Logarithmic Functions Sum and Product of Roots

aloga N = N in ax n + bx n−1 + ⋯ + C = 0:

log N = (log e)(ln N) b

Sum = (−1)n
a
colog N = − log N
C
log 0 = −∞ Product =
a

Factorization Patterns
Remainder/Factor Theorem
a8 − b8 = (a4 + b4 )(a2 + b2 )(a + b)(a − b)
f(x)
7 7 6 5 4 2 3 3 2 4 5 6) for ; remainder = f(r)
a + b = (a + b)(a − a b + a b − a b + a b − ab + b x−r
a7 − b7 = (a − b)(a6 + a5 b + a4 b2 + a3 b3 + a2 b4 + ab5 + b6 )
Mean Proportion

Quadratic Equation – General Form a: x = x: b

ax 2 + bx + c = 0 Third Proportion

Quadratic Formula a: b = b: x

−b ± √b 2 − 4ac Fourth Proportion

x=
2a a: b = c: x
Discriminant and Nature of Roots

b2 − 4ac > 0 ; two distinct

Binomial Theorem
2
b − 4ac = 0 ; two equal
r th term = (nCr−1 )(x n−r+1 )(y r−1 )
b2 − 4ac < 0 ; complex unequal
Sum of Exponents
a+b n
(n)(n + 1) ; for (x a + y b )
Sum and Product of Roots 2
b c Sum of Coefficients
− and ; ax 2 + bx + c = 0
a a substitute 1 to variables
Arithmetic, Geometric and Harmonic Mean Relation 2 tan θ
tan 2θ =
1 − tan2 θ
(AM)(HM) = (GM)2
cot 2 θ − 1
Solving the Numbers cot 2θ =
2 cot θ
a or b = AM ± √AM 2 − GM 2 Half Angle Formulas

GCF, LCM and Number Relationship 1

sin2 θ = (1 − cos 2θ)
2
GCF ∙ LCM = x ∙ y
1
cos2 θ = (1 + cos 2θ)
2
Other Trigonometric Functions 1 − cos 2θ
tan2 θ =
1 + cos 2θ
versine θ = 1 − cos θ
1 − cos 2θ sin θ
coversine θ = 1 − sin θ tan θ = =
sin θ 1 + cos 2θ
1 − cos θ
haversine θ =
2
Power Reduction Formulas
exsecant θ = sec θ − 1
1
sin3 θ = (3 sin θ − sin 3θ)
4
Cofunction Relations 1
cos3 θ = (3 cos θ + cos 3θ)
4
sin(90 − θ) = cos θ
tan(90 − θ) = cot θ
Product Formulas
sec(90 − θ) = csc θ
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
Double Angle Formula
tan A ± tan B
sin 2θ = 2 sin θ cos θ tan(A ± B) =
1 ∓ tan A tan B
cos 2θ = cos2 θ − sin2 θ

cos 2θ = 2 cos2 θ − 1

cos 2θ = 1 − 2 sin2 θ
Addition/Subtraction Formulas Hyperbolic Function
1 ex − e−x
sin A cos B = + [sin(A + B) + sin(A − B)] sinh x =
2 2
1 ex + e−x
cos A cos B = + [cos(A + B) + cos(A − B)] cosh x =
2 2
1 ex − e−x
sin A sin B = − [cos(A + B) − cos(A − B)] tanh x =
2 ex + e−x
Factoring Formulas cosh2 x − sinh2 x = 1
A±B A∓B tanh2 x + sech2 x = 1
sin A ± sin B = +2 sin cos
2 2
coth2 x − csch2 x = 1
A+B A−B
cos A + cos B = +2 cos cos cosh x ± sinh x = e±x
2 2
A+B A−B
cos A − cos B = −2 sin sin
2 2
Other Trigonometric Identities*

sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C

Sine and Cosine Law
tan A + tan B + tan C = tan A tan B tan C
a b c
= =
sinA sinB sinC
c 2 = a2 + b2 − 2abcosC Angle Measurement Units and Conversion

