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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 =