
1 cos 1 hay cos 1
α α
− ≤ ≤ ≤

π
α α π
∀ ≠ +tg xaùc ñònh

2
k

cotg xaùc ñònh k
α α π
∀ ≠
0/EF2G,
• 0+
(
α
N0
(
α
H4
Gv: Hồ Thanh
Tuấn
4(
g
~
j
g
J
J
~
Y
α
O
W
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
•
3!
N0
0+
!
≠=
αα
α
α
T
4>!"!N
0+
N0
!!
≠=
αα
α
α
T α#NαH4
• NαH
α

4
T αH
α
N
4
T 4
(

αH
α
(
N0
4
• 4N
(
αH
α
(
0+
4

C/+ H2G5 : 
vaø -
α α


α α α α
α α α α
− = − =−
− =− − =−
cos( ) cos ; sin( ) sin ;
tan( ) tan ; cot( ) cot
( k@+N0
D/+2G5 I 
vaø -
α π α

π α α π α α π α α π α α

− = − − = − = − − = −cos( ) cos ; sin( ) sin ; ( ) ; cot ( ) cottg tg g g
JB0+
J/+2G5 ;6 
π

vaø
α π α
+

π α α π α α π α α π α α
+ = − + = − + = + =cos( ) cos ; sin( ) sin ; ( ) ; cot ( ) cottg tg g g
j%-*
π
"N
K/+2G5 ;6 
(
π

vaø
2
π
α α
+

π π π π
α α α α α α α α
+ = − + = + = − + = −cos( ) sin ; sin( ) cos ; ( ) ; cot ( ) t
2 2 2 2
tg cotg g g
L/+2G5 M 

vaø
2
π
α α
−


π π π π
α α α α α α α α
− = − = − = − =cos( ) sin ; sin( ) cos ; ( ) ; cot ( ) t
2 2 2 2
tg cotg g g
lN
#$#%&#'
()*")V+*0@U$\+/D0@N
Gv: Hồ Thanh
Tuấn
42
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
€<3!
!
;38
!

4=
2
π
−

9
48
(
π
()+" ANN0H
2
8
−
&'4>!
!
(=!
!
#V0+""N
AN
α
H
2
;
&'
2
(
π
π α
< <
#)VN
α
"0+
α
"N0
α

(),"ANINH4&'!
!
3!
!
#)V+U$\+0+"N0""
N
()@" E970+8!
!
#N0€2!!
!

 AN!
!

α
3!
!
#97/0+
α
3!
!

()A"AN!
α

(
π
#E97+FG
N0
 

α π
+

 
α π
+
0+
(
8
π
α
 
+
 ÷
 
9N0
2
>
π
α
 
−
 ÷
 
()H"eiL+FG

(
(N0 4
0+ N0
A

x x
−
=
+

( (
0+ 4 N  N0 4  B x x x= + + +
()I")V+/+FG

N 
N 
A
α α
α α
+
=
−
+0+
α
H
2
8
&'!
α

(
π
AN
 2
α

=
#)V
(0+ 2N0
;0+ 8N0
α α
α α
+
−
T
2 2
20+ (N0
80+ ;N0
α α
α α
−
+
()J"AG*+RG0
Gv: Hồ Thanh
Tuấn
4;
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát

0+ 4 N0 (
4 N0 0+ 0+
x x
x x x
+
+ =

+
0+
;
N0
;
H4I(0+
(
#N0
(


4 N0

N0 4 0+
x
x
x x
− =
+
90+
<
N0
<
H4I20+
(
#N0
(

:
( (

( (
( (
N0 0+
0+ #N0
N 
x x
x x
x x
−
=
−

(
(
(
4 0+
4 ( 
4 0+
x
x
x
+
= +
−
#M.#K$#
B!.##C
&/>8
α β α β α β α β α β α β
α β α β β α α β α β β α
α β

α β α
α β
+ = − − = +
+ = + − = −
−
−
cos( ) cos .cos sin .sin ; cos( ) cos .cos sin .sin
sin( ) sin .cos sin .cos ; sin( ) sin .cos sin .cos
tg +tg
tg( + ) = ; tg(
1 .tg tg
α β
β
α β
−
+
tg tg
) =
1 .tg tg
*/>N>
α α α
α α α α α
α
α
α
=
= − = − = −
=
−
2 2 2 2

