Cheat_Sheet

Theoretical Computer Science Cheat SheetTrigonometry Matrices More Trig.Multiplication:C(0,1)n∑C = A · B, c i,j = a i,k b k,j .bb a(cos θ, sin θ)k=1hCAθDeterminants: det A ≠0iffA is non-singular.(-1,0) (1,0)A c Bdet A · B = det A · det B,Law of cosines:c a(0,-1)Bdet A = ∑ n∏c 2 = a 2 +b 2 −2ab cos C.sign(π)a i,π(i) .Pythagorean theorem:πArea:i=1C 2 = A 2 + B 2 .2 × 2 and 3 × 3 determinant:Definitions:∣ a bA = 1 2 hc,c d∣ = ad − bc,= 1 2ab sin C,sin a = A/C, cos a = B/C,csc a = C/A, sec a = C/B,a b c∣ ∣ ∣ ∣∣∣ tan a = sin acos a = A B , cos acot a =sin a = B d e fA . ∣ g h i ∣ = g b c∣∣∣ e f ∣ − h a c∣∣∣ d f ∣ + i a b= c2 sin A sin B.d e∣2 sin CHeron’s formula:aei + bfg + cdhArea, radius of inscribed circle:=− ceg − fha− ibd.A = √ s · s a · s b · s c ,12 AB, ABA + B + C .Permanents:s = 1Identities:perm A = ∑ 2(a + b + c),n∏s a = s − a,a i,π(i) .sin x = 1csc x , cos x = 1sπsec x ,i=1b = s − b,Hyperbolic Functionss c = s − c.tan x = 1cot x , sin2 x + cos 2 x =1,Definitions:More identities:√1 − cos x1 + tan 2 x = sec 2 x, 1 + cot 2 x = csc 2 sinh x = ex − e −x, cosh x = ex + e −x, sin xx,2 = ,222sin x = cos ( π2 − x) tanh x = ex − e −x1, sin x = sin(π − x),e x , csch x =+ e−x sinh x ,cos x = − cos(π − x),tan x = cot ( π2 − x) ,cot x = − cot(π − x), csc x = cot x 2− cot x,sin(x ± y) = sin x cos y ± cos x sin y,cos(x ± y) = cos x cos y ∓ sin x sin y,tan x ± tan ytan(x ± y) =1 ∓ tan x tan y ,cot x cot y ∓ 1cot(x ± y) =cot x ± cot y ,sin 2x = 2 sin x cos x, sin 2x = 2 tan x1 + tan 2 x ,cos 2x = cos 2 x − sin 2 x, cos 2x = 2 cos 2 x − 1,cos 2x =1− 2 sin 2 x,cos 2x = 1 − tan2 x1 + tan 2 x ,tan 2x =2 tan x1 − tan 2 x , cot 2x = cot2 x − 12 cot x ,sin(x + y) sin(x − y) = sin 2 x − sin 2 y,cos(x + y) cos(x − y) = cos 2 x − sin 2 y.Euler’s equation:e ix = cos x + i sin x, e iπ = −1.v2.02 c○1994 by Steve Seidensseiden@acm.orghttp://www.csc.lsu.edu/~seidensech x = 1cosh x , coth x = 1tanh x .Identities:cosh 2 x − sinh 2 x =1, tanh 2 x + sech 2 x =1,coth 2 x − csch 2 x =1, sinh(−x) =− sinh x,cosh(−x) = cosh x, tanh(−x) =− tanh x,sinh(x + y) = sinh x cosh y + cosh x sinh y,cosh(x + y) = cosh x cosh y + sinh x sinh y,sinh 2x = 2 sinh x cosh x,cosh 2x = cosh 2 x + sinh 2 x,cosh x + sinh x = e x , cosh x − sinh x = e −x ,(cosh x + sinh x) n = cosh nx + sinh nx, n ∈ Z,2 sinh 2 x 2 = cosh x − 1, 2 cosh2 x 2= cosh x +1.θ sin θ cos θ tan θ0 0 1 0π6π4π3π12√22√32√ √3 32 3√221√12 321 0 ∞...in mathematicsyou don’t understandthings, youjust get used tothem.– J. von Neumann√1 + cos xcos x 2 = ,2√1 − cos xtan x 2 = 1 + cos x ,= 1 − cos xsin x ,= sin x1 + cos x ,√1 + cos xcot x 2 = 1 − cos x ,= 1 + cos xsin x ,= sin x1 − cos x ,sin x = eix − e −ix,2icos x = eix + e −ix,2tan x = −i eix − e −ixe ix + e −ix ,= −i e2ix − 1e 2ix +1 ,sinh ixsin x = ,icos x = cosh ix,tanh ixtan x = .i