some basics of ma thematic calculations

8/3/2019 Some Basics of Ma Thematic Calculations

1/25

Quadratic Formula

if ax^2 + bx + c = 0 then

-b SQRT(b2- 4ac)

x = -------------------

2a

Sum and Difference of Cubes

a^3 +or- b^3 = (a +or- b)(a^2 +or- ab + b^2)

Difference ofSquares

a^2 - b^2 = (a+b)(a-b)

Equation of a Circle

x^2 + y^2 = r^2

Distributive Law

a(b + c) = ab + ac

Cummutative Law (Addition)

a + b = b + a

Associative Property (Addition)

(a + b) + c = a + (b + c)

Area of a Square

A = s^2 where s is the length of any side.

Area of a Rectangle

A = xy where x and y are adjacent sides (not opposite sides)

Area of a Circle

A = pi * r^2 where r is the radius of the circle


2/25

Area of a Triangle

1/2 * b * h where b is the length of the base and h is the height

Volume of a Cube

V = s^3 where s is the length of any side

Volume of a Sphere

(4/3) * PI * r^3 where r is the radius of the sphere

Surface Area of a Cube

Surface Area = 6s^2 where s is the length of any side

Surface Area of a Sphere

4 * PI * r^2 where r is the radius of the sphere

Complex (Imaginary) Numbers

i = SQRT(-1)

i^2 = -1

1/i = -i

SQRT(i) = SQRT(1/2) + SQRT(1/2)i

Midpoint Formula (of a line segment)

M = midpointM(x,y)=(1/2(x1+x2),(1/2(y1+y2))

slope intercept

y=mx+b

Fundamental Trig Identities

(sin^2 x) + (cos^2 x) = 11 + (tan^2 x) = (sec^2 x)1 + (cot^2 x) = (csc^2 x)tan x = (sin x)/(cos x)cot x = (cos x)/(sin x)cot x = 1/(tan x)


3/25

sec x = 1/(cos x)csc x = 1/(sin x)sin (-x) = -sin xcos (-x) = cos xtan (-x) = -tan x

Biology: Hardy-Weinberg equilibrium

p^2 + 2pq + q^2 = 1

p = recessive alleleq = dominant allele

p^2, q^2, and 2pq are percentages.

Vertex of a Parabola

When f(x)=ax^2+bx+cVertex coordinates=(-b/2a, f(-b/2a))

Approximation for e.

This approximates e when x = 100, 1000, 10,000 , etc. the high you go,the closer it resembles e. 1 x 10^n = x. Theoretically, it should equal eas n approches infinity. However, on my calculator (TI-86), after n > 10^13 itstops working, and always equals 1. Funny how that happens. So it's apeculiarity. The formula, however, is

((x+1)/(x-1))^(x/2)

Oddly simple, but it's a large amount of fun. Good calculating.

Sum of Interior angles of a n polygon

(n-2)(180)This formula will find the total sum of the angles of a convex polygon.

Volume of a cylinder

Volume = PI x r^2 x h

where h is height of cylinder and r is radius of its cross-section

area of a trapezoid

(base one + base two) / 2 * height


4/25

Law Of Cosines

Given: two sides and INCLUDED angle, find third side works with any triangle:A triangle with sides a,b,c and angles opposite sides A, B, C (the upper caps are angles)

C^2=a^2+b^2 +2abcosC

If you notice, this Pythagorean "extended", cosine of 90 degrees is 0, therefore, if you havea right triangle, this works out, and now with ANY triangle. Cool huh?

UPDATE:Dear Fellow Students, I goofed on my first "Law of Cosines" entry, the following is updated:

Given ANY triangle and the Side-Angle-Side situation, one can find the length of the thirdside. Ok, given three sides a, b, c, and opposite angles A, B, C (A is opposite side a, forexample) the following is true:

c^2 = a^2 + b^2 -2ab cosineC, and of course, if the included angle is 90 degrees, then the"2abCosC" term goes away, because Cosine 90=0. There ya go.

