Algebra formulas and theorems Flashcards
a^n - b^n = ?
(a - b)(a^n-1 + a^n-2b + … + ab^n-2 + b^n-1)
a^n + b^n = ?
(a + b)(a^n-1 - a^n-2b + … - ab^n-2 + b^n-1)
x^3+y^3+z^3-3xyz = ?
1/2 (x+y+z)((x-y)^2+(y-z)^2+(z-x)^2)
x^3-y^3 = ?
(x-y)^3 - 3xy(x-y)
x^3+y^3 = ?
(x+y)^3 - 3xy(x+y)
(x+y+z)^3 = ?
x^3+y^3+z^3+3(x+y)(y+z)(z+x)
(x+y+z)^2 = ?
x^2 + y^2 + z^2 + 2(xy + yz + zx)
(x^2+y^2)(z^2+t&2) = ?
(xz+yt)^2 + (xt-yz)^2
(x^2+y&2)(z^2+t^2) = ?
(xz-yt)^2 + (xt+yz)^2
Simon’s favorite factoring trick: ab+ma+nb+k = ?
(a+n)(b+m)+k-mn
Lemma: LOOK AT THE BOOK. I DON’T REALLY KNOW IT WELL
1.1.3.11
Newton Sums
1.1.3.12
De Moivre
1.1.4.1
Euler’s formula
e^(ix) = cos(x) + i * sinx
Roots of unity: z^n = 1
z = cos(2pik/n) + isin(2pik/n)
HM GM AM QM inequality
1.1.5.1
Sedrakyan’s inequality for squares
1.1.5.3
Volume of a cone
pir^2x/3
Area of cone
piry+pir^2
Volume of a sphere
4/3 pir^3
Area of a sphere
4pir^2
Area of regular pentagon
sqrt(5(5+2sqrt(5)))a^2/4
Area of regular hexagon
3sqrt(3)a^2/2
Area of regular octagon
2(1+sqrt(2))a^2
Area of regular decagon
5sqrt(5+2sqrt(5))a^2/2
Area of regular dodecagon
3(2+sqrt(3))a^2
Area of regualr polygon
na^2/4tan(180/n)
sin(x+y)
sinx cosy + cosx siny
sin(x-y)
sinx cosy - cosx siny
cos(x+y)
cosx cosy + sinx siny
cos(x-y)
cosx cosy - sinx siny
tan(x+y)
(tan x + tan y)/(1 - tanx tany)
tan(x-y)
(tan x - tan y)/(1 + tanx tany)
sinx + siny
2sin((x+y)/2)cos((x-y)/2)
sinx - siny
2sin((x-y)/2)cos((x+y)/2)
cosx + cosy
2cos((x+y)/2)cos((x-y)/2)
cosx - cosy
-2cos((x+y)/2)sin((x-y)/2)
sinxsiny
1/2(cos(x-y) - cos(x+y))
cosxcosy
1/2(cos(x-y) + cos(x+y))
sinxcosy
1/2(sin(x-y) sin(x+y))
sin2x
2sinxcosx
cos2x
cos^2x-sin^2x
tan2x
2tanx/(1-tan^2x)
sin(x/2)
+/- sqrt((1-cosx)/2)
cos(x/2)
+/- sqrt((1+cosx)/2)
tan(x/2)
sinx/(1+cosx)
