III MATEMATI^KE FORMULE
1. ARITMETIKA I ALGEBRA 1.1. RAZMERE ⎧a : c = b : d ⎪ Ako je a : b = c : d tada je: ⎨d : b = c : a ; ⎪d : c = b : a ⎩
⎧(a ⋅ m) : (b ⋅ m) = c : d ⎪(a : m) : (b : m) = c : d ⎪ , m je proizvoqan faktor; Ako je a : b = c : d tada je: ⎨ ( ) : ( ) : a ⋅ m b = c ⋅ m d ⎪ ⎪⎩(a : m) : b = (c : m) : d ⎧( a ± b ) : ( c ± d ) = a : c = b : d ⎪( a + b ) : ( c + d ) = ( a − b ) : (c − d ) ⎪ Iz a : b = c : d sledi: ⎨ ; ( ) : ( ) ( ) : ( ) a m b n c m d n a m b n c m d n ⋅ + ⋅ ⋅ + ⋅ = ⋅ − ⋅ ⋅ − ⋅ ⎪ ⎪⎩(a ⋅ m ± b ⋅ n) : (c ⋅ m ± d ⋅ n) = (a ⋅ p ± b ⋅ q) : (c ⋅ p ± d ⋅ q )
Zlatni presek:
a : x = x : (a − x) ,
(a > x) ;
x1 + x2 + ... + xn ; n
Aritmeti~ka sredina:
x =
Geometrijska sredina:
Gn ( x) = n x1 ⋅ x2 ⋅ ... ⋅ xn ;
Harmonijska sredina:
H n ( x) =
n 1 1 1 + + ... + x1 x2 xn
,
(xi > 0 , i = 1,2,..., n) ;
1.2. STEPENI a m ⋅ a n = a m+ n ; a m : a n = a m−n ; a 0 = 1 , (a ≠ 0) ; a − m = a m ⋅ b n = ( a ⋅ b) m ; a m : b n = (
1 , (a ≠ 0) ; am
a m ) ; (a m )n = a m⋅n ; b
1.3. KORENI ( a) = a m
p
m
m n
m
a a: b= ; b m
m
p
;
a= n
m
n m
a
n⋅ p
a = p
m⋅n
a
a = m⋅n a ;
m n
p ⋅n
= a m
⋅b = a ⋅ b ; p
n
p n
;
−m
a p = m a− p ;
a ⋅ m b = m a ⋅b ; n
m n
a = a , (m ∈ Z ; n ∈ N ) ; m
1.4. RACIONALISAWE IMENIOCA a ⋅ n b n−r p p ⋅ (a ⋅ b c ⋅ d ) = ; = ; n r b a2 ⋅ b − c2 ⋅ d a⋅ b ±c⋅ d b a
1.5. KVADRAT I KUB BINOMA I RAZLIKA KVADRATA ( a ± b) 2 = a 2 ± 2 ⋅ a ⋅ b + b 2 ; ( a ± b)3 = a 3 ± 3 ⋅ a 2 ⋅ b + 3 ⋅ a ⋅ b 2 ± b 3 ; a 2 − b 2 = ( a − b) ⋅ ( a + b) ;
1.6. RASTAVQAWE U FAKTORE POZNATIJIH IZRAZA a 3 − b3 = ( a − b) ⋅ ( a 2 + a ⋅ b + b 2 ) ; a 4 − b 4 = (a − b) ⋅ (a 3 + a 2 ⋅ b + a ⋅ b 2 + b3 ) = (a 2 − b 2 ) ⋅ (a 2 + b 2 ) ; a 3 + b 3 = ( a + b) ⋅ ( a 2 − a ⋅ b + b 2 ) ; a 5 + b5 = (a + b) ⋅ (a 4 − a 3 ⋅ b + a 2 ⋅ b 2 − a ⋅ b3 + b 4 ) ;
a 2 + b 2 = (a 2 + 2 ⋅ a ⋅ b + b 2 ) − 2 ⋅ a ⋅ b = (a + b − 2 ⋅ a ⋅ b ) ⋅ (a + b + 2 ⋅ a ⋅ b ) ; a 4 + b 4 = (a 4 + 2 ⋅ a 2 ⋅ b 2 + b 4 ) − 2 ⋅ a 2 ⋅ b 2 = (a 2 + b 2 − a ⋅ b ⋅ 2) ⋅ (a 2 + b 2 + a ⋅ b ⋅ 2) ; x 2 + p ⋅ x + q = ( x + m) ⋅ ( x + n ) , ( p = m + n ; q = m ⋅ n) ; a 2⋅n − b 2⋅n = (a − b) ⋅ (a 2⋅n−1 + a 2⋅n−2 ⋅ b + ... + a ⋅ b 2⋅n−2 + b 2⋅n−1 ) = (a n − b n ) ⋅ (a n + b n ) ; a 2⋅n+1 ± b 2⋅n+1 = (a ± b) ⋅ (a 2⋅n a 2⋅n−1 ⋅ b + a 2⋅n−2 ⋅ b 2 ... + a 2 ⋅ b 2⋅n−2 a ⋅ b 2⋅n−1 + b 2⋅n ) ;
itd.
