maths ~ vulpine.zone

Maths

This is a document containing genuinely useful math formulas and information (in my opinion) for when internet is not available!

note: this page can use mathjax to render equations in svg format! unfortunately mathjax does require javascript, so if you'd like this functionality you will have to enable it.

Solving
Economics
1. Compound Interest
2. Continuous Growth
Trigonometry
1. General Relationships
2. Law of (Co)Sines
Geometry
Physics
1. Motion
2. Energy
3. Forces
Other Misc. Formulas
1. Sequences

Solving [top]

Quadratic Formula [top]

Finds points where x = 0 for a quadratic equation in standard form ( $y = a x^{2} + b x + c$ ).

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Exponent Properties [top]

b^{m} b^{n} = b^{m + n}

(b^{m})^{n} = b^{m n}

b^{- n} = \frac{1}{b^{n}}

Logarithm Properties [top]

b^{p} = a \equiv {log}_{b} a = p

{log}_{b} x y = l o g_{b} x + l o g_{b} y

{log}_{b} \frac{x}{y} = {log}_{b} x - l o g_{b} y

{log}_{b} x^{k} = k {log}_{b} x

{log}_{b} x = \frac{{log}_{a} x}{{log}_{a} b}

Economics [top]

$P$ = initial amount
$r$ = annual rate (decimal)
$n$ = compounding periods per year
$t$ = # years

Compound interest [top]

A = P (1 + \frac{r}{n})^{n t}

Continuous growth [top]

A = P e^{r t}

Trigonometry [top]

General Relationships [top]

On a right triangle with a given angle...

$A$ = side adjacent to angle
$O$ = side opposite to angle
$H$ = side opposite to right angle

$x, y$ = coords on a grid
$r$ = radius of an origin-centered circle on a grid

sin θ = \frac{O}{H} = \frac{y}{r}

cos θ = \frac{A}{H} = \frac{x}{r} = sin (θ + \frac{π}{2})

tan θ = \frac{O}{A} = \frac{y}{x} = \frac{sin θ}{cos θ}

csc θ = \frac{1}{sin θ}

sec θ = \frac{1}{cos θ}

cot θ = \frac{1}{tan θ}

Law of (Co)Sines [top]

$A, B, C$ = angles
$a, b, c$ = side lengths

\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}

a^{2} = b^{2} + c^{2} + [- 2 b c cos A]

A = c o s^{- 1} (\frac{a^{2} - c^{2} - b^{2}}{- 2 b c})

Geometry [top]

Perimeter / Area (2D) [top]

$A$ = area
$b$ = base length
$h$ = height
$s$ = side length

Triangle

A = \frac{1}{2} b h

Parallelogram

A = b h

p = 2 b + 2 h

(duh)

Regular Polygon

$r$ = apothem
$n$ = # sides

s = 2 r tan \frac{π}{n}

p = n s

A = n r^{2} tan \frac{π}{n} = \frac{1}{4} n s^{2} cot π n

Trapezoid

A = \frac{1}{2} h (b_{1} + b_{2})

Surface Area / Volume (3D) [top]

For composite figures, subtract the area of the surface where objects connect.

$L A$ = lateral area (area of non-bases)

Rectangular Prism

S A = 2 l w + 2 l h + 2 w h

V = l w h

Generic Prism

$h$ = height between bases
$p$ = perimeter of base
$B$ = area of base

L A = h p

S A = h p + 2 B

V = B h

Cylinder

$r$ = radius of cylinder

L A = 2 π r h

S A = 2 π r^{2} + 2 π r h

V = π r^{2} h

Pyramid

$l$ = slant height
$p$ = base perimeter
$B$ = base area

L A = \frac{1}{2} l p

S A = \frac{1}{2} l p + B

V = \frac{1}{3} b h

Cone

L A = π r l

S A = π r^{2} + π r l

V = \frac{1}{3} π r^{2} h

Sphere

S A = 4 π r^{2}

V = \frac{4}{3} π r^{3}

Hemisphere

S A = 3 π r^{2}

V = \frac{2}{3} π r^{3}

Circles [top]

$C$ = circumference
$A_{s}$ = area of sector
$a$ = arc length
$l$ = length of slice in radians

C = 2 π r

A = π r^{2}

A_{s} = π l r^{2}

s = 2 π l r

Physics [top]

$a$ = acceleration ( $m / s / s$ )
$F$ = force ( $N$ ( $k g * m / s^{2}$ ))
$m$ = mass ( $k g$ )
$p$ = momentum (typically $k g * m / s$ )
$t$ = time ( $s$ )
$v$ = velocity ( $m / s$ )

Motion [top]

v_{x} = v_{x 0} + a_{x} t

x = x_{0} + v_{x 0} t + \frac{1}{2} a_{x} t^{2}

v_{x}^{2} = v_{x 0}^{2} + 2 a (x - x_{0})

{\vec{x}}_{c m} = \frac{\sum m_{i} {\vec{x}}_{i}}{\sum m_{i}}

{\vec{a}}_{s y s} = \frac{\sum \vec{F}}{m_{s y s}} = \frac{{\vec{F}}_{n e t}}{m_{s y s}}

{\vec{v}}_{c m} = \frac{\sum {\vec{p}}_{i}}{\sum m_{i}} = \frac{\sum m_{i} {\vec{v}}_{i}}{\sum m_{i}}

\vec{p} = m \vec{v}

Energy [top]

$d$ = distance ( $m$ )
$K$ = kinetic energy ( $J$ )
$W$ = work ( $J$ )

K = \frac{1}{2} m v^{2}

W = F_{∥} d = F d cos θ

Δ K = \sum W_{i} = \sum F_{∥, i} d_{i}

Forces [top]

$g$ = earth's gravity ( $\approx 9.8 m / s^{2}$ )

F_{g} = m g

F_{n e t} = \frac{Δ \vec{p}}{Δ t} = m \frac{Δ \vec{v}}{Δ t} = m \vec{a}

Other misc. formulas [top]

Sequences [top]

$a_{n}$ = value at index $n$ in arithmetic seq.
$g_{n}$ = value at index $n$ in geometric seq.
$k$ = index of given term
$r$ = rate of change

a_{n} = a_{k} + r (n - k) = a_{0} + r n

g_{n} = g_{k} r^{(n - k)} = {g_{0}}^{r^{n}}

vulpine.zone est. 2635