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February 3, 2017

Math Formulas

Filed under: Tips and Tricks — Tyler @ 10:43 pm
Math Formulas

Math Formulas

Pre-Calculus Formulas:

Formula for an ellipse:

[(x2) / (a2)] + [(y2) / (b2)] = 1

NOTE: x is the center x coordinate for the center of the ellipse. y is the center y coordinate of the ellipse.
If there are modifiers such as (x-1)2 that would mean to move the center point accordingly which would bring the “x” value to zero.

As for the “a” and “b” aspects of the formula:
The “a” is the x consideration for the ellipse. To determine the size of the x “semi-axis” one must take the square-root of the “a” value etc.

Determine the Foci length of an Ellipse

If a > b:

f = sqrt( a2 – b2 )

 

ALGEBRA FORMULAS:

Factoring Polynomials: 

 

(x + a) (x + b) == x^2 + (ab)*x + ab;

 

 


TRIG QUADRANT DATA: 
QUADRANT I (ONE):

(+) sine
(+) cosine
(+) tangent
(+) cosecant
(+) secant
(+) cotangent

QUADRANT II (TWO):

(+) sine
(-) cosine
(-) tangent
(+) cosecant
(-) secant
(-) cotangent

QUADRANT III (THREE):

(-) sine
(-) cosine
(+) tangent
(-) cosecant
(-) secant
(+) cotangent

QUADRANT IV (FOUR):

(-) sine
(+) cosine
(-) tangent
(-) cosecant
(+) secant
(-) cotangent


 

BASIC TRIGONOMETRY FORMULAS:

Law of Cosines

c2 = a2 + b2 – 2*a*b*cos(theta);

In this scenario “c^2” is referring to the length of a triangle “side” which is opposite to the angle “theta”;

Law of Sines

[(sin(A)) / a] == [(sin(B)) / b] == [(sin(C)) / c]


 

Trig Identities

Pi Subtraction Identities (Supplements):

sin(Θ) == sin(π-Θ);

-cos(Θ) == cos(π-Θ);

-tan(Θ) == tan(π-Θ);

Pi Addition Identities (Working Periods):

sin(Θ) = sin(Θ + 2π)

cos(Θ) = cos(Θ + 2π)

tan(Θ) = sin(Θ + π)

PYTHAGOREAN IDENTITY:

(sin Θ)2 + (cos Θ)2 = 1

Other Sum and Difference Identities (Ptolemy):

sin(x + y) == sin(x) * cos(y)+cos(x) * sin(y)

sin(x – y) == sin(x) * cos(y) – cos(x) * sin(y)

cos(x + y) == cos(x) * cos(y) – sin(x) * sin(y)

cos(x – y) == cos(x) * cos(y) + sin(x) * sin(y)

Other Identities Relating to tan(Θ) :

tan(α – β) = ( tan(α) – tan(β) ) / (1 +  tan(α) * tan(β) )

tan(α + β) = ( tan(α) + tan(β) ) / (1 –  tan(α) * tan(β) )

tan(2 * Θ) = ( 2 * tan(Θ) ) / ( 1 – tan(Θ))

Double Up Combo Identity:

sin(2*Θ) == 2*sin(Θ)*cos(Θ)

cos(2*Θ) == 2*cos(Θ)2-1


 

Special Angles and their Unit Circle Data

note = For the degree measurements, your graphing calculator or cal needs to be set to degree or regular depending on the calculator.

For the radians or (pi) based measurements, you need to have your graphing calculator set to radians.

If you use radians when in degree mode, you can get different results after using cosine etc. on angles.


Zeroth Quadrant – QUADRANT 0 (ZERO) – [ALONG x – AXIS]

0° or 0π —> cos(0° or 0π) == 0 , sin(0° or 0π) == 1;


QUADRANT I (ONE) – [TOP RIGHT]

30° or (π/6) —> cos(30° or (π/6)) == (sqrt(3)/2) , sin(30° or (π/6)) == 1/2 , tan(30° or (π/6)) == (sqrt(3) / 3 );

45° or (π/4) —> cos(45° or (π/4)) == (sqrt(2)/2) , sin(45° or (π/4)) == (sqrt(2)/2) , tan(30° or (π/4)) == (1);

60° or (π/3) —> cos(60° or (π/3)) == (1/2) , sin(60° or (π/3)) == (sqrt(3)/2), tan(60° or (π/3)) == (sqrt(3)) ;


Zeroth Quadrant – QUADRANT 0 (ZERO) – [ALONG y – AXIS]

90° or (π/2) —> cos(90° or (π/2)) == 0 , sin(90° or (π/2)) == 1;


QUADRANT II (TWO) – [TOP LEFT]

120° or (2π/3) —> cos(120° or (2π/3)) == (-1/2) , sin(120° or (2π/3)) == (sqrt(3)/2);

135° or (3π/4) —> cos(135° or (3π/4)) == (-sqrt(2)/2) , sin(135° or (3π/4)) == (sqrt(2)/2);

150° or (5π/6) —> cos(150° or (5π/6)) == (-sqrt(3)/2) , sin(150° or (5π/6)) == 1/2;


Zeroth Quadrant – QUADRANT 0 (ZERO) – [ALONG x – AXIS]

180° or (π) —> cos(180° or (π)) == -1 , sin(180° or (π)) == 0;


QUADRANT III (THREE) – [BOTTOM LEFT]