1.7. LOGARITMI Logaritam pozitivnog realnog broja x sa osnovom a ( a > 0 ) je eksponent kojim treba stepenovati osnovu a da bi se dobio broj x .Taj tra`eni eksponent ozna~ava se sa log a x i va`i: a log x = x , za x = a je a log a = a ⇒ log a a = 1 ; Osnova tzv. prirodnih logaritama (oznaka ln x ) je e = 2,7182818284590... a osnova tzv. dekadnih logaritama (oznaka log x ) je 10. Veza izme|u dekadnih i prirodnih logaritama je: log x = ln x ⋅ log e ; log e = 0, 434294... = M , ( M - modul Briggs-ovih logaritama); a
ln x = log x ⋅
a
1 1 ; = 2,3025... = M −1 , ( M −1 - modul prirodnih logaritama); log e log e
1.7.1. OSOBINE LOGARITAMA x = log a x − log a y ; y 1 log a x k = k ⋅ log a x , (k ∈ R ; x ∈ R + ) ; log a n x = ⋅ log a x ; n log a ( x ⋅ y ) = log a x + log a y ; log a
1.8. JEDNA^INE 1.8.1. ^ISTA KVADRATNA JEDNA^INA a ⋅ x2 + c = 0 ;
x1,2 = ± −
c ; a
1.8.2. OP[TA KVADRATNA JEDNA^INA −b ± b 2 − 4 ⋅ a ⋅ c a ⋅ x + b ⋅ x + c = 0 ; x1,2 = ; 2⋅a 2
b c Za p = i q = je: a a
p x + p ⋅ x + q = 0 ; x1,2 = − ± 2 2
p2 −q ; 4
⎧ p2 ⎫ Diskriminanta je: D = {b − 4ac} odnosno D = ⎨ − q ⎬ . Va`i: a) D > 0 ⇒ ⎩4 ⎭ realni razli~iti koreni, b) D = 0 ⇒ dvostruki realni koreni i v) D < 0 ⇒ 2
kowugovano kompleksni koreni.
2. GEOMETRIJA 2.1. PLANIMETRIJA 1.
3.
2.
zbir uglova u trouglu
zbir uglova u mnogouglu
uglovi u pravilnom mnogouglu
α + β + γ = 1800
α1 + α 2 + ... + α n =
α = [(n − 2) ⋅ 1800 ]/ n β = 3600 / n
(n − 2) ⋅ 1800
4.
5.
periferijski ugao β=
α 2
6.
tetivni ugao β=
α
tetivni ~etvorougao α + γ = β + δ = 1800
2
2.2. TROUGLOVI I ^ETVOROUGLOVI
1.
2.
3.
pravougli trougao
jednakostrani~ni trougao
a 2 = p ⋅ c ; b2 = q ⋅ c ;
h = (a ⋅ 3) / 2
a 2 + b2 = c 2 h2 = p ⋅ q ; c ⋅ h = a ⋅ b ;
r = (a ⋅ 3) / 3
d = a⋅ 2 a = (d ⋅ 2) / 2
ρ = (a ⋅ 3) / 6
6.
5.
