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ALGEBRA 1
LOGARITHM
x
b
x = log N → N =b
Properties
log 1
log
log log
log log
log log log
log( ) log log
=
=
=
= −
= +
a
b
x x
x n x
x y y
x
xy x y
a
b
n
REMAINDER AND FACTOR THEOREMS
Given:
x r
f x
Remainder Theorem: Remainder = f(r)
Factor Theorem: Remainder = zero
QUADRATIC EQUATIONS
A
B B AC Root
Ax Bx C
2
4
0
2
2
− ± −
Sum of the roots = - B/A
Products of roots = C/A
MIXTURE PROBLEMS
Quantity Analysis: A + B = C
Composition Analysis: Ax + By = Cz
WORK PROBLEMS
Rate of doing work = 1/ time
Rate x time = 1 (for a complete job)
Combined rate = sum of individual rates
Man-hours (is always assumed constant)
2
2 2
1
1 1
..
( ker )( )
..
( ker )( )
quantityof work
Wor s time
quantityofwork
Wor s time
ALGEBRA 2
UNIFORM MOTION PROBLEMS
S = Vt
Traveling with the wind or downstream:
1 2
V V V
total
Traveling against the wind or upstream:
1 2
V V V
total
DIGIT AND NUMBER PROBLEMS
100 h + 10 t+u → 2-digit number
where: h = hundred’s digit
t = ten’s digit
u = unit’s digit
CLOCK PROBLEMS
where: x = distance traveled by the
minute hand in minutes
x/12 = distance traveled by the hour
hand in minutes
PROGRESSION PROBLEMS
a 1 = first term
an = n
th term
am = any term before an
d = common difference
S = sum of all “n” terms
ARITHMETIC PROGRESSION (AP)
- difference of any 2 no.’s is constant
- calcu function: LINEAR (LIN)
[ 2 ( 1 ) ]
1
1
2 1 3 2
a n d
n
S
a a
n
S
d a a a a etc
a a n m d
n
n m
GEOMETRIC PROGRESSION (GP)
- RATIO of any 2 adj, terms is always constant
- Calcu function: EXPONENTIAL (EXP)
−
r n
r
a
S
r
r
a r
S
r
r
a r
S
a
a
a
a
r
a a r
n
nm n m
n
1
1
1
2
3
1
2
HARMONIC PROGRESSION (HP)
- a sequence of number in which their reciprocals
form an AP
- calcu function: LINEAR (LIN)
Mean – middle term or terms between two terms
in the progression.
COIN PROBLEMS
Penny = 1 centavo coin
Nickel = 5 centavo coin
Dime = 10 centavo coin
Quarter = 25 centavo coin
Half-Dollar = 50 centavo coin
DIOPHANTINE EQUATIONS
If the number of equations is less than the
number of unknowns, then the equations are
called “Diophantine Equations”.
ALGEBRA 3
Fundamental Principle:
“If one event can occur in m different ways, and
after it has occurred in any one of these ways, a
second event can occur in n different ways, and
then the number of ways the two events can
occur in succession is mn different ways”
PERMUTATION
Permutation of n objects taken r at a time
n r
n
nP
r
Permutation of n objects taken n at a time
nP n! n
=
Permutation of n objects with q,r,s, etc. objects
are alike
qr s
n
P =
Permutation of n objects arrange in a circle
P= ( n− 1 )!