Point Slope

Y-Y1=M(x-x1)

Sum-to-Product Formulas

sin u + sin v = 2(sin((u + v)/2))(cos((u - v)/2))

sin u - sin v = 2(cos((u + v)/2))(sin((u - v)/2))cos u + cos v = 2(cos((u + v)/2))(cos((u - v)/2))cos u - cos v = -2(cos((u + v)/2))(sin((u - v)/2))

Volleyball Statistics

For those of you who are into sports, you may find the following interesting (there are acouple of formulas for more advanced v-ball stats included):

K = killsE = attack errors (including being blocked)TA = total attacks (every attempt for a kill or side out)APCT or A% = attack percentage = ((K - E)/TA)A = assistsSA = service acesSE = service errorsRE = reception errorsD = digsBS = solo blocksBA = assisted blocks [total blocks = (BS + (.5 * BA))]


5/25

BE = block errorsBHE = ball handling errors

Frustum of Right Circular Cone

Volume = (1/3)PI(r^2 + rR + R^2)h

Lateral Surface Area = (PI)s(r + R)

Total Surface Area = PI[r(r+s)+R(R+s)]

r = small radiusR = large radiush = heights = slant height = SQRT[(R-r)2+h2]

Inscribed " Y " perimeter of thickness "T" inside of square

with side "S"

perimeter of " Y " =( (sqrt(8)+1)xS) -((sqrt(8)-2))xT

area of " Y "=((((sqrt(8)+1))/2 )xSxT)-((T/(sqrt(8)-2))^2

Volume of an Ellipsoid

(4/3)*pi*r1*r2*r3

Volume of a Pyramid

(b*h)/3 where b is the base, and h is the height

Area of an Ellipse

pi * r1 * r2 where r1 is the verticle radius and r2 is the horizontal radius

Area of a Parallelogram

b*h, b is base, h is height

Negative Exponents

n^-2 = 1/(n^2)

Law ofSines

SinA/a = SinB/b = SinC/c


6/25

In an ASA or an AAS triangle, find the remaining sides. In a SSA triangle, find an angleopposite a given side and then find the right side. Notice that 0,1, or 2 triangles arepossible.

Equation of a Vertical Line

x = k (where k is a constant)

General Linear Equation

Ax + By + C = 0

(A and B not both 0)

Slope of a Line

m = rise/run = (y2 - y1)/(x2 - x1)

cosine and sin

In a triangle:Sides a,b, and cAngles A,B,C directly opposite their matching sides. I.E. A across from a

sin2A=2sinAcosA

c^2=a^2+b^2-2ab (cosC)

when trying to find 'b' also use this formula.........

b^2=a^2+c^2-2ac (cos B)

notice angle B

Herron's Formula

Finds the area of any triangle given the length of its 3 sides:

A=(s(s-a)(s-b)(s-c))^(1/2) where s= (a+b+c)/2

Surface Area of a Right Prisim

(2*L*W)+(P*H) L=length W=width P=perimeter of one side H=height


7/25

Average Deviations

Average value = measured values added together then divided by how many values thereare.

10.123+10.013+9.987/3= 10.041

Deviations= measured values subtracted from Average value.

10.041-10.123=0.08210.041-10.013=0.02810.041-9.987=0.054

Average deviation= deviation values added together and divided by number of deviationsinvolved. Then, place a -+ along with that number to the right of the average value.

0.082+0.028+0.054/3=0.01

Average deviation= 10.041 -+0.01

Temp. Formulas and Boiling & Freezing Points

Kelvin= Celsius +273Celsius= Kelvin -273Celsius= Farenhieght-32/1.8Farenheight= (1.8)* Celsius +32

Freezing points (in degrees)Kelvin -------273

Celsius --------0Farenheight ---32

Boiling points (in degrees)Kelvin -------373Celsius ------100Farenheight --212

difference of cubes

a^3-b^3=(a-b)(a^2+ab+b^2)

Volume of a " Cubic Yoid "

V = (355/2712)* [(SQRT{72})*S^2*T-9*S*T^2+(SQRT{2}+4)*T^3]S = Dimension of edge of cube which encloses the Yoid polyhedronT = Dimension of uniform cross sectional thickness*Note that S >= T*[(SQRT{5}+1)/2]


8/25

Area of Cubic Yoid

355 [ SQRT{72}xS^2xT - 9xSxT^2 + ( SQRT{2} + 4 )xT^3 ]x[( SQRT{8} + 1 )xS - (SQRT{8} - 2 )xT ]AREA = ------ x ----------------------------------------------------------------------------------------------------------------2712 [(( SQRT{8} + 1 )xSxT/2 ) - (( T^2 )/( 12 - SQRT{128}))]

S = width or height of the Cubic Yoid

T = edge or cross section depth

#note that T =< Sx[( SQRT{5} - 1 )/2 ]

cross product of two vectors

(used to determine a vector perpindicular to a and b)

(aXb)=(a2b3-a3b2, a3b1-a1b3, ab2-a2b1)where a and b are vectors and the values 1, 2, 3 represent components for the respectivevector.