4.
kvadrat
pravougaonik
kvadrat
paralelogram
O = 2 ⋅ ( a + b)
O = 4⋅a d = a⋅ 2 A = a2
O = 2 ⋅ ( a + b) A= a⋅h
d = a 2 + b2 A = a ⋅b
7.
9.
8.
nejednakostrani~ni trougao
a ⋅ ha b ⋅ hb c ⋅ hc = = 2 2 2 O = a +b + c ; s = O/2 A=
A = s ( s − a )( s − b)( s − c)
nejednakostrani~ni trougao
s = O / 2 = ( a + b + c) / 2 A= ρ ⋅s a ⋅b⋅c A= 4⋅r
jednakostrani~ni trougao O = 3⋅ a a2 ⋅ 3 A= 4
12.
11.
10.
trapez
~etvorougao sa normalnim dijagonalama
tangencijalni mnogougao
O =a+b+c+d A= s⋅h a+c A= ⋅h 2
O =a+b+c+d d ⋅d A= 1 2 2
O = a + b + c + ... O⋅r A= 2
2.3. KRU@NICA I KRUG
2.
1.
kru`nica i luk O = 2 ⋅ r ⋅π r ⋅π ⋅α s= 1800
3.
krug i ise~ak A = r2 ⋅π r2 ⋅π ⋅α r ⋅ s AI = = 3600 2
2.4. GEOMETRIJSKA TELA
kru`ni venac i ise~ak A = (r 2 − r12 ) ⋅ π (r 2 − r12 ) ⋅ π ⋅ α AI = 3600 s + s1 A= ⋅ (r − r1 ) 2
2.
1.
3.
pravougli paralelopiped
kocka
prizma
d 2 = a 2 + b2 + c 2 A = 2 ⋅ (a ⋅ b + a ⋅ c + b ⋅ c) V = a ⋅b⋅c
d = a⋅ 3 A = 6 ⋅ a2 V = a3
A = 2 ⋅ B + Aomota~a V = B⋅h
5.
4.
piramida A = B + Aomota~a B⋅h V= 3
7.
6.
zarubqena piramida B : B1 = h 2 : x 2 A = B + B1 + Aomota~a
A = r ⋅ π ⋅ (r + s ) r2 ⋅π ⋅ h V= 3
A = 2 ⋅ r ⋅ π ⋅ ( r + h) V = r2 ⋅π ⋅ h
V = [h1 ( B + B1 + BB1 )]/ 3
9.
8.
kupa
vaqak
zarubqena kupa
lopta
A = π ⋅ [r 2 + r12 + (r + r1 ) s ] π ⋅h 2 2 V= ⋅ (r + r1 + r ⋅ r1 ) 3
A = 4 ⋅ r2 ⋅π 4 ⋅ r3 ⋅π A= 3
3. TRIGONOMETRIJA 3.1. SVOJSTVA TRIGONOMETRIJSKIH FUNKCIJA Definicije funkcija: ako su a i b du`ine kateta pravouglog trougla, c du`ina hipotenuze, α i β o{tri uglovi naspram stranica BC i AC onda je: a b = cos β ; cos α = = sin β ; c c a b tg α = = ctg β ; ctg α = = tg β ; b a
sin α =
Osnovni odnosi su: sin 2α + cos 2α = 1 ; tg α ⋅ ctg α = 1 ; sin α cos α tg α = ; ctg α = ; cos α sin α
Znaci funkcija po kvadrantima i vrednosti funkcija za neke uglove su:
prvi kvadrant drugi kvadrant tre}i kvadrant ~etvrti kvadrant
ugao 00 ugao 300 ugao 450 ugao 600 ugao 900 ugao 1800 ugao 2700 ugao 3600
sin α
cos α
tg α
ctg α
+ + −
+ − −
+ − +
+ − +
−
+
−
−
sin α
cos α
tg α
ctg α
0 1/2
1
0
( 3) / 2
( 3) / 3
±∞ 3
( 2) / 2
( 2) / 2
1
1
( 3) / 2
1/2 0
±∞
0
−1
0
±∞
0 1
±∞
0
0
±∞
1 0 −1
0
3
( 3) / 3
