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Level 534

Parametric & Polar Equations & Graphs


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Y-axis Cardiod Equation
r = a +- a sinθ
X-axis Cardiod Equation
r = a +- a cosθ
|2a|
Lengthwise distance of cardiod
Y-axis Limacon Equation
r = a +- b sinθ
X-axis Limacon Equation
r = a +- b cosθ
|a| < |b|
There is an inner loop in a limacon if
|a| > |b|
There is NO inner loop in a limacon if
|a| - |b|
Limacons: inner loop or kidney bean shape of what expression
|a| + |b|
Lengthwise distance of Limacon
r = a sin (bθ)
Y-axis symmetric Rose (where b≠1)
r = a cos (bθ)
X-axis symmetric Rose (where b≠1)
Odd
An integer n is defined to be odd if n=2k+1 for some integer k.
Even
An integer n is defined to be even if n=2k for some integer k.
r = a
Polar Curves: Circle - centered at pole
r = a θ
Spirals (θ is in radians)
θ = constant
Line through pole equation
r = constant
Circle with center at pole
Y-axis shifted Circle
r = d sin θ
X-axis shifted Circle
r = d cos θ
|d|
diameter of circle
d>0
circle located on positive side of the axis
d<0
circle located on negative side of the axis
distance formula: between 2 points (r₁, θ₁) and (r₂, θ₂)
d² = r₁² + r₂² - 2 r₁ r₂ cos (θ₂ - θ₁)
circle (general formula)
a² = r₀² + r² - 2 r₀ r cos (θ - θ₀)
r = 2acos(θ-θ₀)
circle through pole
skew lines (not through pole)
d = r cos (θ - α) (or r = d sec (θ - α))
vertical line
r = h sec θ (or rcosθ = h or x = h or r = h/cosθ)
horizontal line
r = k csc θ (or rsinθ = k or y = k or r = k/sinθ)
y = y₁ + ∆yt
PARAMETRICS: in general: if (x, y) is pt. on line + if ∆y/∆x is slope then the x + y equations are:
PARAMETRICS: circle equations
y = k + r sin T
PARAMETRICS: ellipse equations
y = k + b sin T
PARAMETRICS: hyperbola equations
y = k + b tan T
r = √x² + y²
polar "toolbox" - r =
y = r sin θ
polar "toolbox" - y =
x = r cos θ
polar "toolbox" - x =
tan θ = y/x
polar "toolbox" - tan θ =