Cross-Product

where a and b are vectorsa=(a1,a2,a3)b=(b1,b2,b3)aXb=(a2b3-a3b2, a3b1-a1b3, a1b2-a2b1)

Finds a vector perpindicular to a and b.

distributive property w/ parents

a=b(c+d)+3 a=bc+bd+3

Difference of 2 cubes

x^3-y^3 = (x-y)(x^2+xy+y^2)

Heron's Formula

S= [s(s-a)(s-b)(s-c)]^0.5

tan,sin,cos (by Cornelius M.)

tan x = sin x / cos xsin^2 x + cos^2 x = 1


9/25

sin^2 x / cos^2 x = tan^2 x(tan x)(cos x) = sin x

by the way......1 + 1 = 21 + 1 = 10

Volume of a Rectangular Prism

L * W * H

Perimeter of a circle

(P=2[[pi]]r)

Perimeter of a Rectangle

(l+w)*2

Area of a Regular (Equalateral) Polygon

x(s) = n s^2/(4 tan(180/n))

Where:

n=number of sidess=length of side

Parallel Lines Equation

For parallel lines do (x-1)(x+1)=0. They only solutions are -1 and 1. If you want moreparallel lines (x-2)(x+2)=0 the solutions are -2 and 2. You can also do (x-1)(x+1)(x-2)(x+2)=0 and this will give you the solutions -1, 1, -2, and 2.

Takes 1-year 1-Year for the Earth to revolve around the Sun

Proof of1-Year or "the Earth around the Sun

Radius=93,000,000 miles X 2 = 186,000,000 miles X Pi =584,337,600 miles

in circumference.

Earth Travels at 66,698 miles per hour around the Sun. SO---------

Circumference C/V Velocity = 876 Hours/24 = 365 Days or 1 Year


10/25

Euler's Formula

My (and Euler's) favorite formula is

i(pi)e +1=0

This equation (which includes the 3 basic operations, exponentation, multiplication, andaddition, and also 5 big constants, e, i, pi, 1, adn 0) is gotten from the identity

ixe =cos(x)+sin(x)i

By the way, complex numbers can be written in polar form using

i(theta)re =r


11/25

Laplace's Equation in spherical coordinates

^2V = (1/r^2)/r(r^2*V/r)+ (1/r^2*sin(theta))/(theta)(sin(theta)V/(theta))+ (1/r^2*sin^2(theta))^2V/(phi)^2

Revolution of Reuleaux

V=[(710/339)-(126025/76614)]*{R}^3

V = volume of the solid of revolution where R = radius of one of the three arcs that formthe perimeter of the Reuleaux triangle or the length of one of the sides of the inscribedequilateral triangle.

Area of a trapezoid

0.5(a+b)h

a=longest baseb=shortest baseh=hieght of the shape

Reuleaux's Revolution Redux

A = [(710/113)-(252050/76614)]x{R}^2

A = area of the solid of revolution where R = radius of the arc(s) forming the perimeter oftheReuleaux triangle or the length of the side(s) of the inscribed equilateral triangle.

[13-({12}^0.5x(355/113))]O = ------------------------- x {R}[16-(1420/113)]

O = centroid of the solid of revolution where R =radius of the arc(s) forming the perimeter of theReuleaux triangle or the length of the side(s) of the inscribed triangle.

