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# Formule matematice cls. v viii

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### Formule matematice cls. v viii

1. 1. FORMULE UTILE PENTRU CLASA A VIII A www.mateinfo.ro FORMULE DE CALCUL PRESCURTAT (a+b)2=a2+2ab+b2 ; (a-b)2=a2-2ab+b2 ; a2-b2=(a+b)(a-b) ; (a+b+c)2=a2+b2+c2+2ab+2ac+2bc ; (a+b)3=a3+3a2b+3ab2+b3 ; (a-b)3=a3-3a2b+3ab2-b3 ; a3+b3=(a+b)(a2-ab+b2) ; a3-b3=(a-b)(a2+ab+b2) ; PROPRIETATILE PUTERILOR an·am=an+m ; an:am=an -m ; (an)m=an·m ; (a·b)n=an·bn ; (a:b)n=an:bn ; a0=1 ; 0n=0 ; 1n=1 PROPRIETATILE RADICALILOR a⋅b = a ⋅ b ; a / b = a / b ; x2 = x ; ( y) 2 = y ; a≥0 ; b≥0 ; y≥0 ; exemple: 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2 ; 5 3 = 25 ⋅ 3 = 25 ⋅ 3 = 75 ; ( 6) (− 3)2 = −3 = 3 ; 2 = 6. MODULUL Definitie : |X|=X daca X≥0 si |X|= -X daca X≤0 ; Proprietati : |X|≥0 ; |a·b|=|a|·|b| ; |a+b|≤|a|+|b| ; Exemple : |-5|= -(-5)=5 ; |7|=7 ; |-2|= -(-2)=2 ; |+4|=4 ; FUNCTIA LINIARA f :R R , f(x)=ax+b P(x,y) ∈ Gf daca si numai daca f(x)=y ; A(x,y) ∈ Gf∩ox daca f(x)=y si y=0 ; B(x,y) ∈ Gf∩oy daca f(x)=y si x=0 ; Daca f si g sunt doua functii atunci Q(x,y) ∈ Gf∩Gg daca f(x)=g(x)=y ; A(-b/a , 0) si B(0 , b) MULTIMI DE NUMERE Multimea numerelor naturale notata cu N : 0,1,2,3,4,…∞ Multimea numerelor intregi notata cu Z : -∞ … ,-4,-3,-2,-1,0,1,2,3,4,…+∞ Multimea numerelor rationale notata cu Q: exemple -3/4 ;5/2 ;-12/4 ;0,23 ;-5,(24) ;4,20(576) ; Multimea numerelor reale notata cu R ; exemple : -3/4 ;5/2 ;-1/4 ; 7 5 ; − 6 ; -5,(24) ; 4,20(576) ; 0,202002000200… ;-5,2323323332333323… ;
2. 2. Avem urmatoarele relatii de incluziune intre aceste multimi : N ⊂ Z ⊂ Q ⊂ R. FIGURI PLANE REMARCABILE BC ⋅ AD AB ⋅ AC ⋅ sin A = 2 2 PΔABC= AB+BC+CA AΔABC= AABCD= (AB + CD ) ⋅ BE 2 PABCD= AB+BC+CD+DA AABCD= AB·BC AABCD= CD·AE PABCD= 2·(AB+BC) AABCD= AC2=AB2+BC2 PABCD= 2·(AB+BC) AC ⋅ BD 2 PABCD= 4·AB poligoane regulate : l=latura poligonului ; a=apotema poligonului ; A=aria ; P=perimetrul ; P=3·l l2 3 l 3 ;a = A= 4 6 l=R 3 h= l 3 3R = 2 2 P=4·l A = l2 ; a = l=R 2 d=l 2 =2R l 2 P=6·l 3l 2 3 l 3 A= ;a = 2 2 l=R
3. 3. TRIUNGHIUL DREPTUNGHIC Teorema catetei: b2=a·n Teorema inaltimii: c2=a·m ; h2=m·n ; h= b⋅c a Teorema lui Pitagora: a2=b2+c2 ; c2=h2+m2 si b⋅c a ⋅h = A= 2 2 Aria tr. dreptunghic: b2=h2+n2 FUNCTII TRIGONOMETRICE functia 45° 1 2 cos c.o. AB = i. AC 60° sin sin x°= 30° 3 2 3 2 1 2 2 2 2 2 cos x°= c.a. BC = i. AC tg x°= functia 30° 60° 45° tg 3 3 3 1 3 3 3 1 ctg c.o. AB = c.a. BC ctg x°= c.a. BC = c.o. AB TRIUNGHIURI ASEMENEA ,TEOREMA LUI THALES rezulta: AB AC BC = = MN MP NP rezulta: AB AC = AM AN CERCUL Lc= 2π R ; Ac= π R 2 ; Daca m ∠AOB = x o atunci : π Rx o LAB= 180 o πR 2xo AOAB= 360 o
4. 4. www.mateinfo.ro FORMULE - CORPURI GEOMETRICE I. POLIEDRE CUBUL PARALELIPIPEDUL DREPTUNGHIC Al = 4l 2 ; At = 6l 2 ; V = l 3 At = 2 ⋅ ( L ⋅ l + L ⋅ h + l ⋅ h ) ; V = L ⋅ l ⋅ h d f = l 2; d = l 3 d = L2 + l 2 + h 2 PRISMA REGULATĂ TRIUNGHIULARĂ PATRULATERĂ Al = Pb ⋅ h At = Al + 2 ⋅ Ab HEXAGONALĂ V = Ab ⋅ h PIRAMIDA REGULATĂ TRIUNGHIULARĂ Al = PATRULATERĂ Pb ⋅ a p 2 At = Al + Ab HEXAGONALĂ V= Ab ⋅ h 3
5. 5. www.mateinfo.ro TRUNCHIUL DE PIRAMIDĂ REGULATĂ TRIUNGHIULARĂ Al = PATRULATERĂ ( PB + Pb ) ⋅ at At = Al + AB + Ab 2 HEXAGONALĂ V= ( ht ⋅ AB + Ab + AB ⋅ Ab 3 II. CORPURI ROTUNDE CILINDRUL CONUL Al = π RG At = π R ( G + R ) Al = 2π RG At = 2π R ( G + R ) V= V =πR H 2 TRUNCHIUL DE CON At = π Gt ( R + r ) + π R + π r π Ht 2 2 V= ( R + r + Rr ) 3 3 SFERA Al = π Gt ( R + r ) 2 π R2 H 2 A = 4π R 2 4π R 3 V= 3 )