Polar Coordinates

Distance and direction describe a point just as well as across and up.

Two ways to send someone to the same doorstep.

One: go three blocks east, then two blocks north. That is the street grid, and it is what (3, 2) means.

same place, two sets of directions 3 across 2 up 34° 3.6 street address radar address

Two: face 34° north of east and walk 3.6 blocks. Same doorstep, no turns.

Air traffic control uses the second kind. So does radar, and so does anything that spins.

Both are honest addresses for one point. Which you choose is a matter of what the problem is shaped like — and circles are far easier in the second.

A polar grid, and a point given by distance and direction 60° r = 3 (3, 60°) rings count distance, spokes count direction

Converting polar to rectangular

Drop a right triangle from the point to the horizontal axis and the conversion writes itself.

x = rcosθ and y = rsinθ. That is all there is to it.

For (4, 60°): x = 4cos 60° = 2 and y = 4sin 60° = 2√3.

It is the unit circle, scaled

On the unit circle a point is (cosθ, sinθ). At distance r it is that, multiplied by r.

Converting rectangular to polar

r = √x2+y2 by Pythagoras, and tanθ = yx from the same triangle.

But arctan only returns angles between -90° and 90°, so it cannot see quadrants II and III.

Sketch the point first, then adjust: if it lies left of the axis, add 180° to what the calculator said.

Arctangent loses half the plane

Both (3, 4) and (-3, -4) give tanθ = 43. Only the sketch tells you which one you have.

One point, many addresses

Adding a full turn changes nothing, so (3, 40°) and (3, 400°) are the same point.

A negative r means walk backwards, so (-3, 40°) is the same as (3, 220°).

Rectangular coordinates have no such freedom, which is one thing they are better at.

Unique going one way only

Every polar pair names one point, but every point has endless polar names. That is fine until you try to solve equations, where it needs care.

Equations that polar form makes easy

r = 5 is a circle of radius 5. In rectangular form that needs x2 + y2 = 25 and a square root to graph.

θ = 30° is a straight line through the origin, and no arithmetic at all.

Anything built around a centre — circles, spirals, petals — is simpler in polar form.

Match the coordinates to the shape

Grids suit straight lines and rectangles. Rings and spokes suit circles and rotation. Neither is more correct.

Complex numbers in polar form

A complex number a + bi is a point, so it has a polar form too: r(cosθ + isinθ).

Here r = |z| is the distance from the origin and θ is called the argument.

Multiplying two complex numbers multiplies the distances and adds the angles, which is far tidier than expanding brackets.

Why engineers live here

Repeated multiplication becomes repeated rotation. Raising to a power is then one multiplication of angles, not n expansions.

Worked examples

Polar to rectangular

Convert (6, 150°) to rectangular form, exactly.

  1. x = 6cos 150°. The reference angle is 30° and quadrant II makes cosine negative. Size then sign.
  2. x = 6(-√32) = -3√3. Left of the axis, as quadrant II requires.
  3. y = 6sin 150° = 6(12) = 3. Sine is positive in quadrant II.

(-3√3, 3)

Rectangular to polar

Convert (-4, 4) to polar form with 0° ≤ θ < 360°.

  1. r = √16+16 = √32 = 4√2. Pythagoras on the components.
  2. tanθ = 4-4 = -1, and the calculator returns -45°. Arctangent cannot see quadrant II.
  3. The point is up and to the left, so add 180°: θ = 135°. The sketch settles it.

(4√2, 135°)

Recognising a polar equation

Describe the graph of r = 4.

  1. Every point on it is exactly 4 from the origin. r is the distance.
  2. The angle is unrestricted, so all directions are included. No condition on θ.
  3. That is a circle of radius 4 centred at the origin. In rectangular form, x2 + y2 = 16.

A circle of radius 4 at the origin

Words to know

Polar coordinates
A point given as (r, θ): distance from the origin, and direction.
Pole
The origin, where r = 0.
Polar axis
The ray pointing along positive x, from which θ is measured.
Rectangular form
The ordinary (x, y) description.
Modulus
The distance r, also written |z| for a complex number.

Mistakes to watch for

Trusting arctangent for the quadrant. Both (3,4) and (-3,-4) give tanθ = 43, and the calculator returns the same angle for both. Sketch the point and add 180° when it lies left of the y-axis.

Swapping sine and cosine in the conversion. It is x = rcosθ and y = rsinθ — x goes with cosine, alphabetically in both. Swapping them reflects every point across y = x.

Treating the polar name of a point as unique. (3, 40°), (3, 400°) and (-3, 220°) are all the same point, which matters when checking whether a solution has already been found.

Forgetting that r can be negative. A negative r means measuring backwards along the ray, landing the point in the opposite direction.

Converting an equation term by term. r = 2cosθ is a circle, not a cosine wave. Multiply through by r first to get x2+y2 = 2x, which is recognisable.

A rose curve like r = cos(3θ) takes one line to write in polar form and is a nightmare in x and y.

Newton was using polar coordinates by the 1670s to describe planetary orbits, where distance from the sun is the natural variable.

Every complex number has a polar form too, and multiplying in that form just adds the angles — which is why polar coordinates run so much of engineering.

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