Tangent Law 1 rev = 1 cycle = 360° = 2π rad = 400 grad = 6400 mils

A−B
a − b tan 2
= Slope
a + b tan A + B
2
y2 − y1 A
Mollweide’s Equation* m= = tan θ = − ∗ partial derivative
x2 − x1 B
c
c sin 2 Angle Between Lines
=
a±b a∓b tan θ2 − tan θ1 m2 − m1
cos 2
m = tan θ = =
1 + tan θ2 tan θ1 1 + m2 m1
Line to Point Distance General Equation for Conics
Ax + By + C Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
d= ; Ax + By + C = 0 form
√A2 + B2
B 2 − 4AC < 0 ; ellipse
for (+)A ; +d = point at right ; −d = point at left
B 2 − 4AC = 0 ; parabola
for (+)B ; +d = point above ; −d = point below
B 2 − 4AC > 0 ; hyperbola
Parallel Lines Distance
∗ partial derivative for center (vertex for parabola)
C2 − C1
d= ; Ax + By + C = 0 form Polar Equation of Conics
√A2 + B 2
ed
for (+)A ; +d = line2 at right ; −d = line2 at left r= ; directrix x = ±d
1 ± cos θ
for (+)B ; +d = line2 above ; −d = line2 below
ed
r= ; directrix y = ±d
Normal Form of Linear Equation 1 ± sin θ

x cos θ + y sin θ ∓ p = 0 ecircle = 0 , eellipse < 1

A B C eparabola = 1 , ehyperbola > 1

cos θ = ; sin θ = ; p =
k k k
Angle of Rotation
k= √A2 + B2
B
tan 2θ =
Translation of Axes A−C
x = x′ + h
y = y′ + k Circle Standard Equation

where: (h, k) = new origin (x − h)2 + (y − k)2 = r 2

Rotation of Axes Radical Axis

x = x ′ cos θ − y ′ sin θ (D1 − D2 )x + (E1 − E2 )y + (F1 − F2 ) = 0

y = x ′ sin θ + y ′ cos θ Parabola Standard Equation

Incenter of Triangles (x − h)2 = ±4a(y − k) ; opens up/down

a(xa , ya ) + b(xb , yb ) + c(xc , yc ) (y − k)2 = ±4a(x − h) ; opens right/left

(xi , yi ) =
a+b+c
LR = 4a ; e = 1
where: a = side length opposite (xa , ya ) and so on …
Ellipse Standard Equation 2b2 c a
LR = e= d=
a a e
(x − h)2 (y − k)2
+ = 1 ; horizontally flattened Asymptote: y − k = ±m(x − h)
a2 b2
(y − k)2 (x − h)2 b a
+ = 1 ; vertically flattened mhorizontal = ± ; mvertical = ±
a2 b2 a b
Properties of Ellipse

2a = major/transverse axis Triangle

2b = minor/conjugate axis 1
A = ab sin θ ; ∠θequilateral = 60°
2
2b2 a
LR = d= a2 = b2 + c 2 Equilateral Triangle
a e
c c 3(a4 + b4 + c 4 + x 4 ) = 2(a2 + b2 + c 2 + x 2 )
e= e′ =
a b
Heron’s Formula
a−b a−b
f= f′ = a+b+c
a b A = √s(s − a)(s − b)(s − c) ; where s =
2
Length of Median
Hyperbola Standard Equation
1
(x − h) 2
(y − k) 2 mCc = √2a2 + 2b 2 − c 2
2
2
− =1
a b2
Triangle Inscribed to a Circle
opens left & right (horizontal: opposite directions to parabola)
abc
2 2 A=
(y − k) (x − h) 4r
2
− =1
a b2 Triangle Circumscribing a Circle
opens up & down (vertical: opposite directions to parabola)
a+b+c
A = rs ; s =
Properties of Hyperbola 2
2a = transverse axis (major) Triangle Escribed to a Circle

a = distance from center to a vertex A = r(s − a)