Volume and Area of a rectangle

LxWxH=VL w and h refer to lenth times width times height. The find the area, you go lxw which islenth times width

TRIGNOMETRY EQUATIONS

In Trig, it helps to rememebr the equasions by SohCahToa which means Sin (Soh) isOpposite divided by Hypothesis; Cosin (Cah) is Adjacent divided by Hypothesis; Tan (Toa) is


12/25

Opposite divided by Adjacent.Heres an example for Tan (if the degree is 10* and the opposite is 48cm)

Tan 10*= o/aTan 10*= 48/aTan 10*a= 48/a x a

Tan 10*a= 48Tan 10*=0.158384440324536293838883092694366or 0.1583844Tan 10*A/0.1583844a = 48/0.1583844A=303.060149863244107374211096547387=303.06015Approximatly.

Addition/Subtraction Of Fractions

An way easier way to add/Subt. fractions:

B DA C ___________--- --- WHERE AS : A D + C BB D

It works on every fraction, and is easier than trying to find the common multiple, etc...

sequencial occurance of squares

Any square is determined by the difference of the prior two squares added to the the root ofthe square prior to the result.(x-1)(x-1) - x(x)= X+1(x+1)

formulae of an equilateral triangle

Area of circle using diameter

(pi*d^2)/4

surface area of a rectangular prism

2lh+2lw+2whl=lenght, h=height, w=width

Temperature Formulas

hese are common temperature formulas.


13/25

Fahrenheit To Centigrade:5/9 * (Fahrenheit - 32); note: .55555 = 5/9

Centigrade To Fahrenheit:(1.8 * Centigrade) + 32; note: 1.8 = 9/5

Centigrade To Kelvin:Centigrade + 273;

Kelvin To Centigrade:Kelvin - 273;

Fahrenheit To Kelvin:(5/9 * (Fahrenheit - 32) + 273 ); note: .55555 = 5/9

Kelvin To Fahrenheit:((Kelvin - 273) * 1.8 ) + 32; note: 1.8 = 9/5

Carpenter's Arches

Least Square Method

Given a set of data points, we need to fit a straight line y = a + bx to these set of pointssuch that the sum of squares of the distances to this straight line, y = a + bx, from thegiven set of point is a minimum.

The normal equation is given by

an + b*(Summation)x = (Summation)ya*(Summation)x + b*(Summation)x = (Summation)xy

From these normal equations, the specific values of a and b can be determined.

To figure x^2 without multiplying (x*x)

Intended for use when base is a large number

x^2 = (x/[exponent])*(x+x), when x is even

x^2 = x * (x+1) - x, when x is odd

576^2= (576/2) *(576+576) = 288 * (1152) = 331776576^2 = 331776

573^2 = 573 * (573+1) - 573 = 573 * 574 - 5733238902 - 573


14/25

= 328329573^2 = 328329

Now this may seem like more work but it will help to explain the mechanical action behindpowers and roots.

Please if you have any inpute on how to calulatex^3 for both even and odd numbers email me.

0 power

A^0=1

Trig Identity

sin(A + B)= sinA cosB + cosA sinB

Log Rules

log(A*B) = logA + logB

log(A/B) = logA - logBlog A^n = n*logA

Quadratic Formula

ax+bx+c=d

Triangle Ratios

Triangles and their ratios:SQRT means the Square root sign30-60-90 right triangle| sides are in a 1:SQRT3:2 side ratio

Isoceles right triangle| sides are in a 1:1:SQRT2 side ratio

3-4-5 Triangle and other multiples | 3:4:5 ratio

The six Trigonometric Ratios

The 6 Trig ratios and their abbreviations:

1. sine \ sin2. tangent \ tan3. secant \ secThe opposites (in corresponding order):1. cosine \ cos


15/25

2. cotangent \ cot3. cosecant \ csc

Area of a Rhombus

The area of a Rhombus:1\2 times the product of its diagonalsor in other words1\2(d1+d2)=area of a rhombus

Division by zero

Division by zero is not allowed, giving a non-number.

Boolean Addition

Used with binary numbers 0's &1's, the language of computers.

1+1=11+0=10+1=10+0=0

Rational Numbers

rational number is one that can be written as the ratio of two integers. For example 3=3/1,-17, and 2/3 are rational numbers.Most real numbers (points on the number-line) areirrational (not rational). The rational numbers are those which have repeating decimalexpansions (for example 1/11=0.09090909..., and 1=1.000000...=0.999999...). They arealso those which have terminating continued fraction expansions. Finally, the real number xis rational if and only if there are finitely many solutions to |x - a/b| < 1/b2.

