Phân loại phương pháp giải đại số giải tích 11

512.0076
PH121L
JH
TRUONG
-
TRAN
VAN
THUONG - NGUYEN PHU KHANH
HANH KY
-
NGUYiN MINH NHIEN
-
NGUYEN
TAT
THU
NGUY6N TAN
SIENG
-
DO
NGOC THUY
(Nhom giao vien chuyen Toan THPT)
^
1,-0
z
PHUONG PHAP
GIA
(Tai
ban
c6 sda
chOa
bo

sung)
Danh
cho hoc
sinh
Icip
11 on
tap
va
nang
cao
kien
thifc
^
Bien
soan
theo
npi
dung
sach
giao khoa
cua
Bp
GD&OT
Sin
a
xsm
p
•sin
NHA XUAT
BAN DAI HOC

QUOC
GIA HA NOI
NQiUYtN IAN
bItNQi
- t)(J N(3QC THUy
(Nh6m
gi4o vien chuyen To^n THPT)
PHUONG
PHAP
GIA
DAI
Sfi
-
6IAI
TiGll
(Jal
ban
c6
sda
chOa
bo
sung)
e-
Danh cho hoc sinh Idp 11 on tap va nang cao kien Mc
©•
Bien soan theo noi dung sach giao khoa cua Bo GD&OT
',3
t
. '
NHA

XUAT
BAN
DAI
HQC
QUOC
GIA
HA
NQI
Cdc em hoc sinh than men!
"Phdn loai vd phttcfng phdp gidi Dai so -
Gidi
tich 11" Id
nipt
trong nliQng cuon
thuoc bo sdch "Phan loai vd phucfng phdp gidi theo chuyen de: Idp 10, II, 12",
do nhor i tdc gid chuyen todn
THPT
bien
soan.
Vdi
cdch viet
khoa
hoc vd sinli dgng giup ban doc tiep can vdi man Todn mot cdch tu
nhien,
khong
dp luc. Ban doc trd nen tu tin vd rmng dgng haii,
hieu
ro bdn dmt,
biet
cdch

pMn
tich de tim ra trgng tdm cm vdn de vd biit gidi thich, lap ludn cho ticng bdi todn.
Su
da dgng cua lie thong bdi tap vd tinh huong giup ban doc ludn
hvcng
thu khi gidi todn.
Tdc
gid cliu trgng
bien
sogn
Tilivcng
cdu lidi md,
ripi
dung ca bdn bdm
sdt
sdch gido
klioa
vd cdu true de thi dgi hgc, dong tlidi plidn bdi tap tlmnJi cdc dcmg todn c6 Idi gidi chi tiet.
Hien
nay, dd thi dgi ligc kJiong klio, to hap cua nliieu vdn de don
gian,
nliung chvca nliieu
cdu hoi md neu klidng ndm
elide
ly thuyct se lung tung trong vice tim Idi gid/ bdi
todn.
Vdi
mot bdi todn, Uidng nen tlioa man
ngay
vdi mot Idi gidi

rmriii
tim dugc ma plidi cd
gdng tim nliieu cdch gidi nlidt cho bdi todn do.
Klii
gidi mot bdi todn,
thay
vi dung tlidi gian de luc Igi tri nlid, thi ta can plidi suy nghi
plidn
tich de tim ra phuang plidp gidi quyet bai todn dd. Mon Todn ddi hoi plidi kien nlidn
vd bin bi
ngay
tit
nliixng
bdi tap dan gidn nlidt,
nlivcng
kien thvcc ea bdn nlidt. Vi chinli
nliQng
kien
thvcc
ca bdn
yndi
giup ban dgc
hieu
dugc
nliUng
kien thuc ndng cao sau nay.
Gid
day, cliung tdi
chat
rJid tdi cdu noi cua Ludwig Van Beetlioven: "Gigt nudc c6 die

lam man tdng dd, klidng plidi vi giot nudc c6 sicc ingnli, ina do nudc clidy lien tuc
ngay
dem.
Chi
CO
su plidn dau klidng met inoi indi dem Igi tdi ndng. Do do ta c6 die klidng dirJi, klidng
nliich
ti^ng
budc thi
khong
bao gid c6
thedi
xa ngdn ddm".
Mac
dutdcgiddd ddnli nliieu tdm liuyet cho cudn sdch, song su sai s6t Id dieu klid trdnli
klioi. Chung tdi rat mong niidn dugc su plidn
bien
vd gop y quy bdu cua quy dgc gid de
nliiing
Ian tdi bdn sau cuon sdch dugc hodnthienhmi.
Thay mat nhom bien soan
Chu
bien:
Nguyen
Phu
Khdnh
Nhd
sdch
Khang
Viet

xin
trdn
trong
gi&i
thieu
t&i Quy doc gid vd xin
long
nghe
moi y
kien
dong
gop,
decuon
sdch
ngay
cdng
hay Hon,
botch
horn.
Thttxinguive:
Cty
TNHH
Mpt
Thanh
Vien
-
Dich
vu Van hoa
Khang
Vi?t

