1 Cac cdng thuc hiqng giac co ban:
* sin 2
a + cos 2
a = 1
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:
•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
m khi I al s; 1 tanx = a, cotx = a c6 nghie m vdi Va
* smu+smv=2sm cos
u+v u-v
* cosu-cosv=-2sm sm
* smu-sm v = 2cos sm
12 Vai ti so' hrong giac thong dung:
10 Cong thuc bien dfii tich thanh tfing:
1
* cos a cos b = -[ cos (a+ b) + cos ( a - b) J
2
* sin a sin b = _ _!_ [ cos (a+ b )- cos( a - b) J
2
* sin a cos b = _!_ [ sin (a+ b) + sin ( a - b) J
2
11 Cong thuc bien dfii t6ng thanh tich:
* cosu+ cosv = 2cos cos
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<=> k e Z
x = Jr - a + k2Jr
-
3
-
3
2tana
*tan2a= -
l= tan" a
* cos(Jr-a)=-cosa
') 1-cos2a
*sin- a=
cos( a + ; ) = - sin a
co{ a + ; ) = - tan a
a r k m.k e Z
1[
a -:1:-+kJr,k E Z
2
* sin ( Jr - a) = sin a
=cos 2a = cos ' a-sin2 a
') 1
* 1 + tan - a = ') - ,
cos- a
') 1
* l s-cot" a= ') '
sm-a
= 2cos2 a-1
= 1-2sin2 a
* Cos3a = -lcos'a - 3cosa
* Sin 3a = 3sina - -lsirr'a
9.Cong thuc ha bac:
') 1 +cos2a
*COS- a=
* tan(Jr-a) = -tan a * cos(Jr-a) = -cota
4 Gia tr] hiqng giac cua cac cung hon kern Jr :
* sin(a+Jr)=-sina * cos(a+Jr)=-cosa
* tan(a+Jr) = tana * cot(a+Jr) = cota
5 Gia tr] luong giac ciia cac cung phu nhau:
6.Gia tr] lu'qng giac ciia cac cung hon kem;
sin(
a + ; ) = cos a
7 Cong thuc d)ng:
* cos ( a - b) = cos a cos b + sin a sin b
* 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
( + b) tan a± tan b
1 + tan a tan b
8 Cong thuc nhan doi va nhan ha:
=sin'La = 2sina.cosa
(4) � cotx = cot a<=> x =a+ kst , kEZ Chu y: sin x = a, cos x = a c6 nghie
111
lcoNG THuc cAN NHd LdP GROUP: 2 0 0 1 THỦ KHOA
Trang 2D�o ham t6ng ,Hi�u,Tich va ThuO'ng
( cotx) = -2- ( cotu) = -2-
n
ca + b r = L c� an-k bk
k=O
* (k.u )' = k.u'
(k la hang so)
(cos u )' = -u' sin u
,
( II) _ u -n.u u 11-J f
,
(sin u) = u' cosu
, u' (tanu) =-')-
cos- u
(Fu)'= u�
2'\/U
,
* ( u.v) = u'.v + u.v'
V6'i u ta m9t ham s6
,
(sin x) = cosx (cosx)' = -sin x , 1 (tanx) =-')-
cos- x
,
* ( u ± v) = u' ± v'
,
* (l_! ) = u'.v�u.v'
, (xii) = n.x'""
(c)' = 0 (C: hang so') (x)'=l
( C.x)' = C
(Fx)' = \- [x c- O]
2'\/X
,
23.Bang cong thirc d�o ham
* PTTT cua d6 thi hs :y=f(x) te;1i diSm M(xo;Yo):
y = y(x 0 ).(x-x 0 )+ Yo
22.Cong tlurc nh] tlnrc Niu-To'n
( a+ b)n = co na n + cl na n-1b + + ck n-kbk na + + en: n
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 ¢:>
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:
M'=D0(M)=(x';y'),khi d6 {x:= -x
y =-y
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!
k!(n -k)!
14.Phtidng trinh b�c nhat do'i vdi sinx va cosx
* a sin x ± b cos x = c <=> -J a 2 + b 2 sin( x ± a) = c
* acosx±bsinx =c <=>-Ja2 +b2 cos(x+a) =c
(cos nhtr t/Ji dilu)
01 cos a= 1 ,sma = 1
Ca hai PT tren muon tim a barn shif cos -J a' + b'
Ch , , C , PT " , hi " 2 b2> 2
!dll!.J! : ac tren co ng iern <=> a + _ c
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:
atan x +btanx +c = ') -
cos- x
¢=> atan 'x +btanx +c =d(l +tanx )
¢:> (a-djtanx +btanx +c -<l= o day la ptb2 da biSt
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
Khi d , 1 o smxcosx = t 2 - l h ' (7) d
- 2- t ay vao ta iroc pt:
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 ma 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)!
Tinh chftt: