Equivalent Polar Coordinates

One Point, More Than One Coordinate Pair

A point in the plane has one location, but it can be described by more than one polar coordinate pair. Turning through a full revolution brings a direction back to itself, and a negative radius points in the direction opposite to the stated angle. These facts mean different pairs can specify exactly the same point.

The goal is to distinguish two things: a point, which is a location in the plane, and a coordinate pair, which is one way of describing that location. The equivalence rules follow directly from the Cartesian coordinates of a polar pair.

Definition: A polar pair \((r,\theta)\), where \(r\) is a real number and \(\theta\) is an angle in radians, represents the Cartesian point \[ (x,y)=(r\cos\theta,\;r\sin\theta). \] Two polar pairs are equivalent if they represent the same Cartesian point.

In this convention, \(r\) is allowed to be negative. The pair \((r,\theta)\) therefore gives a signed distance along the line making angle \(\theta\) with the positive horizontal axis. This definition also works when \(r=0\): the point is then the origin, regardless of \(\theta\).

Why a Full Turn Gives the Same Point

Sine and cosine repeat after a full turn. For every integer \(k\), \(\cos(\theta+2k\pi)=\cos\theta\) and \(\sin(\theta+2k\pi)=\sin\theta\). Substituting these identities into the Cartesian coordinate formula gives

\begin{aligned} x'&=r\cos(\theta+2k\pi)=r\cos\theta=x,\\ y'&=r\sin(\theta+2k\pi)=r\sin\theta=y. \end{aligned}

Thus, \((r,\theta)\) and \((r,\theta+2k\pi)\) represent the same point. The integer \(k\) can be positive, negative, or zero: it counts how many full turns are added to the angle. For example, \(k=1\) adds one full revolution, while \(k=-2\) subtracts two.

Worked Example: Adding a Full Turn

Find the Cartesian point represented by \((3,\pi/6)\), then check the pair \((3,13\pi/6)\).

For the first pair, use \(\cos(\pi/6)=\sqrt{3}/2\) and \(\sin(\pi/6)=1/2\):

x=3\cos(\pi/6)=\frac{3\sqrt{3}}{2}, \qquad y=3\sin(\pi/6)=\frac{3}{2}.

Since \(13\pi/6=\pi/6+2\pi\), the second angle differs by one full turn. Directly, its trigonometric values are the same:

3\cos(13\pi/6)=\frac{3\sqrt{3}}{2}, \qquad 3\sin(13\pi/6)=\frac{3}{2}.

Both pairs therefore represent \((3\sqrt{3}/2,3/2)\), confirming the equivalence.

Changing the Sign of the Radius

Changing the sign of \(r\) reverses both Cartesian coordinates if the angle stays fixed: \[ (-r\cos\theta,\,-r\sin\theta). \] To recover the original point, the angle must also turn by \(\pi\), which reverses both trigonometric values. The identities \(\cos(\theta+\pi)=-\cos\theta\) and \(\sin(\theta+\pi)=-\sin\theta\) give

\begin{aligned} (-r)\cos(\theta+\pi)&=(-r)(-\cos\theta)=r\cos\theta,\\ (-r)\sin(\theta+\pi)&=(-r)(-\sin\theta)=r\sin\theta. \end{aligned}

Consequently, a negative radius can be replaced by its positive magnitude if the angle is shifted by \(\pi\). Additional full turns may be added without changing the point.

Equivalent-coordinate rules: For any real \(r,\theta\) and integer \(k\), \((r,\theta)\) and \((r,\theta+2k\pi)\) represent the same point. Also, \((r,\theta)\) and \((-r,\theta+(2k+1)\pi)\) represent the same point. In particular, \((r,\theta)\) is equivalent to \((-r,\theta+\pi)\).

Worked Example: Replacing a Negative Radius

Convert \((-4,\pi/3)\) to an equivalent pair with positive radius, and verify both pairs in Cartesian coordinates.

Changing the radius from \(-4\) to \(4\) requires adding \(\pi\) to the angle: \[ (-4,\pi/3)\quad\longleftrightarrow\quad(4,\pi/3+\pi)=(4,4\pi/3). \] For the original pair, \(\cos(\pi/3)=1/2\) and \(\sin(\pi/3)=\sqrt{3}/2\), so

x=-4\left(\frac12\right)=-2, \qquad y=-4\left(\frac{\sqrt{3}}2\right)=-2\sqrt{3}.

For the new pair, \(\cos(4\pi/3)=-1/2\) and \(\sin(4\pi/3)=-\sqrt{3}/2\). Therefore,

x=4\left(-\frac12\right)=-2, \qquad y=4\left(-\frac{\sqrt{3}}2\right)=-2\sqrt{3}.

The Cartesian coordinates agree, so the pairs represent the same point.

All Equivalent Pairs for a Nonzero Point

The rules above do more than produce some equivalent pairs: for a point other than the origin, they describe every possible equivalent pair. This can be proved by comparing the Cartesian coordinates of two representations.

Theorem: Suppose \((r,\theta)\) and \((s,\phi)\) represent the same point, and that point is not the origin. Then either \(s=r\) and \(\phi=\theta+2k\pi\), or \(s=-r\) and \(\phi=\theta+(2k+1)\pi\), for some integer \(k\). These conditions are also sufficient for the pairs to represent the same point.

Proof. Equality of the represented points means \(r\cos\theta=s\cos\phi\) and \(r\sin\theta=s\sin\phi\). The distance of the point from the origin is \(\sqrt{x^2+y^2}\). For the first pair, this distance is

\sqrt{(r\cos\theta)^2+(r\sin\theta)^2} = \sqrt{r^2(\cos^2\theta+\sin^2\theta)} = |r|.

For the second pair it is \(|s|\). Because the points are the same and are not the origin, \(r\) and \(s\) are nonzero and \(|r|=|s|\). Therefore \(s=r\) or \(s=-r\).

If \(s=r\), equality of coordinates and division by the nonzero \(r\) give \(\cos\theta=\cos\phi\) and \(\sin\theta=\sin\phi\). Angles with the same sine and cosine differ by an integer number of full turns, so \(\phi=\theta+2k\pi\).

If \(s=-r\), dividing the coordinate equalities by \(r\) gives \(\cos\theta=-\cos\phi\) and \(\sin\theta=-\sin\phi\). Equivalently, the cosine and sine of \(\phi\) are those of \(\theta+\pi\). Thus \(\phi=\theta+\pi+2k\pi=\theta+(2k+1)\pi\). This proves necessity. Sufficiency follows by substituting either stated form into the Cartesian coordinate equations, as shown above.

Worked Example: Checking Two Proposed Pairs

Determine whether \((2,-\pi/4)\) and \((-2,3\pi/4)\) represent the same point.

The radii are opposites: the second radius is \(-2\). The angle difference is

\frac{3\pi}{4}-\left(-\frac{\pi}{4}\right) = \frac{4\pi}{4} = \pi.

This is exactly the negative-radius rule: the radius changes sign and the angle increases by \(\pi\). Verify in Cartesian coordinates. Since \(\cos(-\pi/4)=\sqrt{2}/2\) and \(\sin(-\pi/4)=-\sqrt{2}/2\), the first pair gives

x=2\left(\frac{\sqrt{2}}2\right)=\sqrt{2}, \qquad y=2\left(-\frac{\sqrt{2}}2\right)=-\sqrt{2}.

For the second pair, \(\cos(3\pi/4)=-\sqrt{2}/2\) and \(\sin(3\pi/4)=\sqrt{2}/2\), so

x=-2\left(-\frac{\sqrt{2}}2\right)=\sqrt{2}, \qquad y=-2\left(\frac{\sqrt{2}}2\right)=-\sqrt{2}.

The Cartesian coordinates match, confirming that the pairs are equivalent.

The Special Case of the Origin

The theorem for nonzero points needs an exception at the origin. If \(r=0\), then \[ (0\cos\theta,0\sin\theta)=(0,0) \] for every angle \(\theta\). Thus \((0,\theta)\) and \((0,\phi)\) represent the origin for any real angles \(\theta\) and \(\phi\); their angles do not need to differ by a full turn or a half-turn.

Worked Example: Different Angles at the Origin

Compare \((0,\pi/5)\) and \((0,7\pi/6)\). For the first pair,

x=0\cos(\pi/5)=0, \qquad y=0\sin(\pi/5)=0.

For the second pair,

x=0\cos(7\pi/6)=0, \qquad y=0\sin(7\pi/6)=0.

Both pairs represent \((0,0)\), even though the difference between their angles is not an integer multiple of \(\pi\). The zero radius makes the direction irrelevant.

A Common Mistake and Its Correction

A frequent error is to change the sign of the radius without adjusting the angle. For instance, claiming that \((3,\pi/6)\) and \((-3,\pi/6)\) are equivalent is incorrect. The first pair represents \((3\sqrt{3}/2,3/2)\), while the second represents \((-3\sqrt{3}/2,-3/2)\). Those are opposite points, not the same point.

The correction is to shift the angle by \(\pi\) when reversing the radius: \[ (3,\pi/6)\quad\longleftrightarrow\quad(-3,7\pi/6). \] Indeed, \(\cos(7\pi/6)=-\sqrt{3}/2\) and \(\sin(7\pi/6)=-1/2\), so the corrected pair has coordinates \[ (-3)(-\sqrt{3}/2)=3\sqrt{3}/2, \qquad (-3)(-1/2)=3/2. \] They match the original point. The reliable check is always to convert both pairs to Cartesian coordinates.

Check Your Understanding

Use the Cartesian coordinate definition to check each claim before deciding whether it is true.

Write an equivalent pair to \((5,\theta)\) whose angle is \(\theta-2\pi\).
Write \((-6,\theta)\) as an equivalent pair with positive radius.
Are \((4,\pi/3)\) and \((-4,4\pi/3)\) equivalent? Explain the role of the angle difference.
Why do \((0,\alpha)\) and \((0,\beta)\) represent the same point for arbitrary real \(\alpha\) and \(\beta\)?
State the two forms of equivalent pairs for a nonzero point, including the integer parameter.

What You'll Learn