Locating a Point with Polar Coordinates
A point can be described by its distance and direction from a chosen reference point. In polar coordinates, the reference point is called the pole. A polar coordinate pair is written \((r,\theta)\): \(r\) is the signed radius, and \(\theta\) is an angle measured from a fixed initial direction.
Unless a problem specifies otherwise, angles in polar coordinates are measured in radians. Positive angles turn counterclockwise from the initial direction, and negative angles turn clockwise. When \(r\) is positive, move a distance \(r\) from the pole along the ray at angle \(\theta\). This locates the point represented by \((r,\theta)\).
Unlike a point, a coordinate pair is a particular description of a point. Different pairs can describe the same point. This happens because a full turn brings a ray back to its original direction, and because changing the sign of the radius reverses the direction of travel.
Angles That Differ by Full Turns
Turning through \(2\pi\) radians makes one complete revolution. Therefore, the angle \(\theta+2\pi\) specifies the same direction as \(\theta\). The same is true after any integer number of full turns, including turns in the clockwise direction.
Here \(k\) can be any integer. Angles related in this way are called coterminal. For a fixed nonzero radius, coterminal angles give equivalent polar representations. For instance, \(\pi/3\), \(7\pi/3\), and \(-5\pi/3\) differ by whole turns, so each points along the same ray.
Worked Example: Add or Subtract Full Turns
Find two polar coordinate pairs equivalent to \((4,\pi/3)\), using one positive angle greater than \(2\pi\) and one negative angle.
To obtain an angle greater than \(2\pi\), add \(2\pi\):
Thus \((4,7\pi/3)\) represents the same point as \((4,\pi/3)\). To obtain a negative angle, subtract \(2\pi\):
Therefore \((4,-5\pi/3)\) is also equivalent. In both calculations the radius is unchanged, and the angle changes by exactly one full turn. The points are the same because their rays and distances from the pole are the same.
What a Negative Radius Means
A negative radius does not mean that distance from the pole is negative; distance is always nonnegative. Instead, the sign tells you to travel in the direction opposite the ray specified by the angle. Reversing a ray is equivalent to turning its angle by \(\pi\) radians, which is half a revolution.
Consequently, a negative radius can be replaced by a positive radius if the angle is shifted by \(\pi\). Conversely, a positive radius can be replaced by a negative one by shifting the angle by \(\pi\). Further full turns can then be added or subtracted without changing the point.
For either pair, the distance from the pole is \(|r|\). The second pair points in the opposite direction from angle \(\theta\), and its negative radius reverses that direction again. It therefore reaches the same point as the first pair.
Worked Example: Replace a Negative Radius
Rewrite \((-3,\pi/6)\) first with a positive radius and then with a negative radius and a coterminal angle.
To change the radius from \(-3\) to \(3\), add \(\pi\) to the angle:
Thus \((-3,\pi/6)\) and \((3,7\pi/6)\) describe the same point. The positive radius now travels in the direction opposite the original angle, for the same distance of \(3\).
To give an equivalent pair that retains the negative radius, add one full turn to its angle:
So \((-3,13\pi/6)\) is also equivalent to \((-3,\pi/6)\). In every representation, the distance is \(3\), and the actual direction of travel accounts for the negative sign.
Check the positive-radius result by reversing its direction once more: \((3,7\pi/6)\) becomes \((-3,7\pi/6-\pi)=(-3,\pi/6)\). This recovers the original pair.
A Test for Equivalent Polar Pairs
The same-point test can be stated precisely. For a nonzero radius, first account for the sign of the radius: the point is a distance \(|r|\) from the pole, and its actual direction is \(\theta\) when \(r>0\), or \(\theta+\pi\) when \(r<0\). Two nonzero pairs represent the same point exactly when their distances are equal and their actual directions differ by a whole number of turns.
The conditions can be checked in two stages: compare the distances \(|r|\) and \(|s|\), then compare the actual directions. The two possible signs of \(s\) give the two cases in the theorem.
For a nonzero point, the distances from the pole must agree, so \(|r|=|s|\). This means \(s=r\) or \(s=-r\).
The signs agree, so both pairs use the same rule for travel direction. Their angles must differ by \(2k\pi\): \(\phi=\theta+2k\pi\).
The signs differ, so one pair travels in the direction opposite its stated angle. The stated angles must differ by an odd multiple of \(\pi\): \(\phi=\theta+(2k+1)\pi\).
Only radius zero reaches the pole. If both radii are zero, their angles do not affect the point.
These steps prove both directions of the test. If the conditions hold, the pairs have equal distances and the same actual direction, so they locate the same point. Conversely, if two nonzero pairs locate the same point, their distances must agree and their actual directions must agree up to full turns. Equal distances give \(s=r\) or \(s=-r\); matching directions then gives the corresponding angle condition above.
Worked Example: Test Two Representations
Determine whether \((5,-\pi/4)\) and \((-5,3\pi/4)\) represent the same point.
First compare the distances from the pole:
The radii have opposite signs, so the theorem requires the angles to differ by an odd multiple of \(\pi\). Their difference is:
This is an odd multiple of \(\pi\), so the pairs are equivalent. As a direct check, changing the radius \(5\) in the first pair to \(-5\) requires adding \(\pi\) to its angle:
This produces exactly the second pair, \((-5,3\pi/4)\).
The Pole and Nonunique Coordinates
At the pole, every angle leads to the same point because moving a distance zero does not leave the pole. Thus \((0,0)\), \((0,\pi)\), and \((0,17\pi/6)\), for example, all represent the pole. No angle adjustment is needed to compare such pairs; the radii alone show that they both locate the pole.
For a point away from the pole, there are infinitely many equivalent pairs as well. Starting from any one representation, you can add any integer multiple of \(2\pi\) to its angle. You can also reverse the radius and shift the angle by \(\pi\), then add any number of full turns. A convention that restricts radii or angles may select a preferred representation, but that restriction is an extra rule for choosing coordinates; it does not change which point the pair describes.
Common Mistake: Changing the Radius Sign Alone
A frequent error is to change a negative radius to positive while leaving the angle unchanged. For example, one might incorrectly claim \((-2,\pi/3)=(2,\pi/3)\). This is wrong: the first pair travels opposite the ray at \(\pi/3\), while the second travels along it. They are on opposite sides of the pole.
To correct the representation, change the radius sign and shift the angle by \(\pi\):
The radius now has the desired positive value, and the angle specifies the opposite direction. The same care applies in reverse: changing a positive radius to a negative one also requires a half-turn in the angle.
Why Equivalent Representations Matter
A point may be given in a form that is not immediately convenient for the task at hand. Recognizing equivalence lets you replace a negative radius with a positive one, or choose an angle in a preferred interval, without changing the point. It also prevents a common interpretation error: coordinate pairs that look different are not necessarily different locations.
Keep the distinction clear: \(r\) records signed travel, \(\theta\) records an angle, and the point is the result of following both instructions. Compare distances and actual directions, not just the written numbers. The pole is the special case where distance zero makes every angle irrelevant.
Check Your Understanding
Use full-turn changes and the sign of the radius to decide what each pair represents.
- Give an equivalent pair for \((3,5\pi/6)\) with a negative angle and the same radius.
- Rewrite \((-4,2\pi/3)\) with a positive radius and an angle between \(0\) and \(2\pi\).
- Do \((2,7\pi/4)\) and \((-2,3\pi/4)\) represent the same point? Justify your answer using the equivalence test.
- How many distinct points are represented by \((0,\theta)\) as \(\theta\) varies over all real numbers?
- State the conditions under which two nonzero polar coordinate pairs represent the same point.
