AP Precalculus Unit 3 Notes: Polar Coordinates, Polar Graphs, and Change

Polar coordinates

A coordinate system that represents a point by its directed distance from the origin and an angle: (r, θ).

Pole

The origin in the polar coordinate system; the point from which the radius r is measured.

Polar axis

The reference ray for measuring θ; typically the positive x-axis.

r (radius) in polar coordinates

The directed distance from the pole to the point; can be positive or negative.

θ (theta) in polar coordinates

The direction angle measured from the polar axis, usually counterclockwise.

Negative radius (negative r) interpretation

If $r$ is negative, the point is $|r|$ units from the origin in the direction opposite $\theta$ (equivalently, add $\pi$ to the angle).

Angle periodicity in polar coordinates

Angles that differ by 2πk (k an integer) represent the same direction, so (r, θ) and (r, θ + 2πk) are the same point.

Equivalent polar representations

Different (r, θ) pairs that locate the same point, e.g., (r, θ), (r, θ+2πk), (−r, θ+π), and (−r, θ+π+2πk).

Polar-to-Cartesian conversion

Formulas to convert (r, θ) to (x, y): x = r cos(θ) and y = r sin(θ).

Cartesian-to-polar radius formula

The relationship r² = x² + y² (so r = $\sqrt{x^2 + y^2}$, often taken as nonnegative for a standard polar form).

Cartesian-to-polar angle relationship

tan(θ) = $\frac{y}{x}$ (when x ≠ 0), used with quadrant reasoning to choose the correct θ.

Quadrant issue with arctan

Inverse tangent alone may give the wrong θ because tan repeats; you must use signs of x and y (or a quadrant-aware function) to pick the correct quadrant.

Converting a polar equation to Cartesian

A method using substitutions r² = x² + y², r cos(θ) = x, and r sin(θ) = y to rewrite a polar equation in x and y.

Completing the square (circle recognition)

An algebra technique often used after converting to Cartesian form to identify circle center and radius, e.g., x² − 4x + y² = 0 → (x−2)² + y² = 4.

Polar function

A function of the form r = f(θ), where each angle θ determines a radius r for plotting points.

Graphing with negative r values

When $r < 0$ for some $\theta$, the plotted point is reflected through the origin (equivalently plotted at angle $\theta + \pi$ with positive radius).

Key angles (in polar graphing)

Angles with well-known trig values (e.g., 0, π/6, π/4, π/3, π/2) used to quickly compute r and sketch polar curves.

Symmetry about the polar axis (x-axis) test

Replace θ with −θ; if the equation is unchanged, the graph is symmetric about the polar axis.

Symmetry about the line θ = $\frac{\pi}{2}$ (y-axis) test

Replace $\theta$ with $\pi - \theta$; if the equation is unchanged, the graph is symmetric about the y-axis.

Symmetry about the pole (origin) test

Replace θ with θ + $\pi$; if the equation is unchanged, the graph has origin (pole) symmetry.

Rose curve

A petal-shaped polar graph given by r = a cos(nθ) or r = a sin(nθ), where a controls petal length and n controls petal count.

Rose curve petal count rule

For integer n: if n is odd, the rose has n petals; if n is even, it has 2n petals.

Limaçon

A snail-shaped polar curve of the form r = a ± b cos θ or r = a ± b sin θ; its shape depends on the relationship between |a| and |b|.

Cardioid

A special limaçon with a cusp that occurs when |a| = |b| (e.g., r = 1 + cos(θ)).

Average rate of change of r with respect to θ

For r = f(θ), the average change in radius per change in angle: $\frac{f(θ_2) - f(θ_1)}{θ_2 - θ_1}$; interpreted as “in/out” change per radian (or per degree), not slope in the xy-plane.