3.2. FUNKCIJE ZBIRA I RAZLIKE sin (α ± β ) = sin α ⋅ cos β ± cos α ⋅ sin β cos(α ± β ) = cos α ⋅ cos β sin α ⋅ sin β tgα ± tgβ ctgα ⋅ ctgβ 1 tg(α ± β ) = ; ctg(α ± β ) = ; 1 tgα ⋅ tgβ ctgβ ± ctgα
3.3. FUNKCIJE DVOSTRUKOG UGLA I FUNKCIJE POLOVINE UGLA sin (2 ⋅ α ) = 2 ⋅ sin α ⋅ cos α ; cos (2 ⋅ α ) = cos 2 α − sin 2α ; sin α = 2 ⋅ sin
α
⋅ cos
α
; cos α = cos 2
α
− sin 2
α
2 2 2 2 2 2 ⋅ tg α ctg α − 1 tg (2 ⋅ α ) = ; ctg (2 ⋅ α ) = ; 2 1 − tg α 2 ⋅ ctg α
tg α =
α
2 ⋅ tg 1 − tg
2 ; ctg α = 2
α
1 + cos α = 2 ⋅ cos sin
α 2
=±
2
2α
2
ctg 2
α
2
2 ⋅ ctg
;
−1
α
;
2
; 1 − cos α = 2 ⋅ sin 2
α 2
;
1 − cos α α 1 + cos α α 1 − cos α ; cos = ± ; tg = ± ; 2 2 2 2 1 + cos α
3.4. FORMULE ZA TRANSFORMACIJU sin α + sin β = 2 ⋅ sin
α +β
⋅ cos
α −β
; sin α − sin β = 2 ⋅ cos
α +β
⋅ sin
α −β
; 2 2 2 2 α +β α −β α +β α −β ⋅ cos ⋅ sin cos α + cos β = 2 ⋅ cos ; cosα − cos β = −2 ⋅ sin ; 2 2 2 2
3.5. PRAVOUGLI TROUGAO Ako su a i b du`ine kateta pravouglog trougla, c du`ina hipotenuze, α i β o{tri uglovi naspram stranica BC i AC onda je:
a = c ⋅ sin α = c ⋅ cos β ; b = c ⋅ sin β = c ⋅ cos α ; a = b ⋅ tg α = b ⋅ ctg β ; b = a ⋅ tg β = a ⋅ ctg α ; a a b b c= = ; c= = ; sin α cos β sin β cos α
Povr{ina pravouglog trougla je: a⋅c b⋅c a2 b2 c2 A= ⋅ sin β = ⋅ sin α = ⋅ tg β = ⋅ tg α = ⋅ sin (2 ⋅ α ) ; 2 2 2 2 2
3.6. PROIZVOQAN TROUGAO Ako su a , b i c du`ine stranica proizvoqnog trougla a α , β i γ naspramni uglovi u wemu onda je: Sinusna teorema: a : b : c = sin α : sin β : sin γ
Kosinusna teorema: a 2 = b 2 + c 2 − 2 ⋅ b ⋅ c ⋅ cos α ; b 2 = a 2 + c 2 − 2 ⋅ a ⋅ c ⋅ cos β ; c 2 = a 2 + b 2 − 2 ⋅ a ⋅ b ⋅ cos γ ;
4. ANALITI^KA GEOMETRIJA 4.1. JEDNA^INE 1. Jedna~ina prave:
y = a ⋅ x + b , a = tg α je koeficijent smera , b je odse~ak na y koordinatnoj osi ;
2. Jedna~ina kru`nice: x 2 + y 2 = r 2 , r je polupre~nik kru`nice ; ( x − p)2 + ( y − q)2 = r 2 ; 3. Op{ta jedna~ina kru`nice: x2 y 2 b ⋅ x + a ⋅ y = a ⋅ b ili + =1 ; 4. Osna jedna~ina elipse: a 2 b2 x2 y2 2 2 2 2 2 2 b ⋅ x − a ⋅ y = a ⋅ b ili − =1 ; 5. Osna jedna~ina hiperbole: a 2 b2 y 2 = 2 ⋅ p ⋅ x , 2 ⋅ p je parametar . 6. Temena jedna~ina parabole: 2
2
2
2
2
2