2b = conjugate axis (minor) a = side tangent to circle

d2 − d1 = 2a c 2 = a2 + b2
Parallelograms Centroid – Medians
1 Incenter – Angle Bisector
A = bh = ab sin θ = d1 d2 sin α
2
Circumcenter – Perpendicular Bisector
θ = angle at sides (90° for rectangle)
Orthocenter – Altitudes
α = angle at diagonals (90° for rhombus)
Euler Line – Line containing the center points

Trapezoid
Elliptic Cone
na2 + mb 2 x2 y2 z2
x=√ + + =0
m+n a2 b 2 c 2
x = a "median" not in the middle Ellipsoid

x2 y2 z2
+ + =1
a2 b 2 c 2
Non-cyclic Quadrilateral
Hyperboloid (1 sheet)
√(s − a)(s − b)(s − c)(s − d)
A= x2 y2 z2
abcd cos 2 θ + − = 1 → one term has different sign to 1
a2 b 2 c 2
A+C B+D
θ= = Hyperboloid (2 sheets)
2 2
Cyclic Quadrilateral (Inscribable into a Circle) x2 y2 z2
− − + = 1 → two terms have different sign to 1
a2 b 2 c 2
√(s − a)(s − b)(s − c)(s − d)
A=
1 abcd cos2 θ
A + C = B + D = 180° ; θ = 90° Circle

d1 d2 = ac + bd → Ptolemy’s Theorem θ θ
A = πr 2 × = 2πrh ×
360° 360°
√(ab + cd)(ac + bd)(ad + bc)
r= Pappus theorem: A = 2πrh
4A
h = distance of instantaneous centroid to center
Circumscribing Quadrilateral

A = rs = √abcd
 Spherical Triangle/Polygon & Pyramid
h

x y E
Apolygon = πr 2 ×
180°
4 E
h2 = ab Vpyramid = πr 3 ×
3 720°
E = ∑θ − (n − 2)180°
Parabolic Segment & Spandrel Spherical Cone
2 1 2
Asegment = bh ; Aspandrel = bh V = πr 2 h
3 3 3
Pentagram & Hexagram Spherical Lune
A = 1.1226r 2 ; A = r 2 √3 θ
A = 4πr 2 ×
360°
Spherical Wedge
Prismatoid
4 θ
h V = πr 3 ×
V = (A1 + A2 + 4Am ) 3 360°
6
Frustrum of Pyramid
Paraboloid, Ellipsoid, Hyperboloid
h
V = (A1 + A2 + √A1 A2 ) 2 3
3 Vparaboloid = 2π × rh × r ; Pappus ′ theorem
s 3 8
LA = (P1 + P2 ) ; s = slant height
2 4
Vellipsoid = πabc
3
h
Spherical Zone/Segment Vhyperboloid = (πR2 + πR2 + 4πr 2 ) ; Prismatoid
6
h Conoid
Vzone = (3πRh − πh2 )
6
πr 2 h
h V=
Vsegment = (3A1 + 3A2 + πh2 ) 2
6
LA = Azone = 2πRh
Polyhedron f v e Face SA Volume Radius
 √2 3 √6 3
Tetrahedron 4 4 6 √3s 2 12
s 12
s
s Trigonometric and *Hyperbolic Function Derivatives
Hexahedron 6 8 12 6s 2 s3 2
√2 3 √6 d du
Octahedron 8 6 12 2√3s 2 s s sin u = cos u
3 6 dx dx
Dodecahedron 12 20 30 20.64s2 7.66s3 1.11s d du du
cos u = − sin u ; ∗ + sinh u
Icosahedron 20 12 30 8.66s 2
2.18s 3 0.76s dx dx dx
d du
tan u = sec 2 u
dx dx
Leibniz’s Notation
d du
cot u = − csc 2 u
dy d2 y dn y dx dx
, 2 ,…, n
dx dx dx d du du
sec u = sec u tan u ; ∗ − sech u tanh u
Euler’s Notation dx dx dx

Dy , D2 y , … , Dn y d du
csc u = − csc u cot u
dx dx
Lagrange’s Notation
Trigonometric Function Integrals
′ (x) ′′ (x) n (x)
f ,f ,… ,f
∫ tan u du = − ln|cos u| + C
Jacobi’s Notation