2
sin2 2sin .cos
cos2 cos sin 2cos 1 1 2sin
2
2
1
tg
tg
tg
0/>%
( ( (
4 N0( 4 N0( 4 N0 (
N0 T0+ T
( ( 4 N0(
tg
α α α
α α α
α
+ − −
= = =
+
C/>3.
[ ] [ ]
[ ]
α β α β α β α β α β α β
α β α β α β
= + + − = − − +
= + + −
1 1
cos .cos cos( ) cos( ) ; sin .sin cos( ) cos( )

2 2
1
sin .cos sin( ) sin( )
2
D/>.3
Gv: Hồ Thanh
Tuấn
48
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
α β α β α β α β
α β α β
α β α β α β α β
α β α β
α β α
α β α β
α β
+ − + −
+ = − = −
+ − + −
+ = − =
+
+ = − =
cos cos 2cos .cos ; cos cos 2sin .sin
2 2 2 2
sin sin 2sin .cos ; sin sin 2cos .sin
2 2 2 2
sin( ) sin(
;

cos cos
tg tg tg tg
β
α β
− )
cos cos
#$#%&#'
()*")V+U$\+/

4(
π

8
4(
π

=
4(
π
()+"AG*+m
π π π π
α α α α α α α α
+ = − = + − = − = − +)sin cos 2 cos( ) 2 sin( ); b)sin cos 2 sin( ) 2 cos( )
4 4 4 4
a
()," J+{+'{+FG
xxA 2N0#8N0
=
)V+/+FG
4(

=
0+
4(
8
N0
ππ
=B
()@" J+{+'V+FG
20+(0+0+ ++= xA
()A")V
N0
2
π
α
 
−
 ÷
 

4(
0+
42
α
= −
&'
2
(
(
π
α π

< <
()H"AG*+m

4 

4  ;
x
x
x
π
−
 
= −
 ÷
+
 

4 

4  ;
x
x
x
π
+
 
= +
 ÷
−
 

()I")V+/+FG

0+ #N0 #N0 #N0
(; (; 4( <
A
π π π π
=

( ) ( )
! ! ! !
N048 0+48 # N048 0+48C = − +
Gv: Hồ Thanh
Tuấn
4<
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát

( !
(N0 =8 4B = −
()JN"WZ9B6U$\+"V+/+FG
0

( 2
N0 N0 N0
= = =
P
π π π
= − +


( ; <
N0 N0 N0
= = =
Q
π π π
= + +
()O"eiN+FG

0+ ( 0+
4 N0( N0
A
α α
α α
+
=
+ +

(
(
;0+
4 N0
(
B
α
α
=
−
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4 N0 0+
4 N0 0+

α α
α α
+ −
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()*5"AG*++FG0-Zl&'N
" 
α β

0+ < #N2 N0<
α α α
−

  N   #
α β α β α β
− − −

(
N  #
2 2 2
α α α
 
−
 ÷
 
PH N HÌNH H CẦ Ọ
.#KP$#Q'$#
 !
&/<2GO 6
AN*+gJADJAH"gAH"gJH"
1gYH

a
m
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b
m
"AYH
c
m
12$O9

(
H
(

(
I(#N0gT 
(
H
(

(
I(#N0JT

(
H
(

(
I(#N0A
P<Q,

Gv: Hồ Thanh
Tuấn
4=
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
N0gH
bc
acb
(
(((
−+
N0JH
ac
bca
(
(((
−+
N0AH
ab
cba
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 12$9
C
c
B

b
A
a
0+0+0+
==
H(e&?+eU'-V$QaN`+
+*+gJA
*/18(.RS5  6
;
(
;(
((((((
(
acbacb
m
a
−+
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+
=
T
;
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bcabca
m
b
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=−
+
=
;
(
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((((((
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cabcab
m
c
−+
=−
+
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
0/>3(<3 6
4qbH
(
4
a
a
H
(
4
b
b
H
(
4

c
c
2/

bH
(
4
#0+AH
(
4
#0+gH
(
4
#0+J
2qbH
R
abc
;
;qbH bH
 cpbpapp −−−
&?+H
(
4

#$#%&#'"
()*"AN
∆
ABCDH28"H(!"gH<!
!
#)V


TeT
()+"AN
∆
ABCDgJH4!"gAH;&'gH<!!
!
#)V&+/
∆
ABC"
VA
(),"AN
∆
ABCDgH<!
!
"`AgH>*"`gJH8*
a) )VJA )V9+.V
∆
ABC
b) E:*DJB1Lu
c) )V9'+$QNgj
d) )Ve
Gv: Hồ Thanh
Tuấn
4>
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
()@")N
∆
ABC"+IH4"gH2!

