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Level 535

Polar Coordinates & Parametric Equations


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(x, y)
Rectangular Form (Coordinates)
r is...
the distance from the origin to a point
θ is...
the angle between the polar axis
x = r cos θ
To change from POLAR to RECTANGULAR, you use...
r² = x² + y²
To change from RECTANGULAR to POLAR, you use...
x = ? is a vertical line
If only ONE variable in rectangular is found x or y...
If no angle of θ is determined...
the graph is a circle. The "r" value found is the radius of the circle.
Polar Curves: Circles
r = a
Polar Curves: Circles
r = a sin θ
Polar Curves: Circles
r = a cos θ
Polar Curves: Spiral
r = aθ
a < b (Limaçon with inner loop)
*Orientation depends on the trigonometric function (sine or cosine) and the sign of "b."
Limaçons
a = b (cardioid)
Limaçons
a > b (dimpled limaçon)
Limaçons
a ≥ 2b (convex limaçon)
2n-leaved if n is even
r = a cos 2θ (4-leaved rose)
roses
r = a cos 3θ (3-leaved rose)
roses
r = a cos 4θ (8-leaved rose)
roses
r = a cos 5θ (5-leaved rose)
Figure-eight-shaped curves
r² = a² sin 2θ
Lemniscates
r² = a² cos 2θ
r = a
Polar Curves: Circle - centered at pole
r = a sin Θ
Polar Curves: Circle - shifted along the y-axis
r = a cos Θ
Polar Curves: Circle - shifted along the x-axis
Limaçons
r = a ± b sin θ
Limaçons with inner loop
a < b in
Limaçons Cardioid (heart-shaped)
a = b in
Dimpled Limaçon
a > b in
Convex Limaçons
a ≥ 2b in
roses
r = a sin nθ
4-Leaved Roses
r = a cos 2θ
8-Leaved Roses
r = a cos 4θ
5-Leaved Roses
r = a cos 5θ
Figure-eight-shaped curves
r² = a² cos 2θ
Pole
The point in a polar coordinate system used as a reference. Analogous to the origin in a Cartesian coordinate system.
parameter
Number that describes a population