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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