!
"

H(#)Vb+J
()A"AN
∆
ABCDH42*"H4;*"H48*
 )V9+.V
∆
ABC
 5DJB1Lu)VJ
)V-Ve"
9)V9'+$Q1*

()H"AN
∆
ABCDH42*"H4;*"H48*
)V9+.V
∆
ABC
5DJB1Lu)VJ
)V-V$Qae"
9)V9'+$Q1
()I"AN
∆
ABCDJAH4("AgH42"1gYH>#)V9+.V
∆
ABC u)VDJu
()J"AN
∆

gJAD2`3T8T&'=#)VD/*+u)V
-N6sgJA
()O"AG*+mN
∆
ABCUZDZG
( ( (
N
;
b c a
A
S
+ −
=
()*5"AN
∆
ABC
AG*+mb+JHb+gA
ANgH<!
!
"JH=8
!
"gJH("V`aU`+/
∆
gJA
()**"AN
∆
ABCD5U'L*#5L+HJA"HAg"HgJ#AG
*+m
5g
(

5J
(
5A
(
H
( ( (
4
 
2
a b c+ +
()*+")*+ABCDJAH"AgH"gJH#
AG*+mHb#N0Cc#NB
()*,")*+ABCDJAH"AgH"gJH&'$Q1gYH
HgJ#AG*+m
 a
2
= ((b
2
– c
2
)
 b+
(
AH(b+
(
BIb+
(
C
()*@AG*+mN*+ABCD
b

2
– c
2
Hab#N0CIc#N0B b
2
– c
2
N0AHac#N0CIb#N0B)
0+AHb+gN0J0+JN0g
()*AAG*+mN*+ABCD
NANBNCH
( ( (
a b c
R
abc
+ +
Gv: Hồ Thanh
Tuấn
43
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
()*HYgJAhD+1gJH"AhH&'
·
BCD
α
=
#
)V-V/$QaN`++#
()*I")V9+.V/

∆
ABC, +&+*+m("D
µ
g
H
;8
!
"
µ
J
H<!
!
#
()*JN"AG*+mD/
∆
ABC[*d+,-+.0+JH
(0+g#N0A"
∆
D#
()*ON"AG*+RGi&?+*L+
∆
ABC

( ( (
; #Na b c S A= + −

0+ 0+      !a B C b sinC sinA C sinA sinB− + − + − =

( ( ( ( ( (
 # N0g # N0J # N0AH!bc b c c a c b c− − −

()+5")V9'+*

"+mH4"H2"
·
BAC
H<!
!
RS
 !"
&/T 69:5 RF
∆




+=
+=
(!
4!
tuyy
tuxx
&?+Y
!!
T yx
∈∆&'
T
(4
uuu =

U'&:%

r$%C)A
*/TQ5 RF
∆

I
!
x
1I
!
y
H!11H!
&?+HI
!
x
I
!
y
&'
(

(
≠! NDY
!!
T yx
∈∆&'
T ban =

U'&:%1C))
• TUV78:WX7TYUZ78:T[78\]:T^):W_\:`^Ya`+++F*
gT!&'J!TU'

4=+
b
y
a
x

• TUV78:WX7TYUZ78:T[78Y)b=^Y)cd3
!!
T yx
4\eTfgh8e\
;D9`1I
!
y
H;I
!
x

0/UO,V68)67
!!
T yx
RF
∆

1H!$\V:NZG9YT∆H
((
!!
ba
cbxax
+
++


Gv: Hồ Thanh
Tuấn
(!
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
C/W3:5  RF
4
∆
?
444
cybxa ++
?! &'
(
∆
?
(((
cybxa ++
?!
4
∆
•
(
∆
⇔
4 4
( (
a b
a b

≠
T)L+N+F*/
4
∆
&'
(
∆
U'+.*
/.
4 4 4
( ( (
H!
H!
a x b y c
a x b y c
+ +


+ +

4
∆
⁄⁄
(
∆
⇔
4 4 4
( ( (
a b c
a b c

= ≠
T 
4
∆
≡
(
∆
⇔
4 4 4
( ( (
a b c
a b c
= =
&?+
(
a
"
(
b
"
(
c
-!
#$#%&#'"
%&WRF
()*"SM$%*0@&'{v/$QR
∆
+
 
