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Precalculus Unit 10: Polar coordinates & graphs

Polar coordinates locate a point by distance r and angle θ instead of x and y. Some curves are much simpler in polar form.

Covers: Converting coordinates, polar graphs, rates of change in polar

Study guide

Polar coordinates locate a point by its distance r from the origin and its angle θ from the positive x-axis. Conversion formulas link the two systems: x = r cos θ and y = r sin θ go from polar to rectangular, and r² = x² + y² with tan θ = y/x go back (check the quadrant to choose θ). A point has many polar names, since adding 2π to θ or replacing (r, θ) with (−r, θ + π) gives the same point.

A polar equation describes a curve by giving r as a function of θ. Simple ones: r = a is a circle of radius a centered at the origin, θ = constant is a line through the origin, and r = 2a cos θ is a circle through the origin. Rose curves r = a cos(nθ) or a sin(nθ) have n petals when n is odd and 2n when n is even. Limaçons r = a + b cos θ include cardioids when a = b.

To graph a polar function, make a table of θ and r values or think of it as a function of θ on a rectangular graph, then wrap it around the origin.

Rates of change in polar describe how the distance from the origin changes as the angle increases. When r is positive and increasing, points move away from the origin; when r is positive and decreasing, they move toward it. The average rate of change of r over an interval of θ is [r(θ₂) − r(θ₁)]/(θ₂ − θ₁).

To convert a polar equation, multiply by r if needed and substitute r² = x² + y², r cos θ = x and r sin θ = y. For example, r = 4 sin θ becomes x² + y² = 4y, a circle centered at (0, 2) with radius 2. The greatest distance of a polar curve from the origin is the largest value of |r|. Where r is negative, the point is plotted in the opposite direction, on the ray at θ + π.

Key ideas

• Convert with x = r cos θ, y = r sin θ, r² = x² + y².
• A point has many polar names: (r, θ) = (r, θ + 2π) = (−r, θ + π).
• r = a is a circle; θ = c is a line through the origin.
• r = a cos(nθ) or a sin(nθ) are roses; r = a ± b sin θ are limaçons.

Formulas and rules

Common mistakes

• Using tan θ = y/x without checking the quadrant.
• Treating negative r as impossible.

Words to know

Polar to rectangular

x = r cos θ, y = r sin θ

r² in rectangular terms

x² + y²

Graph of r = 3

A circle of radius 3 centered at the origin.

Rose curve

r = a cos(nθ): n petals if n is odd, 2n if n is even.

Polar coordinates (other names)

(r, θ) = (r, θ + 2πk) = (−r, θ + π)

Polar coordinates

(r, θ): distance from the origin and angle from the positive x-axis.

Rectangular to polar

r² = x² + y², tan θ = y/x (check the quadrant).

Limaçon

r = a + b cos θ; a cardioid when a = b.

Circle (polar)

r = a, or r = 2a cos θ through the origin.

Rate of change in polar

How r changes as θ increases; r increasing means moving away from the origin.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.