n
th term
Common
difference
Sum of ALL
terms
Sum of ALL
terms
Sum of ALL
terms, r >
Sum of ALL
terms, r < 1
n
th term
ratio
Sum of ALL
terms,
r < 1 , n = ∞
PLANE
TRIGONOMETRY
ANGLE, MEASUREMENTS &
CONVERSIONS
1 revolution = 360 degrees
1 revolution = 2π radians
1 revolution = 400 grads
1 revolution = 6400 mils
1 revolution = 6400 gons
Relations between two angles (A & B)
Complementary angles → A + B = 90°
Supplementary angles → A + B = 180°
Explementary angles → A + B = 360°
Angle (θ) Measurement
NULL (^) θ = 0°
ACUTE (^) 0° < θ < 90°
RIGHT θ = 90°
OBTUSE (^) 90° < θ < 180°
STRAIGHT θ =180°
REFLEX (^) 180° < θ < 360°
FULL OR PERIGON (^) θ = 360°
Pentagram – golden triangle (isosceles)
36 °
72 ° 72 °
TRIGONOMETRIC IDENTITIES
A
A A
A
A A
A A A
A A B
A B
A B A B
A B
A B A B
A B A B A B
A B A B A B
A A
A A
A A
2 cot
cot 1 cot 2
1 tan
2 tan tan 2
cos 2 cos sin
sin 2 2 sin cos
cot cot
cot cot 1 cot( )
1 tan tan
tan tan tan( )
cos( ) cos cos sin sin
sin( ) sin cos cos sin
1 tan sec
1 cot csc
sin cos 1
2
2 2
2
2 2
2 2
2
2
−
−
=
= −
=
±
± =
± ± =
± =
± = ±
m
m
m
SOLUTIONS TO OBLIQUE TRIANGLES
SINE LAW
C
c
B
b
A
a
sin sin sin
= =
COSINE LAW
a
2
= b
2
+ c
2
b
2
= a
2
+ c
2
c
2
= a
2
+ b
2
AREAS OF TRIANGLES AND
QUADRILATERALS
TRIANGLES
- Given the base and height
Area bh
- Given two sides and included angle
sin θ
2
1 Area = ab
- Given three sides
2
( )( )( )
a b c s
Area s s a s b s c
=
= − − −
- Triangle inscribed in a circle
r
abc Area
4
=
- Triangle circumscribing a circle
Area = rs
- Triangle escribed in a circle
Area =r( s−a )
QUADRILATERALS
- Given diagonals and included angle
sin θ
1 2
Area = dd
- Given four sides and sum of opposite angles
2
2 2
( )( )( )( ) cos
2
a b c d s
A C B D
Area s a s b s c s d abcd
=
=
=
= − − − − −
θ
θ
- Cyclic quadrilateral – is a quadrilateral
inscribed in a circle
4 ( )
( )( )( )
2
( )( )( )( )
Area
ab cd ac bd ad bc r
a b c d s
Area s a s b s c s d
=
=
= − − − −
d d = ac+bd →
1 2 Ptolemy’s Theorem
- Quadrilateral circumscribing in a circle
Area = rs
a b c d
s
Area abcd
THEOREMS IN CIRCLES
TRAPEZOID
A (a b) h
PARALLELOGRAM
sin
sin
1 2
A dd
A bh
A ab
RHOMBUS
sin α
2
1 2
A a
A dd ah
SOLIDS WITH PLANE SURFACE
Lateral Area = (No. of Faces) (Area of 1 Face)
Polyhedron – a solid bounded by planes. The bounding
planes are referred to as the faces and the intersections
of the faces are called the edges. The intersections of the
edges are called vertices.