71,
Dinh
Tien
Hoang, P. Dakao,
Quan
1, TP. HCM
Tel:
(08)
39115694
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39111969
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39111968
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39105797
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39110880
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Email:

CHd DE 1: HAM SO
LUONG
GIAC
VA
PHUONG
TRINH
LUONG
GlAC
CHl/ONG
1: , HAM s6

LUOMG
GIAC
i
A. TOM TAT LI
THUYET
I.
CAC
CONG
THLfC
Ll/ONG
GIAC
1. Cac
hSng
dang thuc;
*
sin^ a + cos^ a =
1
voi moi a , . , i ' •/j
*tana.cota
= l voi moi a — , ,
2
' ' ,
*
1
+ tan^ a = — voi moi a ^ k2n
cos a '
*
1
+ cot'^ a = —\ voi moi a^kn
sin'

a
2. thiic cac cung dac
bif
t , ,
a. Hai
ciin^
ddi nhau: a va -a
cos(-a)
=
cosa
sin(-a)
=-sina
tan(-a)
=-tana
, •
cot(-a)
=-cota
b. Hai cung phu nhau: a va
—
-a
cos( —-a) = sina
sm(—a)
= cos a
2 2 .
>•
ii'i'-
tan(^-a)
=
cota
cot(^-a)

=
tana
c. Hai cung hii nhau: a va n-a ,.t JI it
sin(7r-a)
= sina
cos(7t
- a)
=-cosa
tan(7i
- a) = - tan a cot(7i-a)
=-cota
d. Hai cung han kem
nhau
n :a va n + a
sin(n + a)
=-sina
cos(7t
+ a)
=-cosa
tan(7r
+ a) = tan a
cot(7r
+ a) =
cota
3. Cdc cdng thuclugng gidc "" ' •
a. Cong thuc cdng ^
cos (a ± b) =
cosa.cosb
±
sina.sinb

sin(a
± b) = sin a. cos b ± cos a. sin b
tan a ± tan b
tan(a±b):
1
± tan a. tan b
b. Cong thiec nhan
sin
2a
= 2
sin
a cos
a
cos2a = cos^ a -
sin'^
a - l-2sin^a= 2cos^a-l
sin3a
= 3sina-4sin''a cos3a = 4cos"'a - 3cosa * *'
c. Cong thiec ha bac
.
2 l-cos2a 2 l + cos2a ^ 2 l-cos2a
sin
a = cos a = tan^ a =
2
2 l + cos2a
d. Cong thiec
bie'n
dot tich thanh to'ng j') '
cosa.cosb = —[cos(a - b)+ cos(a + b)]
sina.sinb

= ^[cos(a-b)-cos(a + b)]
sina.cosb =
^[sin(a
- b) +
sin(a
+ b)]. #
e. Cong thiec Men dot
to'ng
thanh tich
a + b a-b , ^.a + b.a-b
cosa + cosb =
2cos
.cos cosa -cosb =
-2sin
.sin
2
2 2 2
.•
a + b a-b . ., „ a + b.a-b
sina
+ sin b - 2sin .cos
sina
- sinb = 2cos .sm
2
2 2 2
.
i ,
sin(a
+ b) ,
sin(a

- b)
tana + tanb = !^ f- tan a-tan b= ^ ' .
cosacosb cosacosb
II.
TINH
TUAN
HOAN
CUA HAM SO ,
Dinh
nghia: Ham so y = f(x) xac
djnh
tren tap D duoc gpi la ham so tuan
hoan
ne'u c6 so T ?i 0 sao cho vol moi x e D ta c6: x ± T e D va f{x + T) = f(x).
Neu
CO
so T duomg nho nhd't thoa man cac dieu
kien
tren thi ham so do
dupe gpi la ham so tudn hoan vai chu ki T.
III.
CAC HAM SO
Ll/QNG
GIAC
*
1.
Ham so y = sinx ^* , .
•
y.
T^;:*-