∂ ∂2 ∂n ∫ cot u du = ln|sin u| + C
, 2 ,… , n
∂x ∂x ∂x

∫ sec u du = ln|sec u + tan u| + C

Logarithmic Function Derivatives
d u du ∫ csc u du = − ln|csc u + cot u| + C
a = au ln a
dx dx
d 1 du ∫ sec u tan u du = sec u + C
log a u =
dx u ln a dx
∫ csc u cot u du = − csc u + C
Exponential Function Integrals
au
∫ au du = +C ∫ sec 2 u du = tan u + C
ln a
1
∫ log a u du = (u ln u − u) + C ∫ csc 2 u du = − cot u + C
ln a
 d ±1 du
arccosh u =
dx √u2 − 1 dx
Inverse Trigonometric Function Derivatives
d 1 du
d sin ±1 du arctanh u =
arc u= dx 1 − u2 dx
dx cos √1 − u2 dx
d ±1 du
d tan ±1 du arccoth u =
arc u= dx 1 − u2 dx
dx cot 1 + u2 dx
d ±1 du
d sec ±1 du arcsech u =
arc u= dx u√1 − u2 dx
dx csc u√u − 1 dx
2
d −1 du
Inverse Trigonometric Integrals arccosh u =
dx u√1 + u2 dx
du u d du
∫ = arcsin + C g du = sech u
√a2 − u2 a dx dx
du 1 u g du = gudermanion of u = arctan(sinh u)
∫ = arctan + C
a2 +u 2 a a
du 1 u
∫ = arcsec + C Variable with Variable Exponents
u√u2 − a2 a a
du y = uv ; ln y = v ln u
∫ = ln |u + √u2 ± a2 | + C
√u2 ± a2 dy⁄
dx = ln u dv + v du
du 1 a+u u v dx u dx
∫ = ln + C , u 2 < a2
a2 − u2 2a a − u
u a2 u
∫ √a2 − u2 du = √a2 − u2 + arcsin + C Curvature & Radius of Curvature
2 2 a
y ′′
u a2 k= 3 ; R = k −1
∫ √u2 ± a2 du = √u2 ± a2 ± ln |u + √u2 ± a2 | + C [1 + (y ′ )2 ]2
2 2
Center of Curvature (h, k)

Inverse Hyperbolic Function Derivatives y ′ [1 + (y ′ )2 ]

h=x+
y′′
d 1 du
arcsinh u =
dx √u + 1 dx
2 1 + (y ′ )2
k=y+
y ′′
 if m and n is even, else × 1

Errors and Relative Errors

dR dA dV Partial Fractions
dR, dA, dV ; , ,
R A V
1 A B
= +
(x + 1)(x − 1) x + 1 x − 1
1 A B
2
= +
(x − 1) x − 1 (x − 1)2
Trigonometric Transformation (sin and cos)
1 A B(2x + 3) + C
m n m n = + 2
∫ sin u cos u du , ∫ sin u du , ∫ cos u du x(x 2 + 3x + 4) x x + 3x + 4
1 A B(2x) + C D(2x) + E
If at least m or n is odd = + 2 + 2
x(x 2 + 4)2 x x +4 (x + 4)2
sin2 x = 1 − cos 2 x or cos2 x = 1 − sin2 x
If both are even
Algebraic Substitution
1
sin2 x = (1 − cos 2x) xdx
2 ex: ∫ → Let: u = x + 1
(x + 1)8
1
cos2 x = (1 + cos 2x) Trigonometric Substitution
2
1
sin x cos x = 2 sin 2x 1. a2 − x 2 → Let: x = a sin θ

2. a2 + x 2 → Let: x = a tan θ

3. x 2 − a2 → Let: x = a sec θ
Trigonometric Transformation (tan and sec)

∫ tann u du ; ∫ sec n u du

Use: tan2 x = sec 2 x − 1 or sec 2 x = tan2 x + 1

Wallis Formula
π
2 1 9∙7∙5∙3∙1 π
∫ cos10 ax sin4 ax dx = × ×
0 3 (10 + 4) ∙ 12 ∙ 10 ∙ 8 ∙ 6 ∙ 4 ∙ 2 2
Area Under the Curve