PRISM
V = Bh
A(lateral) = PL
A(surface) = A(lateral) + 2B
where: P = perimeter of the base
L = slant height
B = base area
Truncated Prism
numberof heights
heights
V B
PYRAMID
A A B
A A
V Bh
surface lateral
lateral faces
( ) ( )
( )
Frustum of a Pyramid
A 1 A 2 A 1 A 2
h
V = + +
A 1 = area of the lower base
A 2 = area of the upper base
PRISMATOID
1 2 m
A A A
h
V = + +
Am = area of the middle section
REGULAR POLYHEDRON
a solid bounded by planes whose faces are congruent
regular polygons. There are five regular polyhedrons
namely:
A. Tetrahedron
B. Hexahedron (Cube)
C. Octahedron
D. Dodecahedron
E. Icosahedron
Name
Type ofFACE
No. of FACES
No. of EDGES
No. of
VERTICES
Formulas for
VOLUME
TetrahedronHexahedronOctahedron Dodecahedron
Icosahedron
TriangleSquareTrianglePentagonTriangle
4
6
4
(^68)
(^86)
12 12
12
30
20
20
30
12
3
12
2
x
V
=
3 x
V
=
3
(^23)
x
V
=
3
(^66). 7
x
V
=
3
(^18). 2
x
V
=
Where: x = length of one edge
SOLIDS WITH CURVED SURFACES
CYLINDER
V = Bh = KL
A(lateral) = PkL = 2 π r h
A(surface) = A(lateral) + 2B
Pk = perimeter of right section
K = area of the right section
B = base area
L= slant height
CONE
A rL
V Bh
lateral
( )
FRUSTUM OF A CONE
A R r L
A A AA
h
V
lateral
( )
1 2 1 2
SPHERES AND ITS FAMILIES
SPHERE
2 ( )
3
A r
V r
surface π
SPHERICAL LUNE
is that portion of a spherical surface bounded by the
halves of two great circles
°
=
90
(deg)
2
( )
π r θ
A surface
SPHERICAL ZONE
is that portion of a spherical surface between two parallel
planes. A spherical zone of one base has one bounding
plane tangent to the sphere.
A r h zone
2 π ( )
=
SPHERICAL SEGMENT
is that portion of a sphere bounded by a zone and the
planes of the zone’s bases.
2
r h
h
V = −
( 3 3 )
6
( 3 )
6
2 2 2
2 2
a b h
h V
a h
h V
= + +
= +
π
π
SPHERICAL WEDGE
is that portion of a sphere bounded by a lune and the
planes of the half circles of the lune.
°
=
270
(deg)
3 π r θ
V
STRAIGHT LINES
General Equation
Ax + By + C = 0
Point-slope form
y – y 1 = m(x – x 1 )
Two-point form
( ) 1
2 1
2 1
1
x x
x x
y y
y y −
−
−
− =
Slope and y-intercept form
y = mx + b
Intercept form
b
y
a
x
Slope of the line, Ax + By + C = 0
B
A
m =−
Angle between two lines
−
1 2
(^121)
tan
mm
m m
Note: Angle θ is measured in a counterclockwise
direction. m 2 is the slope of the terminal side while m 1 is
the slope of the initial side.
Distance of point (x 1 ,y 1 ) from the line
Ax + By + C = 0;
2 2
1 1
A B
Ax By C
d
Note: The denominator is given the sign of B
Distance between two parallel lines
2 2
1 2
A B
C C d
−
Slope relations between parallel lines:
m 1 = m 2
Line 1 → Ax + By + C 1 = 0
Line 2 → Ax + By + C 2 = 0
Slope relations between perpendicular lines:
m 1 m 2 = –
Line 1 → Ax + By + C 1 = 0
Line 2 → Bx – Ay + C 2 = 0
PLANE AREAS BY COORDINATES
1 2 3 1
1 2 3 1
, , ,.... ,
, , ,.... ,
2
1
y y y y y
x x x x x
A
n
n
Note: The points must be arranged in a counter clockwise
order.
LOCUS OF A MOVING POINT
The curve traced by a moving point as it moves in a
plane is called the locus of the point.
SPACE COORDINATE SYSTEM
Length of radius vector r:
2 2 2 r = x + y + z
Distance between two points P 1 (x 1 ,y 1 ,z 1 ) and
P 2 (x 2 ,y 2 ,z 2 )
2 2 1
2 2 1
2 2 1
d = ( x −x) +(y −y) +(z −z)
ANALYTIC
GEOMETRY 2
CONIC SECTIONS
a two-dimensional curve produced by slicing a plane
through a three-dimensional right circular conical surface
Ways of determining a Conic Section
- By Cutting Plane
- Eccentricity
- By Discrimination
- By Equation
General Equation of a Conic Section:
Ax
2
+ Cy
2
+ Dx + Ey + F = 0 **
Cutting plane Eccentricity
Circle Parallel to base e → 0
Parabola Parallel to element e = 1.
Ellipse none e < 1.
Hyperbola Parallel to axis e > 1.
