TÓM tắt CÔNG THỨC TOÁN 11 THỦ KHOA

lcoNG THuc cAN NHd LdP
GROUP: 2 0 0 1 THỦ KHOA
111

1 .Cac cdng thuc hiqng giac co ban:
* sin 2 a + cos 2 a = 1
1[
')
1
* 1 + tan - a = --)' , a -:1:-+kJr,k E Z.
cos- a
2

*

')

1

l s-cot" a=-.
'

sm-a
')

10. Cong thuc bien dfii tich thanh tfing:
1
* cos a cos b = -[ cos (a+ b) + cos ( a - b)
2

J

* sin a sin b = _ _!_ [ cos (a+ b )- cos( a - b)
2

a r k m.k e Z

* sin a cos b = _!_ [ sin (a+ b) + sin ( a - b)
2

1[

* tana.cota = 1, a* k-,k E Z
2
2. Gia tr] hiqng giac cac cung do'i nhau:
* cos(-a) =cos a
* sin (-a)= -sin a

*

*

tan(-a) = -tan a

cot(-a) = -cot a

3. Gia tr] hiqng giac ciia cac cung bu nhau:
* sin ( Jr - a) = sin a
* cos(Jr-a)=-cosa

*

* tan(Jr-a) = -tan a

cos(Jr-a) = -cota

4. Gia tr] hiqng giac cua cac cung hon kern Jr :

* sin(a+Jr)=-sina
* tan(a+Jr) = tana

* cos(a+Jr)=-cosa
* cot(a+Jr) = cota

5. Gia tr] luong giac ciia cac cung phu nhau:

• sin (; - a) = cos a
• tan(; -a)= cota

*
*

cos (; - a) = sin a
cot-a)= tan a

6.Gia tr] lu'qng giac ciia cac cung hon

sin(

a+

;)

= cos a

tan( a+;)= -cot a

kem;

cos( a + = - sin a
co{ a + ; ) = - tan a

* cos( a+ b) = cos a cosb- sin a sinb
* sin(a-b)=sina cosb-cosa sinb
* sin (a+ b) = sin a cos b + cos a sin b
tan a± tan b
+ b) =----tan ( a_
1 + tan a. tan b
8. Cong thuc nhan doi va nhan ha:
=sin'La = 2sina.cosa
=cos 2a = cos ' a-sin2 a
2tana
= 2cos2 a-1
*tan2a=--l= tan" a
= 1-2sin2 a

*

* Cos3a = -lcos'a - 3cosa
* Sin 3a = 3sina - -lsirr'a
9.Cong thuc ha bac:

')
1 +cos2a
*COS- a=---2

. ')
1-cos2a
*sin- a=---2

J

11. Cong thuc bien dfii t6ng thanh tich:
u+v
u-v
* cosu+ cosv = 2cos--cos-2

2

. u+v . u-v
* cosu-cosv=-2sm--sm-2
2
.
.
. u+v
u-v
* smu+smv=2sm--cos-2
2
.
.
u+v . u-v
* smu-sm v = 2cos--sm-2

2
12. Vai ti so' hrong giac thong dung:
O(rad)

Cung
sin
cos
tang

oo
0

1
0

1[

1[

1[

1[

-

-

-

-

6
30°
1
2

4
45°

60°

-

J2

-

2

2

Jj

J2

-

-

2

2

Jj

1

3

Jj
1
2

2
90°
1
0

-

fj

II

-

;)

7. Cong thuc d)ng:
* cos ( a - b) = cos a cos b + sin a sin b

J

3

cotg

II

fj

1

Jj

0

-

3

13.Phu'ong trmh lu'qng giac co ban :
• sinx = a (1)
neu a la 1 nghiern cua (l),nghia Iasin zz = a
[ x =a+ k2Jr
(1) � sinx = sin a<=>
ke Z
x = Jr - a + k2Jr
•cosx =a (2)
neu a la l nghiern cu a (2),nghia lacosa = a thl

(2) � cosx = cos a<=> x =±a+ k2Jr,k E Z
• tanx = a (3)
neu a la I nghiern cua (3),nghia la tana = a thl
(3) � tan x = tan a <=> x = a + kst .k E Z
•cotx = a (4)
neu rz la l nghiern cu a (4),nghia la cot a= a thi
(4) � cotx = cot a<=> x =a+ kst , kEZ
Chu y: sin x = a, cos x = a c6 nghie m khi I al s; 1
tanx = a, cotx = a c6 nghie m vdi Va


14.Phtidng trinh b�c nhat do'i vdi sinx va cosx

*

2

2

a sin x ± b cos x = c <=> -J a + b sin( x ± a) = c

acosx±bsinx =c <=>-Ja2 +b2 cos(x+a) =c
(cos nhtr t/Ji dilu)
a
.
b
)
01 cos a=
,sma =
(v,.