𝐴 = ∫(𝑦𝑢𝑝𝑝𝑒𝑟 − 𝑦𝑙𝑜𝑤𝑒𝑟 )𝑑𝑥

𝐴 = ∫(𝑥𝑟𝑖𝑔ℎ𝑡 − 𝑥𝑙𝑒𝑓𝑡 )𝑑𝑦

X-center Equation

𝑑𝑥 → 𝑥𝑐 = 𝑥
𝑥𝑟𝑖𝑔ℎ𝑡 + 𝑥𝑙𝑒𝑓𝑡
𝑑𝑦 → 𝑥𝑐 =
2
Y-center Equation
𝑦𝑢𝑝𝑝𝑒𝑟 + 𝑦𝑙𝑜𝑤𝑒𝑟
𝑑𝑥 → 𝑦𝑐 =
2
𝑑𝑦 → 𝑦𝑐 = 𝑦
Area Revolved at x=h

2𝜋 ∫ 𝐴(𝑥𝑐 − ℎ)

Area Revolved at y=k

2𝜋 ∫ 𝐴(𝑦𝑐 − 𝑘)

Centroid

𝑥𝑐𝑒𝑛𝑡𝑒𝑟 𝐴 = ∫ 𝐴𝑥𝑐

𝑦𝑐𝑒𝑛𝑡𝑒𝑟 𝐴 = ∫ 𝐴𝑦𝑐

Moment of Inertia

𝐼𝑥 = ∫ 𝐴𝑥 2 𝑑𝑥

𝐼𝑦 = ∫ 𝐴𝑦 2 𝑑𝑦
 arcsinh x = ln [x ± √1 − x 2 ]
Complex Numbers: Rectangular, Polar, Exponential Form
arccosh x = ln [x ± √x 2 − 1]
j53.13° j0.9273
3 + j4 = 5∠53.13° = 5e = 5e
1 1+x
Exponent of Complex Number arctanh x = ln [ ]
2 1−x
(z∠θ)n = z n ∠nθ

Roots of Complex Number Euler’s Notation

1 1 θ + 360k 1 ejθ − e−jθ
(z∠θ)n = z n ∠ ; k is 0 → (n − 1) sin θ = sinh jθ =
n j j2

ejθ + e−jθ
cos θ = cosh jθ =
Logarithmic Functions of Complex Number 2

log(zejθ ) = log z + jθr log e 1 1 ejθ − e−jθ

tan θ = tanh jθ = ( jθ )
j j e + e−jθ
ln(zejθ ) = ln z + jθr

π j π −π
Trigonometric Functions of Complex Numbers jj = e− 2 , jj = e 2
2
sin jx = j sinh x ↔ sinh jx = j sin x
cos jx = cosh x ↔ cosh jx = cos x Division of Matrices

tan jx = j tanh x ↔ tanh jx = j tan x [A] adj [B T ]

= [A] × [B]−1 = [A] ×
[B] |B|
ex: sin(x + jy) = sin x r cosh y + j cos x r sinh y

Laplace Transforms
Inverse Trigonometric/Hyperbolic Functions of Complex Numbers
f(t) → F(s)
arcsin x = −j ln [jx ± √1 − x 2 ]
C
C →
arccos x = −j ln [x ± √x 2 − 1] s
1
1 1 + jx eat →
arctan x = − j ln [ ] s−a
2 1 − jx
 n!
tn →
s n+1
a
sin at →
s2 + a2
s
cos at →
s + a2
2

a
sinh at →
s 2 − a2
s
cosh at →
s − a2
2

First Shift Theorem

ℒ {eat f(t)} = F(s − a)

Second Shift Theorem
dn
ℒ {t n f(t)} = (−1)n {F(s)} = (−1)n F n (s)
ds n
Third Shift Theorem
∞
f(t)
ℒ { } = ∫ F(0)ds
t 0=s

Poles and Zeroes

s+2
ex: F(s) =
(s + 5)(s − 1)
poles = −5, 1
zeroes = −2