Discriminant **Equation****
Circle B
2
- 4AC < 0, A = C A = C
Parabola B
2
- 4AC = 0
A ≠ C
same sign
Ellipse B
2
- 4AC < 0, A ≠ C
Sign of A
opp. of B
Hyperbola B
2
- 4AC > 0 A or C = 0
CIRCLE
A locus of a moving point which moves so that its
distance from a fixed point called the center is constant.
Standard Equation:
(x – h)
2
+ (y – k)
2
= r
2
General Equation:
x
2
+ y
2
+ Dx + Ey + F = 0
Center at (h,k):
A
E
k
A
D
h
Radius of the circle:
A
F
r = h +k −
2 2 2 or r^ D E^4 F 2
(^1 2 ) = + −
PARABOLA
a locus of a moving point which moves so that it’s always
equidistant from a fixed point called focus and a fixed line
called directrix.
where: a = distance from focus to vertex
= distance from directrix to vertex
AXIS HORIZONTAL:
Cy
2
+ Dx + Ey + F = 0
Coordinates of vertex (h,k):
C
E
k
2
= −
substitute k to solve for h
Length of Latus Rectum:
C
D
LR =
Length of latus rectum:
a
b
LR
2 2
=
Eccentricity:
d
a
a
c
e = =
HYPERBOLA
a locus of a moving point which moves so that the
difference of its distances from two fixed points called the
foci is constant and is equal to length of its transverse
axis.
d = distance from center to directrix
a = distance from center to vertex
c = distance from center to focus
STANDARD EQUATIONS
Transverse axis is horizontal
1
( ) ( )
2
2
2
2
=
− −
−
b
y k
a
x h
Transverse axis is vertical:
2
2
2
2
b
x h
a
y k
GENERAL EQUATION
Ax
2
- Cy
**2
Coordinates of the center:
C
E
k
A
D
h
If C is negative, then: a
2 = C, b
2 = A
If A is negative, then: a
2 = A, b
2 = C
Equation of Asymptote:
(y – k) = m(x – h)
Transverse axis is horizontal:
a
b m =±
Transverse axis is vertical:
b
a m =±
KEY FORMULAS FOR HYPERBOLA
Length of transverse axis: 2a
Length of conjugate axis: 2b
Distance of focus to center :
2 2
c = a +b
Length of latus rectum:
a
b
LR
2
Eccentricity:
d
a
a
c
e = =
POLAR COORDINATES SYSTEM
x = r cos θ
y = r sin θ
2 2 r = x + y
x
y
tan θ =
SPHERICAL
TRIGONOMETRY
Important propositions
- If two angles of a spherical triangle are equal ,
the sides opposite are equal ; and conversely.
- If two angels of a spherical triangle are
unequal , the sides opposite are unequal , and
the greater side lies opposite the greater
angle ; and conversely.
- The sum of two sides of a spherical triangle is
greater than the third side.
a + b > c
- The sum of the sides of a spherical triangle is
less than 360°.
0° < a + b + c < 360°
- The sum of the angles of a spherical triangle is
greater that 180° and less than 540°.
180° < A + B + C < 540°
- The sum of any two angles of a spherical
triangle is less than 180° plus the third angle.
A + B < 180° + C
SOLUTION TO RIGHT TRIANGLES
NAPIER CIRCLE
Sometimes called Neper’s circle or Neper’s pentagon, is
a mnemonic aid to easily find all relations between the
angles and sides in a right spherical triangle.
Napier’s Rules
- The sine of any middle part is equal to the
product of the cosines of the opposite parts.
Co-op
- The sine of any middle part is equal to the
product of the tangent of the adjacent parts.
Tan-ad
Important Rules:
- In a right spherical triangle and oblique angle
and the side opposite are of the same quadrant.
- When the hypotenuse of a right spherical
triangle is less than 90°, the two legs are of the
same quadrant and conversely.
- When the hypotenuse of a right spherical
triangle is greater than 90°, one leg is of the first
quadrant and the other of the second and
conversely.