1
1
-va2+b2
-va2+b2

*

22.Cong tlurc nh] tlnrc Niu-To'n
( a+ b)n = cona n + clna n-1b + ... + ckna n-kbk + ... + n

en:

ca + b

r = L c� an-k bk
n

k=O

23.Bang cong thirc d�o ham

Ca hai PT tren muon tim a barn shif cos -J a' + b'

(c)'

, PT tren
" co, ng hiiern
"
Ch , , C ac
!dll!.J!..:

(x)'=l

15. PT thuan nha'tb�c hai do'i vdi sinx va cosx
Dang: asin2x+bsinxcosx+c cosix = d (6)
each giai:
Bl:thu' vdi cosx=O co thoa (6) khong?
B2:Chia 2 v€ cua (6) cho cosx :;t: 0 ta duce pt:

d

2

atan x +btanx +c = --)'
cos- x
¢=> atan 'x +btanx +c =d(l +tanx )
¢:> (a-djtanx +btanx +c -16. Phuong trinh d6i xung d6i v6'i sinx va cosx
Dang :a(sinx +bcosx)+bsinxcosx =c (7)
Cach giai: D�t t = sinx +cosx dk : ltl s

Ji

2

.
' (7) ta d iroc pt:
Khi1 d o, smxcosx
= t - l th ay vao
-2

at:2 + b

1 \-

I =c day la pt bac hai da biet

17.0ui tic cong:M(>t cong viec duoc hoan thanh boi
1 trong 2 hanh d(>ng.NSu HDl c6 m each thuc hien,
HD2 c6 n each thirc hien khong trung voi bky each
nao cua HD 1 thi cong viec d6 c6m+n each thuc hien
18.0ui tic nhan: Mot cong viec duce hoan thanh boi
2 hanh d<)ng lien tiep.Neu c6 m each thuc hien HDl,
Va irng vo i m6i each d6 c6 n each thtrc hien HD2 thi
c6 m.n each hoan thanh cong viec.
Chu y:Cac qui t�c tren c6 thS
rong cho nhieu HD.
19.Hoan vj:KSt qua cua su s�p xSp n phan tu cua A
theo mot thir tu nao d6 dgl mot hoan vi cua t�p A.
S6 hoan vi cua A ki hieu: Pn ta c6:
P n=n.(n-1 ).(n-2) ... 2.1 =n!
20.Chinh hop: KSt qua viec lfiy k phan tu cua A
(1 s ks n) Va xep theo m(>t tlur tu nao d6 duoc goi la
mdt chinh hop chap k cua n phan tu.
S6 cac chinh hg ch� k cua n p.tir ki hi�u:A\ ta c6 :
n!
Akn =--(n-k)!

ma

21.T6 hQ'P:M(>t t�p con g6m k p.nr cua A
(1 s ks n) duce goi la mot t6 hop chap k cua n p.tir.
S6 cac t6 hgp ch�p k cua n phan tu ki hieu.C" n ta c6 :
n!

c. =--k!(n -k)!

Tinh chftt:

V6'i u

( C.x)' = C

,

,

(xii) = n.x'""

( u II) _-n.u 11-J .u

,

( t) � - :, (
(Fx)' =
,

ta m9t ham s6

\-

f

x .< 0)
[x c- O]

2'\/X

(sin x) = cosx

(Fu)'=

u�
2'\/U

,

(sin u) = u' cosu

(cosx)' = -sin x
(cos u )' = -u' sin u
,
,
u'
1
(tanx)
(tanu)
cos- x
cos- u
'

1
'
u'
(
( cotx) = --cotu) = --. . 2
2
sin x
sin u
D�o ham t6ng ,Hi�u,Tich va ThuO'ng
=-)'

=-')

<=> a 2 + b2>_ c 2

= 0 (C: hang so')

* ( u ± v)
,

* (l_!__)

= u' ± v'

= u'.v�u.v'

v

*

,

,

v-

* ( u.v) = u'.v + u.v'
* (k.u )' = k.u'
(k la hang so)

PTTT cua d6 thi hs :y=f(x) te;1i diSm M(xo;Yo):
y = y(x0).(x-x0)+ Yo

24.Bi�u thrrc t9a dQ cu.a phep tjnh ti�n:
Trong mp oxy cho diSm M(x;y),M'(x';y') va � (a;b)
{x'= x+a
,
T-(M) = M ¢:>
y'=y+b
1'

25. Bi�u thrrc t9a dQ cu.a phep D6i xrrng tr\)C:
• Trong mp oxy cho diSm M(x;y) goi M'(x' ;y')= Dd(M)
{
* NSu ch9n d la t[\lc ox,thi ¢=> �· = x
y =-y
*. NSu ch9n d la t[\lc oy,thi¢=>

{x'�

-x
y=y

26. Bi�u thrrc t9a dQ cu.a phep D6i tam:
• Trong mp oxy cho diSm M(x;y),I(a;b) goi
{x'=2a-x
M'=D1(M)=(x';y'),khi d6
y'= 2b- y

* NSu chon

I la g6c toa do 0(0;0) thi:
{x:=
-x
M'=D0(M)=(x';y'),khi d6
y =-y