210° or (7π/6) —> cos(210° or (7π/6)) == (-sqrt(3)/2) , sin(210° or (7π/6)) == -1/2;

225° or (5π/4) —> cos(225° or (5π/4)) == (-sqrt(2)/2) , sin(225° or (5π/4)) == (-sqrt(2)/2);

240° or (4π/3) —> cos(240° or (4π/3)) ==  (-1/2), sin(240° or (4π/3)) == (-sqrt(3)/2);


Zeroth Quadrant – QUADRANT 0 (ZERO) – [ALONG y – AXIS]

270° or (3π/2) —> cos(270° or (3π/2)) == 0 , sin(270° or (3π/2)) == -1;


QUADRANT IV (FOUR) – [BOTTOM RIGHT]

300° or (5π/3) —> cos(300° or (5π/3)) == 1/2 , sin(300° or (5π/3)) == (-sqrt(3)/2);

315° or (7π/4) —> cos(315° or (7π/4)) == (sqrt(2)/2) , sin(315° or (7π/4)) == (-sqrt(2)/2);

330° or (11π/6) —> cos(330° or (11π/6)) ==  (sqrt(3)/2), sin(330° or (11π/6)) == -1/2;


 

Parabolic Formulas:

FOCUS and Directorix

The Focus coordinates are (a,b) representing the point that is inside the parabola which is equidistant to all the other points on the parabola in the same way that the directorix line is equidistant to the points along the parabola.

The Directorix is the constant – “k” which is a line.

 

y = (1/2*(b-k)) * (x-a)2 + (1/2) * (b+k)

Base Vertex Form:

y = a(x-h)2 + k

vs: Standard-

y = a*x2 + bx + c

vertex h=> standard (-b / 2a) => Reinput into vertex formula as necessary

 

Standard Parabolic Form

This represents the most primative parabola possible. A simple squaring of the x coordinates.

y = x2

Detailed Standard Parabolic (Quadratic) Form

This represents that the details of the parabola can be affected by secondary terms and coefficients.
For example, a negative “a” coefficient can make the parabola “point” in a different direction.

y = a*x2 + bx + c

 


Finite Series ( Arithmetic):

Sn = [(S0 + Sn) / 2] * n

 

Finite Series (Geometric and Exponential Patterns)

 

Finite Geometric Series – Sn

Sn = (a(1-r^n)) / 1 – r

Sn == Evaluated Finite Geometric Series to the “nth” number of  terms or items

a == The first term in the Series

r == the common ratio which everything is acted on by

n == This is the total number of terms – NOTE: This is an exponent in the Geometric Series Formula

 

SIGMA NOTATION
(nfinal)
∑ a*rk
(k=z)

In sigma notation the top number is the final term.

The bottom number k is assigned to  the first term to be used.

NOTE: If the first term is a negative or a zero, take this into account when plugging “n” into the formula

Honestly, sigma notation can be confusing because “k” and “n” are directly related to each other but, they aren’t the same.

What we really want to do is plug “n” into the first Series Formula. But a lot of times, we have “k” instead. So when the bottom of the sigma says, “k = 0” it is really saying, “if k = 0, then the first term is multiplied by 1 in the iteration. This makes sense because anything to the exponent of 0 is one.

Sometimes there are algebraic instructions in the top and/or bottom. These are to help describe the iteration process.

The information to the right of the sigma is important.

Everything except for the exponent is part of the common ratio or “r”.

The exponent is the true “n” from the previous formula.

 

 


 

 

Quadratic Equations:

 

Quadratic Formula:

 

x =   ( -b ±  [#sqrt] (b^2 – 4ac) ) / 2a

 

Standard Form for Quadratic Equations:

 

ax^2 + bx + c = 0

 

PROPERTIES OF LOG (logarithm properties):

Basic property -> converting log to exponent

logab = x

is equivalent to:

a^x == b;

 

Evaluate Properties –> adding log terms to each other

( logca ) + ( logcb )  == ( logca * b )

 

Evaluate Properties –> subtracting log terms to each other

( logca ) – ( logcb )  == ( logca / b )

 

Multiplying Log -> adding additional exponent to log expression

A * logbc == logb(c^A)

Special Property – Evaluating non-base 10 Log using division

logba = ( logca ) / ( logcb ) ;

 

Factors of (i) – complex number theory:

 

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

i ^5 = i

…continue rotation…

 

 

Remainder Formula for Polynomials

Let’s say there is a polynomial x^3 + 2x^2 + 17x + 8

and we need to divide that polynomial by x-9

There is a shortcut where the remainder can be found by substituting (x) for the opposite of -9.

So we would input +9 or 9 into the polynomial for x and evaluate.

If the evaluation ends up as 0. Then we know that x-9 is actually a factor for that polynomial. And that there is no remainder.

If the evaluation ends up with a number or another binomial or polynomial etc, then we have that as the remainder.

 

Standard Form:

Ax + By = C

 

Point Slope Form:

HERE is an Evaluator program for point Slope Form

(y – y{1}) = m(x – x{1})

KEY: y{1} means the y location of a specific point. (I’m using {} in this instance not to show a set but to denote a base.)

x{1} is talking about the same point of (x{1}, y{1})

y refers to another point which has the x location of x.

To think about this one, we are simply making the rise/run of the line very clear. Point Slope Form makes it really obvious that an average of two points is happening.

 

 

 

 

 

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