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Pre-Calculus Parametric and polar

Polar coordinates

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Theory

Polar coordinates locate a point by direction and distance instead of horizontal and vertical position. A point is \((r,\theta)\):
  • \(r\) — the distance from the origin (the pole).
  • \(\theta\) — the angle from the positive \(x\)-axis (the polar axis).

A negative \(r\) means measure in the opposite direction; and adding \(360^\circ\) (or \(2\pi\)) to \(\theta\) names the same point.

Polar points are not unique. \((r,\theta)\), \((r,\theta+360^\circ)\), and \((-r,\theta+180^\circ)\) are all the same point.
Polar coordinates A polar point is located by its distance r from the origin and its angle theta from the positive x-axis. (r, θ) θ
A polar point \((r,\theta)\): distance \(r\), angle \(\theta\).
Polar coordinates Polar coordinates Polar coordinates point: (r, θ) r = distance from origin (pole) θ = angle from positive x-axis
The polar coordinate system.

The polar description and its non-uniqueness:

\[(r,\theta)=(r,\theta+360^\circ k)=(-r,\theta+180^\circ)\]
a polar point r comma theta equals r comma theta plus 360 k, and negative r comma theta plus 180
The pole is \((0,\theta)\) for every angle \(\theta\).

How to plot a polar point

  1. Face the angle \(\theta\) from the positive \(x\)-axis.
  2. Walk \(r\) units along that ray (backward if \(r<0\)).
  3. Recognize equivalent names by adding turns or flipping sign.
Example 1 — Plot a polar point
Describe the location of \((3,\ 60^\circ)\).
Solution

Go out \(3\) units at an angle of \(60^\circ\) from the positive \(x\)-axis.

r=3,\ \theta=60^\circ
the point is 3 units out at 60 degrees
Example 2 — A negative radius
Where is \((-2,\ 30^\circ)\)?
Solution

A negative \(r\) points in the opposite direction, so go \(2\) units at \(30^\circ+180^\circ=210^\circ\).

(-2,30^\circ)\(=\)(2,210^\circ)
negative r flips to 2 units at 210 degrees
Example 3 — Multiple names
Give another polar name for \((4,\ 90^\circ)\).
Solution

Add a full turn to the angle: \((4,\ 90^\circ+360^\circ)\).

(4,90^\circ)\(=\)(4,450^\circ)

A polar point has infinitely many names.

the same point is also 4 comma 450 degrees
Example 4 — The pole
What point is \((0,\ \theta)\) for any \(\theta\)?
Solution

With \(r=0\), you are at the origin regardless of angle.

(0,\theta)\(=\)\text{the pole (origin)}
r equals 0 is the pole, the origin, for any angle

Common pitfalls

Negative \(r\) reverses direction. \((-2,30^\circ)\) sits at \(210^\circ\).
Polar points have many names. Don't assume a single \((r,\theta)\).
Angle measured from the positive \(x\)-axis, counterclockwise for positive \(\theta\).

Frequently asked questions

What are polar coordinates?

A way to locate a point by its distance \(r\) from the origin and its angle \(\theta\) from the positive \(x\)-axis, written \((r,\theta)\).

What does a negative r mean?

Measure the distance in the opposite direction: \((-r,\theta)\) equals \((r,\theta+180^\circ)\).

Why does a polar point have more than one name?

Adding full turns to \(\theta\) or flipping the sign of \(r\) with a \(180^\circ\) shift lands on the same point.

What is the pole?

The origin of the polar system, given by \(r=0\) for any angle.

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