Angular Momentum

To find the angular momentum of a spinning body, the formula is simple:

M=Mass of the objectV=Speed around the orbitR=Radius of the OrbitL=Momentum

L=MVR


16/25

Experimental Probability

Experimental Probability equals:

Number of times a succesful event has occured---------------DIVIDED BY------------------Number of trials

Inradius of a Right Triangle

inradius = (ab) \ (a+b+c)where a,b,and c are sides of the right triangle,c being the hypotenuse

Herrons Formula

A=(s(s-a)(s-b)(s-c))^(1/2) where s= (a+b+c)/2

Division by zero revision

When dividing by zero with a non zero or infinite numerator. the result is infinity

Electro-magnetic Flux

q = (Permitivity)*Flux = (E dA) * (permitivity)

Flux is equal to the integration of the dot product of the Electromagnetic field vector by theArea vector ( normal to the surface)

Kirchhoff's rule's

The total voltage drop around a cloded circuit is zero. (conservation of energy,Potential isindependent of the path)the sum of the current entering a junction is equal to the sum of the current leaving thejunction (conservation of charge)

Boolean Addition

Slope of a Line

the slope of any line at a point is given by the first derivative of the function evaluated atthat point.

f(x,y) = x^2 +y^3 +xy + 6x + 10y + 5partial x derivative:


17/25

d(f(x,y,))/dx = 2x +y + 6

partial y derivativesd(f(x,y))/dx = 3y^2 + x + 10

a unique solution exists iff

d^2f(x,y)/dxdy = d^2f(x,y)/dydx

Potentials

the potential of an electricfield is difinded by

U = - Edr

defined by intergration from initial to final point of the dot product of the electricfield by thedisplacement

remember the potential is independent of the path travelled

Dot Product

the dot product of two vectors is defined as a scalar.... it is the magnitude of vector 1 timemagitude of vector 2 times the cosine of the angle between them...

or if the vectors are given in cartiesian corridates....

AB = AxBx+AyBy+AzBz

Cross Product

there are two ways to find the cross product of 2 vectors.. if you know the magnitude of thevectors the A x B = A*B*sin(x) U

were u is a unit vector in the direction normal to the plane defined by A and B

or if you have the cartesian coordiates of the vector...

AxB =(AyBz-AzBy)i -(AxBz-AzBx)j +(AxBy-AyBx)k

where i,j,k are unit vectors in the direction of the x,y,z axis respectfully


18/25

hexidecimal

Hex number

112 2A 10F 1510 161A 26

y=mx+b

Slope formula

Lever Systems

F1X+F2(d-x)

Kinetic Energy

Kinetic energy is equal to (m*v^2)/2, in which m is mass and v is velocity. To find kineticenergy in Joules, enter mass as kilograms (kg) and velocity as meters per second (m/s).

Volume of a Pyramid

V = LWH/3

whereL = Length of BaseW = Width of BaseH = Height of Pyramid

Infinite Nested Square Roots

2 = sqrt( 2 + sqrt( 2 + sqrt( 2 + sqrt ( .... ) ) ) )3 = sqrt( 6 + sqrt( 6 + sqrt( 6 + sqrt ( .... ) ) ) )5 = sqrt( 20 + sqrt( 20 + sqrt( 20 + sqrt ( .... ) ) ) )

and, in general,

n = sqrt( m + sqrt( m + sqrt( m + sqrt ( .... ) ) ) )

where n is an integer > 1, and m=n(n-1)


19/25

Relative Centrifugal Force

RCF (in g forces) = 1.119x10^-5 x rpm^2 x radius (in cm)

Linear speed on rotating disk

LinearSpeed = Pi(Radius*2)*RevolutionsPerMinute

Radius is distance of interested point from center of disk at whatever unit used.

Number of Diagonals in a n-gon or polygon

[n(n-3)]/2

or times then divided by .

for the number of diagonals from1

vertex, it's:

n-3

Euler Descartes Formula

number of vertices - number of edges + number of faces = 2

summed up: v-e+f=2

E=Mc^2

E = Mc^2E = EnergyM = Massc = the speed of light (2.99792458e8 meter/sec)

In other words:Energy = Mass * 89875517873681764

This implies that if mass could be converted to energy in a controlled environment, 1 gramof a mass would produce 89,875,517,873,681,764 units of energy.