^ , 1 ^^^^
•
Tapxacdjnh: D = R ''
.
• Tapgiactrj: [-l;l],tucla -l<sinx<] Vx e R ''
•
Ham so dong bien tren moi khoang ("^ +
k27i;^
+
k27i),
nghjch bie'n tren
moikhoang
(^ + k27i;Y^ + k27t). ^
•
Ham so y = sinx la ham so le nen do thj ham so nhan goc tpa dp O lam
tam doi xung. *
•
Ham so y = sin x la ham so tuan hoan voi chu ki T = 2:1.
hi \
Cty
TNHH
MTV DWH Khang Vtgt
•
Do thi ham so y =
sin
x .
2. Ham so y = cosx ;
•
Tapxacdjnh: D = R ' ! ' ; _
^''*|-'

•
Tapgiactrj: [-l;l],tuc la-1< cosx <1 Vx e R ^
•
Ham so y = cosx nghjch bie'n tren moi khoang
(k27i;7i
+
k2T:),
dong bie'n
tren moi khoang (-71 +k27t;k27r). %
•
Ham so y = cos x la ham so
chin
nen do thi ham so nhan true Oy lam
true do'i xung.
•
Ham so y = cos x la ham so tuan hoan voi chu ki T =
27i.
•
Do thj ham so' y = cos x. -
Do
thj ham so y = cos x bang each tinh tien do thj ham so y =
sin
x theo
vee to V = (-—;0).
3. Ham so y = tan x
•
Tap xac
djnh:
D = M\\- + kn, ke Z
•

Tapgiatrj: X > •'"
•
La ham sole t - - ' ' '
•
La ham so tuan hoan voi chu ki T =
7t
f
1
lli
'.%h%^'
•1,'
Ham
dong bie'n tren moi khoang
—
+
k7i;
—+
kTt
2
2
I
•
Do thj nhan moi duong thSng x = +
k7t,
k e Z lam mpt duong
tif
m
can.
Phdn loai va phuang phdp gidi Dai so - Gidi tick 11
• Do thi

-5JI
2
57t
2
mi
4. Ham so y = cot x
• Tap xac djnh: D = M\ k e Z
• Tap gia
tri:
M
• La ham so' le
• La ham so tuan hoan voi chu ki T =
TT
• Ham nghich bien tren moi khoang
(k7r;7r
+ kn)
• Do thi nhan moi duong thang x = kn, keZ lam mpt duang ti^m can.
• Dothj
7t \
\
\
\ 71
\t
\
\t
71 \
2 \
-371 \
2 \
o \.

V
B. PHLTONG
PHAP
GIAI TOAN
Van de 1. Tap xac dinh va tap ^a tri ciia ham so v
I.
PHl/ONG
PHAP
V
• Ham so y = 7f(x) c6 nghla o f(x) > 0 va f(x) ton tai
• Ham so y = —5— c6 nghla o f(x) 0 va f(x) ton tai.
f(x)
• sin u(x) ?t 0 o u(x) = kn, k e Z
•
cosu(x)?!:0<=>u(x)^-
+
k7i,
k€.
• -1 < sinx,
cosx
<
1
.
Cty
7WHH
MTV D WH KhangViet
U.
CACVIDV
Vi
d\ 1. Tim tap xac dinh cua ham so sau:

1.
y =
tan(x )
i,:,:"::, !::,
6 ru:.i
2. y =
cot2(^-3x)
Giai
1.
Dieu ki?n:
cos(x
^0 x^~+ kn x + kn
6 6 2 3
TXD: D =
271
+ k7t, k e Z
2. Dieu kien: sin(— - 3x) ^ 0 <=>'^-3x ^kn <:> -k^
TXD: D = IR \
271 , 7t , r„
k-,
ke£
9 3
Vi
2. Tim tap xac djnh ciia ham so sau:
tan2x 71
1.
y = + cot(3x + —)
sin
X +1 6
1.

Dieu ki^n:
sinx ^-1
sin
3x + -
6J
^0
Giai
x^-— + kin
2
71 nTC
X
9t +
18 3
2- y=-
tanSx
sin4x -
cos3x
VgyTXD:
D = #\ + k27i,-Y^ +y;k,n € Z •
2. Ta c6: sin 4x -
cos3x
= sin 4x - sin
3x
(x
f7x Tl\
3x
=
2
cos
— + —

sin
\2
J
K2
AJ
L
2 4;
Dieu ki^n:
cos5x
^ 0
cos
sin
— + —
2 4;
^7x 71
2 4;
^0
• 71 ,71
•,,„;,Q
10 5 .
xii — + n27C
2
71
n27t
X
^ +
14 7
Vay TXD: D = # \
71
^

k7t
7t
^ ^ ^ 2m7i ; n, m e Z L
IO^T'2^"
'^'"TI^ 7 ' J
14 7
4