Cheatography
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Fraction RulesCommon Denominators
a/b + c/d = ad + bc/bd
Multiplication
a/b * c/d = ac/bd
Reciprocal
1/(a/b) = b/a 
Theory of Geometric SeriesS = p + pr + pr^{2} + pr^{3} + ... + pr^{n}
Sr = p + pr + pr^{2} + pr^{3} + ... + pr^{n} + pr^{n+1}
S  Sr = p  pr^n+1
S = ppr^{n+1}/1r 
LinesGiven point (c,d) and slope m, the unique line that satisfies this is the set of all points (x,y) such that
m = yd/xc
slope is change in y/change in x
Parallel, Perpendicular
y=2x+4 and y=2x+3 are parallel
y=2x+4 and 2y=4x+8 are same line
y=2x+4 and y = 1/2x+6 meet perpendicularly
Parallel lines never meet, same slope
Perpendicular lines meet once at right angles and slopes are negative reciprocals 
Geometric FormulasSOHCAHTOA
SOH = sin(x) = a/c
CAH = cos(x) = b/c
TOA = tan(x) = a/b
180 degrees = (pi)rad
Divide arclength by radius to get radian measure
Special Angles
Degree Radian Cosine Sine Tangent
0 0 1 0 0
30 (pi)/6 sqrt3/2 1/2 1/sqrt3
45 (pi)/4 1/sqrt(2) 1/sqrt(2) 1
60 (pi)/3 1/2 sqrt(3)/2 sqrt(3)
90 (pi) 0 1 undefined 
Quadratic Functions: Parabolasf(x) = ax^{2} + bx + c
Zeros given by the quadratic formula:
b +/ sqrt(b^{2}  4ac)/2a
Coordinate point (b/2a,( b^{2} +4ac)/4a) 
Geometric EquationsCircles: Circumference 2(pi)r Area (pi)r^{2}
Cylinders: Surface area 2(pi)r x h + 2(pi)r^{2} Volume: (pi)r^{2}h
Sphere: Surface area 4(pi)r^{2} Volume 4/3(pi)r^{3} 
  Negative Exponentsa^{m} = 1/a^{m}
1/a^{m} = a^{m} 
Trigonometric Identitiescos^{2}x + sin^{2}x = 1
tan^{1}x = arctan x
cot(x) = 1/tan x
e^{10} = cos x + isin x 
Secant, Cosecant, CotangentSecant: 1/cos(x)
Cosecant = 1/sin(x)
Cotangent = 1/tan(x) = cos(x)/sin(x) 
Exponential Rulesexp(a)exp(b) = exp(a+b)
[exp(a)]^b = exp(ab)
exp(a) = 1/exp(a)
Domain: all real numbers
Range: all positive numbers 
Absolute ValuesAbsolute Values often generate "and" and "or" situations.
Examples:
x<1: 1<x and x<1 1<x<1
x>1: x>1 or x<1
2x+3> 1: 2x+3>1 or (2x+3)>1
x>1 or x<2 
Multiplication and Scientific NotationMultiplication and Division:
 Convert into scientific notation
 Add/subtract exponents
 Multiply/divide coefficients
 Convert to scientific notation
 2.3E4 x 9.5E7/1.6E10 = 2.3x9.5/1.6E(4+710) 
  LogarithmsLogarithms are the functional inverses of the exponential
y = b^x is equivalent to logb(y) = x
log(mn) = log(m) + log(n)
log(m^n) = nlog(m)
log(1/m) = log(m)
loga(m) = logb(m)/logb(a)
exp(a)exp(b) = exp(ab)
exp(a) = 1/exp(a)
e^{2.3 = 10, e}12 = e^{2.3*5.2 = (e}2.3)^{5.2 = 10}5.2 = 2E5 
Function VariablesDomain = valid inputs to function
Range = what can the function produce
Zeros or Roots = where is f(x)=0
Intersections = Where is f(x) = g(x)
Local maximum is largest value around itself
Local minimum is smallest value around itself
Global is largest overall 
SineSine is the y component as theta spins
Domain is all real numbers
Range 1</ y </ 1
Maxima at pi/2 + 2kpi, Minima at 3pi/2 + 2kpi
Zeros at kpi
Period 2pi 
CosineCosine is the x component as theta spins
Domain is all real numbers
Range is 1</ y </ 1
Max is 2kpi, Minima (2k + 1)pi
Zeros at pi/2 + kpi
Period is 2pi 
TangentTangent is the slope of the line with angle theta
Domain is all real numbers except pi/2 + kpi
Range is all real numbers
No max or min. asymptotes at undefined points
Zeros at kpi
Period is pi 
Distance Between PointsThe distance between two points on the plane is based on the Pythagorean Theorem
AB = sqrt((XaXb)^{2} +(Ya  Yb)^{2})
A=(Xa,Ya) B=(Xb,Yb) 
Basic FactsTotal human population: 7 billion
USA population: 300 million
Distance from NY to LA: 2500 miles
Distance to the moon: 2.4E5 miles
Distance to the Sun: 1E8 miles
Distance around the equator: 2.5E4 miles
Area of the US: 4E6 square miles
Surface area of the Earth: 2E8 square miles 

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