∆

vYI(T2&'DC))
n
r
H8T4
 
∆
vY(T;&'DC)A
2T;u =
r
()+"SM$%$QR
∆
+
∆
vY(T;&'D.0@
D-H(
(),"AN(+F*g2T!&'J!TI(#C+$%$QRgJ#
()@"AN2+F*gI;T4"J!T("A2TI4
 C+$QRgJ"JA"Ag
 5L+YU'+F*/JA#C+*0@/$QRgY
 C+$%$QR+v+F*g&'*$QaN`+
+
∆
()A"C+$%$QR9+v+N+F*/+$QR
9
4
"9
(
D$%UcU$\U'42I=144H!"43441I3H!&'+F*
Y4T4#
()H"SM$%$QR

∆
+
∆
vg4T(&'0N0N
&?+$QR21I4H!
()I"SM$%$QR
∆
+
∆
vA2T4&'0N0N
$Q+G|/*fRL
()J"AN++F*`/**+U'Y
4
(T4TY
(
8T2TY
2
2TI;#SM$%`/*+D#
()O")N*fRLN*+&?+YI4T4U'+F*/
*`"+`-+D$%U'1I(H!"(<12H!#E
Lr/*+#
Gv: Hồ Thanh
Tuấn
(4
Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
()*5"SM$%/$QRhN$Q\0
 hvY4TI(&'&ZD&?+
∆

21H!# hv
@L&'&ZD&?+
( 8
4
x t
y t
= −


= +

()**"C+$QR+v@L&'+F*Y2T;*
-N6U?7#
()*+"AN*+gJADrg(T(
SM$%`/*++$QN-‚sJ&'A
UcU$\D$%
3I21I;H!&'1I(H!
SM$%$QRvg&'&ZDgA#
()*,"AN
∆
gJAD$%`gJ8I21(H!T$QNv
rg&'JUcU$\U';I214H!T=(1I((H!#SM$%+
`gA"JA&'$QNG#
%*S)(%RF
()*"AN$QR9
2 (
4
x t
y t
= +



= − −

"U'*0@#jd1&+$%
{v/9#
()+"C+$%*0@/$QR(I21I4(H!
(),"C+$%{v"*0@"V•D/lL
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()@" C+$%$Qa*|4T(&'+i&?+$QR
hI(1I(H!
C+$%$Qa*|2T4&'+i&?+$QR
h2;1=H!
Gv: Hồ Thanh
Tuấn

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Đ cương ôn tp hc k II Gv:HồThanh Tuấn

Trường THPT Lưu Tấn Phát
()A")*L+N+F*/$QR
x 1 2t
:
y 2 t
= +

∆

= − +

&'$Qa
AI4
(
1I(
(
H4<
()HN"C+$%$Qa+vg4T4"J!T;&'D*
∈

$QR9I1I(H!
()IN"C+$%$Qa+vg(T4"JI;T4&'D-V
eH4!
()JN"C+$%$Qa+vg2T("J4T;&'+i&?+
lO
()ON"C+$%$Qa+vg4T4"D-VeH
4!

&'D
*m*O
()*5"AN|(TI(#C+$%$Qa*|&'+i&?+
91I;H!
%0\S
()*"SM$%+1&?+$QaC
( (
 4  ( 2<x y− + + =
`++F*Y
N
;T($Qa#
()+"C+$%+1&?+$QaC 
( (
 (  4 42x y− + − =
`++F*Y$QaDN'm
N
H(#
(),"C+$%+1&?+$QaC
( (
( ( 2 !x y x y+ + + − =
&'+v+F*Y(T2
()@"C+$%+1/$QaC
( (
 ; ;x y− + =
-‚
s@L#
()A"AN$QaC
( (
( < 8 !x y x y+ − + + =
&'$QR9(

1I4H!#C+$%+1
∆
+
∆
qq9T)*L++F*#
()H"AN$QaC
( (
 4  ( >x y− + − =
#C+$%+
1&?+C "+m+1Dqq9D$%1I=H!#
()I"C+$%+1&?+$QaC 
( (
8x y+ =
"+m
+1D&ZD&?+$QRI(1H!#
()J"AN$QaC
( (
< ( < !x y x y+ − + + =
&'+F*g4T2
 AG*+mgm*N'+$Qa
 C++1/C-‚sg
Gv: Hồ Thanh
Tuấn
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