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Pre-Calculus Parametric and polar

Polar coordinates

20 practice questions 0 video lessons Theory + worked examples

Polar Coordinates

Texas Precalculus (TEKS) • Standard P.3(D) • Parametric & Polar

Polar Coordinates is a topic in Parametric & Polar in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.3(D), which requires students to graph points in polar coordinates and convert between coordinate systems.

Polar coordinates locate a point by its distance \(r\) from the origin and its angle \(\theta\), written \((r,\theta)\).

Texas Precalculus (TEKS) › Parametric & Polar › Polar Coordinates  —  Standard P.3(D)

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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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