Polar Coordinates

Polar coordinates \((r, \theta)\) locate a point by its distance from the origin and its angle from the positive x-axis. Converting with \(x = r\cos\theta,\; y = r\sin\theta\) lets a single equation \(r = f(\theta)\) describe circles, roses, and spirals that would be a mess in x and y.

By the end you'll be able to convert points and equations between polar and Cartesian form, and recognize the curves — circles, roses, spirals — that simple polar equations \(r = f(\theta)\) trace out.

Predict: for \(r = 4\sin\theta\), sweep \(\theta\) from 0 to \(\pi\) — what shape closes up, and where is its center? Switch to Trace a curve and slide \(\theta\) to check.

In Place a point mode, the sliders set distance r (the solid segment) and angle θ (the dashed ray and arc). Try a negative r: the point jumps to the opposite side of the ray. Hover or focus the point for exact coordinates.

Mode

r = 2.00, θ = 0.79 rad → (x, y) = (1.41, 1.42)

Polar grid — rings of constant r, spokes of constant θ
distance r angle θ

Polar coordinates \((r, \theta)\) name a point by distance and angle from the origin; the conversion \(x = r\cos\theta,\; y = r\sin\theta\) links them to Cartesian coordinates and makes equations \(r = f(\theta)\) a compact language for circles, roses, and spirals.

Visual

Instead of "go right x, then up y," polar directions say "face angle θ (counterclockwise from the positive x-axis), then walk distance r" — exactly what the sliders above do. The origin is called the pole. This viewpoint is a natural fit for anything built around a center: \(r = a\) is a circle (every angle, same distance), \(\theta = \beta\) is a ray (every distance, same angle), \(r = a\theta\) spirals outward as the angle winds around, and \(r = a\cos(k\theta)\) or \(a\sin(k\theta)\) oscillates in and out to trace flower-like rose curves — k petals when k is odd, 2k when k is even. One quirk: names aren't unique. Adding \(2\pi\) to \(\theta\) changes nothing, and a negative r walks backwards — \((-2, 0)\) and \((2, \pi)\) are the same point.

Algebraic

Right-triangle trigonometry gives the dictionary between the two systems: \(x = r\cos\theta,\; y = r\sin\theta\), and in reverse \(r = \sqrt{x^2 + y^2},\; \theta = \tan^{-1}(y/x)\) (adjusted for quadrant). Three substitutions do almost all conversion work: \(r\cos\theta \to x\), \(r\sin\theta \to y\), and \(r^2 \to x^2 + y^2\). A useful trick: when an equation has a lone \(r\) times a trig function, multiply both sides by r first to manufacture those patterns. And notice the conversion is itself a parametrization: a polar curve \(r = f(\theta)\) is the parametric curve \(x = f(\theta)\cos\theta,\; y = f(\theta)\sin\theta\) with parameter \(\theta\) — both are ways of describing curves without solving for y in terms of x.

Worked example

Convert \(r = 4\sin\theta\) to Cartesian form and identify the curve. Multiply both sides by r: \(r^2 = 4r\sin\theta\). Substitute \(r^2 = x^2 + y^2\) and \(r\sin\theta = y\): \(x^2 + y^2 = 4y\). Complete the square (\(y^2 - 4y = (y-2)^2 - 4\)): \(x^2 + (y-2)^2 = 4\) — a circle of radius 2 centered at \((0, 2)\). It passes through the origin, which is why such a simple polar equation could describe a circle that isn't centered at the pole.

Your turn

Same trick, cosine this time: convert \(r = 6\cos\theta\). Multiply by r: \(r^2 = 6r\cos\theta\). Substitute \(r^2 = x^2 + y^2\) and \(r\cos\theta = x\): \(x^2 + y^2 = 6x\). Now complete the square in x: \((x - \_\_)^2 + y^2 = \_\_\) — what circle is this?

Reveal the answer

\(x^2 - 6x = (x-3)^2 - 9\), so \((x-3)^2 + y^2 = 9\) — a circle of radius 3 centered at \((3, 0)\). Cosine shifted the center along the x-axis, where sine (in the worked example) shifted it along the y-axis.

More info — why one point has many polar names

A Cartesian pair \((x, y)\) names exactly one point, and each point has exactly one name. Polar coordinates give up that second half: since \(\theta\) is an angle, \((r, \theta)\) and \((r, \theta + 2\pi)\) point the same way — infinitely many names per point. Allowing \(r < 0\) adds more: \((-r, \theta)\) means "face \(\theta\), walk backwards," landing where \((r, \theta + \pi)\) does. You saw this in the demo when a negative r flipped the point to the opposite side of the dashed ray. This matters later: when finding where two polar curves intersect, a shared point can wear different \((r, \theta)\) names on each curve, so setting the equations equal can miss intersections. Paul's Notes (linked below) works through the conversion identities that keep this straight.

Check your understanding

Question 1 of 4

Convert the polar point \((r, \theta) = (4, \pi/6)\) to Cartesian coordinates.

Question 2 of 4

Convert the polar equation \(r = 8\cos\theta\) to Cartesian form and identify the curve.

Question 3 of 4

Polar names for a point aren't unique. Which pair names the SAME point as \((3, \pi/2)\)?

Question 4 of 4

A polar curve \(r = f(\theta)\) is a parametric curve in disguise. Which parametrization traces it?

Recap

  • \((r, \theta)\) = distance from the origin and angle counterclockwise from the positive x-axis; convert with \(x = r\cos\theta,\; y = r\sin\theta\) and back with \(r = \sqrt{x^2+y^2},\; \theta = \tan^{-1}(y/x)\) (adjusted for quadrant).
  • Polar names aren't unique: \(\theta\) is periodic (add \(2\pi\)) and negative r points the opposite way — \((-2, 0)\) equals \((2, \pi)\).
  • Simple polar equations draw rich curves: \(r = a\) is a circle, \(\theta = \beta\) a ray, \(r = a\theta\) a spiral, \(r = a\cos(k\theta)\) a rose (k petals for odd k, 2k for even k), \(r = a \pm b\cos\theta\) cardioids and limaçons.
  • To convert an equation, manufacture the patterns \(r\cos\theta = x\), \(r\sin\theta = y\), \(r^2 = x^2+y^2\) — often by multiplying both sides by r.

Dive deeper

Sources

  • Polar Coordinates