QUADRANTAL TRIANGLE
is a spherical triangle having a side equal to 90°.
SOLUTION TO OBLIQUE TRIANGLES
Law of Sines:
C
c
B
b
A
a
sin
sin
sin
sin
sin
sin
Law of Cosines for sides:
c a b a b C
b a c a c B
a b c b c A
cos cos cos sin sin cos
cos cos cos sin sin cos
cos cos cos sin sin cos
Law of Cosines for angles:
C A B A B c
B A C A C b
A B C B C a
cos cos cos sin sin cos
cos cos cos sin sin cos
cos cos cos sin sin cos
dx
du
u u
h u
dx
d
dx
du
u u
h u
dx
d
dx
du
u
u
dx
d
dx
du
u
u
dx
d
dx
du
u
u
dx
d
dx
du
u
u
dx
d
dx
du hu hu u
dx
d
dx
du hu hu u
dx
d
dx
du u hu
dx
d
dx
du u hu
dx
d
dx
du u u
dx
d
dx
du u u
dx
d
dx
du
u u
u
dx
d
dx
du
u u
u
dx
d
dx
du
u
u
dx
d
dx
du
u
u
dx
d
dx
du
u
u
dx
d
2
1
2
1
2
1
2
1
2
1
2
1
2
2
2
1
2
1
2
1
2
1
2
1
1
1 (csc )
1
1 (sec )
1
1 (sinh )
1
1 (tanh )
1
1 (cosh )
1
1 (sinh )
(csc ) csc coth
(sec ) sec tanh
(coth ) csc
(tanh ) sec
(cosh ) sinh
(sinh ) cosh
1
1 (csc )
1
1 (sec )
1
1 (cot )
1
1 (tan )
1
1 (cos )
−
−
−
−
−
−
=
−
=
=
=−
=−
=−
=
=
=
−
−
−
=
−
=
−
−
−
−
−
−
−
−
−
−
−
−
− DIFFERENTIAL
CALCULUS
LIMITS
Indeterminate Forms
∞
L’Hospital’s Rule
g x
f x
Lim
g x
f x
Lim
g x
f x
Lim
x →a x→a x→a
Shortcuts
Input equation in the calculator
TIP 1: if x → 1, substitute x = 0.
TIP 2: if x → ∞ , substitute x = 999999
TIP 3: if Trigonometric, convert to RADIANS then
do tips 1 & 2
MAXIMA AND MINIMA
Slope (pt.) Y’ Y” Concavity
MAX (^0) (-) dec down
MIN (^0) (+) inc up
INFLECTION (^) - No change (^) -
HIGHER DERIVATIVES
n
th derivative of x
n
( x ) n!
dx
d
n
n
n
n
th derivative of xe
n
n X
n
n
xe x n e
dx
d
TIME RATE
the rate of change of the variable with respect to time
dt
dx
= increasing rate
dt
dx
= decreasing rate
APPROXIMATION AND ERRORS
If “dx” is the error in the measurement of a quantity x,
then “dx/x” is called the RELATIVE ERROR.
RADIUS OF CURVATURE
[ 1 ( ') ]
2
3 2
y
y
R
INTEGRAL
CALCULUS 1
u udu u C
u udu u C
udu u C
udu u C
udu u C
udu u C
edu e C
C a
a a du
u C u
du
C n n
u u du
f u gu du f udu gudu
adu au C
du u C
u u
u u
n n
=− +
= +
=− +
= +
= +
=− +
= +
= +
= +
=
= +
= +
csc cot csc
sec tan sec
csc cot
sec tan
cos sin
sin cos
ln
ln
..............( 1 ) 1
[ ( ) ( )] ( ) ( )
2
2
1
= −
= + > −
= + < −
=− +
= +
−
= +
= +
= +
=− +
=− +
=− +
= +
= +
= +
+
= −
−
= +
−
= +
= +
−
= − +
= + +
= +
= +
−
−
−
−
−
−
−
−
−
udv uv vdu
C u a a
u
a u a
du
C u a a
u
a u a
du
C u
a
u u a a
du
C a
u
u a
du
C a
u
u a
du
udu u C
udu u C
hu udu hu C
hu udu hu C
hudu u C
hudu u C
udu u C
udu u C
C a
u
au u
du
C a
u
u u a a
du
C a
u
a u a
du
C a
u
a u
du
udu u u C
udu u u C
udu u C
udu u C
coth .......... ....
