Theoretical Computer Science Cheat Sheet Page 4
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10Theoretical Computer Science Cheat Sheet
Trigonometry
Matrices
More Trig.
C
Multiplication:
n
(0,1)
C = A · B,
c
=
a
b
.
a
i,j
i,k
k,j
b
b
h
(cos θ, sin θ)
k=1
C
θ
Determinants: det A = 0 iff A is non-singular.
A
(-1,0)
(1,0)
A
c
B
det A · B = det A · det B,
Law of cosines:
c
a
n
2
2
2
c
= a
+b
2ab cos C.
(0,-1)
B
det A =
sign(π)a
.
i,π(i)
Area:
π
i=1
Pythagorean theorem:
2 × 2 and 3 × 3 determinant:
2
2
2
C
= A
+ B
.
1
A =
hc,
a b
2
= ad
bc,
1
Definitions:
=
ab sin C,
c d
2
sin a = A/C,
cos a = B/C,
2
c
sin A sin B
a
b
c
=
.
b
c
a
c
a b
csc a = C/A,
sec a = C/B,
d
e
f
= g
h
+ i
2 sin C
e f
d f
d e
sin a
A
cos a
B
Heron’s formula:
g h
i
tan a =
=
,
cot a =
=
.
cos a
B
sin a
A
√
aei + bf g + cdh
=
s · s
· s
· s
A =
,
Area, radius of inscribed circle:
ceg
f ha
ibd.
a
b
c
1
AB
1
s =
(a + b + c),
Permanents:
AB,
.
2
2
A + B + C
n
s
= s
a,
a
perm A =
a
.
Identities:
i,π(i)
s
= s
b,
b
π
i=1
1
1
sin x =
,
cos x =
,
s
= s
c.
Hyperbolic Functions
c
csc x
sec x
1
More identities:
2
2
Definitions:
tan x =
,
sin
x + cos
x = 1,
x
x
x
x
cot x
e
e
e
+ e
1
cos x
x
sinh x =
,
cosh x =
,
sin
=
,
2
2
2
2
2
2
2
1 + tan
x = sec
x,
1 + cot
x = csc
x,
2
x
x
e
e
1
tanh x =
,
csch x =
,
1 + cos x
π
x
x
x
sin x = cos
x ,
sin x = sin(π
x),
e
+ e
sinh x
cos
=
,
2
2
2
1
1
sech x =
,
coth x =
.
π
1
cos x
cos x =
cos(π
x),
tan x = cot
x ,
cosh x
tanh x
2
x
tan
=
,
2
1 + cos x
Identities:
x
cot x =
cot(π
x),
csc x = cot
cot x,
1
cos x
2
2
2
2
2
=
,
cosh
x
sinh
x = 1,
tanh
x + sech
x = 1,
sin x
sin(x ± y) = sin x cos y ± cos x sin y,
sin x
2
2
coth
x
csch
x = 1,
sinh( x) =
sinh x,
=
,
1 + cos x
cos(x ± y) = cos x cos y
sin x sin y,
cosh( x) = cosh x,
tanh( x) =
tanh x,
1 + cos x
tan x ± tan y
x
cot
=
,
tan(x ± y) =
2
,
1
cos x
1
tan x tan y
sinh(x + y) = sinh x cosh y + cosh x sinh y,
1 + cos x
cot x cot y
1
=
,
cot(x ± y) =
,
cosh(x + y) = cosh x cosh y + sinh x sinh y,
sin x
cot x ± cot y
sin x
2 tan x
=
,
sinh 2x = 2 sinh x cosh x,
sin 2x = 2 sin x cos x,
sin 2x =
,
1
cos x
2
1 + tan
x
2
2
ix
ix
e
e
cosh 2x = cosh
x + sinh
x,
2
2
2
cos 2x = cos
x
sin
x,
cos 2x = 2 cos
x
1,
sin x =
,
2i
2
x
x
cosh x + sinh x = e
,
cosh x
sinh x = e
,
1
tan
x
ix
ix
2
e
+ e
cos 2x = 1
2 sin
x,
cos 2x =
,
2
cos x =
,
1 + tan
x
n
n ∈ Z,
2
(cosh x + sinh x)
= cosh nx + sinh nx,
2
ix
ix
2 tan x
cot
x
1
e
e
2 x
2 x
tan 2x =
,
cot 2x =
,
tan x =
i
,
2
2 sinh
= cosh x
1,
2 cosh
= cosh x + 1.
1
tan
x
2 cot x
ix
ix
2
2
e
+ e
2ix
2
2
e
1
sin(x + y) sin(x
y) = sin
x
sin
y,
=
i
,
θ
sin θ cos θ tan θ
. . . in mathematics
2ix
e
+ 1
2
2
cos(x + y) cos(x
y) = cos
x
sin
y.
you don’t under-
sinh ix
0
0
1
0
sin x =
,
√
√
stand things, you
1
3
3
i
π
Euler’s equation:
6
2
2
3
just get used to
√
√
ix
iπ
cos x = cosh ix,
e
= cos x + i sin x,
e
=
1.
2
2
π
1
them.
4
2
2
√
tanh ix
√
v2.02 c 1994 by Steve Seiden
– J. von Neumann
3
1
tan x =
.
π
3
i
3
2
2
∞
π
1
0
2
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