20/25

To find the total number of degrees inside any polygon

180 x (n-2)= Total degreesn = the number of sides that the polygon has.

ie Any pentagon has 5 sides.Thus, there are180 x (n-2)= 180 x (5-2)= 180 x 3= 540 degrees in total.

Fractions with Zero as numerator / denominator / both

0/5 = 05/0 = undefined0/0 = indeterminate

Volume of a hexagonal/pentagonal prism

V = Area of base x height

Surface Area of a Cylinder

s= 2(pi)(r2) + (2)(pi)(r)(h)The surface area of a cylinder is a function of the radius r and height h.

rectangular prism

Slope Formula

Stewart's Theorem

Brahmagupta's Formula

surface area of a sphere

sum of squares and cubes of first 'N' natural no`s.

Sum of the Sides of the exterior angles in a polygon

Reliability

sum of arithmetic progression


21/25

Trig identity for tan(x/2)

The Nand and Nor formulas

Double-Angle Formulas

Union

Theory of relativity

Gradient Of A Curve

Ohm's Law

Master Product Method for solving Quadratic Equations

Ideal Gas Law Equation

Friction

y-y1=m(x-x1)

subtraction through binary

Each angle of a regular polygon

ARITHMETIC PROGRESSION FORMULAE

Future Value

AREA OF A REGULAR POLYGON

POWER FORMULA I

POWER FORMULA II

RESISTANCE FORMULA

volume of a prisim

Motion d=rt


22/25

Capacitors in series

Capacitors in parallel

Remainder Theorem

CALCULUS - General Derivatives

Euclidean Distance

Circumference of an Ellipse

Reciprocal of a Fraction Property

Area of a Triangle (3 ways)

Surface Area of a Pyramid

Distance Formula (3 dimensional)

Permutations, Combinations, Factorials

Perimeter of an Ellipse

Series Expansions

Sum of Arithmetic Progression

quadratic

Potential energy

Equation Of A Linear Line

Mulplicative Inverse of 1 and 0

Area of a Pyramid

The Manish Loop

Area of Triangle (when length of sides is known)


23/25

Definition of Factorial

SOHCAHTOA

Formula for work

Relationship between Doolittle and Crout's Method

Sum of Geometric Progression

Slope

Area of a Trapezoid

Pemdas

Eccentricity of an Ellipse

Eccentricity of an Ellipse

Right Triangle Relations

Factorial, Permutations, and Combonations

Googolplex

Discriminant

Midpoint of a Line Segment

Final Velocity

Acceleration

diameter of circle

work

Distance of a Line segment

Cube of a binomial


24/25

Roots of 2nd order linear homogeneous differentialequations

The Trapezoid rule and Simpson's rule: numerical integration

tan-1/1-tan2@+sin@

tan-1/1-tan2@+sin@

Circumference of a circle

Compound Interest

Simple Interest

Area of a Sector of a Circle

right-angled triangle

Holy Grail Formula

Sum of Roots

some trig identity

CORRECTION OF FORMULA FOR WORK

Scientific Method

SUM OF PROGRESSION.( AP & GP)

SUM OF PROGRESSION(AP &GP)

A 100- ray - angle

surface area of cylinder (factored expression)

surface area of cone (factored expression)

Area of a Regular Polygon

Is point in shadow formula (sin)


25/25

Gravitational Potential Energy (Newton)

Cenrtifugal Force (Newtons - Uses SI units)

Height by a stone throw

Transistor Parameters - common emitter mode.

OPAMP - inverting and non-inverting voltage gains

Relation between edges , vertices and faces in a regular solid

oPeRaTiOnS nUmBeRs

MiDpOiNt

Order of Operations

THE Quadratic Formula

Eccentricity of an Elliptical Orbit

Correct Estimate for Perimeter for Ellipse

The speed of light equls time

The speed of light equals time

distributive property

23434%*846585-5726tt-333yyx+66879(6546954-654654)=There is no God!

The area of a ploydodecafolastomageton

Volume of sphere

some basics of ma thematic calculations

Documents