1
tanh ..............
1
sinh
1
cosh
sinh
coth lnsinh
tanh lncosh
csc coth csc
sec tanh sec
csc coth
sec tanh
cosh sinh
sinh cosh
cos 1
2
sec
1
tan
1
sin
csc lncsc cot
sec lnsec tan
cot lnsin
tan lnsec
1 2 2
1 2 2
1
2 2
1
2 2
1
2 2
2
2
1
2
1
2 2
1 2 2
1
2 2
INTEGRAL
CALCULUS 2
TIP 1: Problems will usually be of this nature:
- “Find the area bounded by”
- “Find the area revolved around..”
TIP 2: Integrate only when the shape is IRREGULAR,
otherwise use the prescribed formulas
VOLUME OF SOLIDS BY REVOLUTION
Circular Disk Method
∫
2
1
2
x
x
V π R dx
Cylindrical Shell Method
=
2
1
2
y
y
V π RL dy
Circular Ring Method
∫
= −
2
1
( )
2 2
x
x
V π R r dx
PROPOSITIONS OF PAPPUS
First Proposition: If a plane arc is revolved about a
coplanar axis not crossing the arc, the area of the surface
generated is equal to the product of the length of the arc
and the circumference of the circle described by the
centroid of the arc.
= •
= •
A dS r
A S r
π
π
2
2
Second Proposition: If a plane area is revolved
about a coplanar axis not crossing the area, the volume
generated is equal to the product of the area and the
circumference of the circle described by the centroid of
the area.
= •
= •
V dA r
V A r
π
π
2
2
CENTROIDS OF VOLUMES
∫
2
1
x
x
V x dV x
∫
2
1
y
y
V y dV y
WORK BY INTEGRATION
Work = force × distance
∫ ∫
2
1
2
1
y
y
x
x
W Fdx Fdy; where F = k x
Work done on spring
1
2
2
W = k x −x
k = spring constant
x 1 = initial value of elongation
x 2 = final value of elongation
Work done in pumping liquid out of the
container at its top
Work = (density)(volume)(distance)
Force = (density)(volume) = ρv
Specific Weight:
Volume
Weight
γwater = 9.81 kN/m
2 SI
γwater = 45 lbf/ft
2 cgs
Density:
Volume
mass
ρwater = 1000 kg/m
3 SI
ρwater = 62.4 lb/ft
3 cgs
ρsubs = (substance) (ρwater)
1 ton = 2000lb
MOMENT OF INERTIA
Moment of Inertia about the x- axis:
=
2
1
2
x
x
x
I y dA
Moment of Inertia about the y- axis:
=
2
1
2
y
y
y
I x dA
Parallel Axis Theorem
The moment of inertia of an area with respect to any
coplanar line equals the moment of inertia of the area
with respect to the parallel centroidal line plus the area
times the square of the distance between the lines.
2
I Ix Ad
x o
Moment of Inertia for Common Geometric
Figures
Square
3
bh
I
x
12
3 bh Ixo =
Triangle
3
bh
I x =
36
3 bh Ixo =
Circle
4
4 r I xo
π
Half-Circle
4
r
I
x
Quarter-Circle
4
r
I
x
Ellipse
4
3 ab I x
π
4
3 ab I y
π
FLUID PRESSURE
∫
F wh dA
F whA γ hA
F = force exerted by the fluid on one side of
the area
h = distance of the c.g. to the surface of liquid
w = specific weight of the liquid (γ)
A = vertical plane area
Specific Weight:
Volume
Weight
γwater = 9.81 kN/m
2 SI
γwater = 45 lbf/ft
2 cgs
