From a Cartesian Point to a Polar Point
A Cartesian point is specified by its horizontal and vertical coordinates, \((x,y)\). A polar point describes the same location using a distance from the origin and an angle from the positive \(x\)-axis. Converting from Cartesian to polar therefore requires two decisions: how far the point is from the origin, and which direction from the positive \(x\)-axis reaches it.
The distance is straightforward to calculate. Selecting the angle requires more care: an inverse tangent can give an angle with the right slope but the wrong quadrant. A reliable conversion uses the signs of both coordinates, then checks the proposed polar coordinates by converting them back.
Find the Radius
For a point \((x,y)\), the horizontal and vertical displacements form the legs of a right triangle whose hypotenuse is the point’s distance from the origin. By the Pythagorean theorem, that distance is \(\sqrt{x^2+y^2}\). Thus, for the nonnegative-radius representation, calculate \(r\) using this formula:
The square root is nonnegative, as required by this convention. The formula gives the distance regardless of the point’s quadrant. It does not, by itself, determine the angle.
Choose the Angle Using the Quadrant
For a point not on an axis, first determine its quadrant from the signs of \(x\) and \(y\). Then find the reference angle \(\alpha\), the acute angle between the point’s ray and the nearest horizontal axis. For \(x\neq 0\), it can be calculated as \(\alpha=\arctan\left(\left|y/x\right|\right)\). Use the quadrant to place this reference angle correctly:
| Location of \((x,y)\) | Angle \(\theta\), with \(0\leq\theta<2\pi\) |
|---|---|
| Quadrant I: \(x>0,\ y>0\) | \(\alpha\) |
| Quadrant II: \(x<0,\ y>0\) | \(\pi-\alpha\) |
| Quadrant III: \(x<0,\ y<0\) | \(\pi+\alpha\) |
| Quadrant IV: \(x>0,\ y<0\) | \(2\pi-\alpha\) |
These rules give a standard angle between \(0\) and \(2\pi\), not including \(2\pi\). A coterminal angle outside that interval, such as \(-\pi/3\) instead of \(5\pi/3\), is also valid. On an axis, use the direction of that axis rather than the reference-angle formula: positive \(x\)-axis, \(\theta=0\); positive \(y\)-axis, \(\theta=\pi/2\); negative \(x\)-axis, \(\theta=\pi\); negative \(y\)-axis, \(\theta=3\pi/2\). These axis rules assume \(r>0\).
Why the Radius and Angle Formulas Work
The polar-coordinate equations explain why the conversion formulas are correct. If \((r,\theta)\) represents \((x,y)\), then \(x=r\cos\theta\) and \(y=r\sin\theta\). Squaring these equations and adding them gives the distance formula:
Here the last equality uses \(\cos^2\theta+\sin^2\theta=1\). Since the usual convention requires \(r\geq 0\), taking the nonnegative square root gives \(r=\sqrt{x^2+y^2}\). For a non-axis point, the direction of the point’s ray determines its quadrant, and the ratio of the vertical to horizontal displacement determines its reference angle. The quadrant rules then place that angle in the correct direction. This is why both the radius calculation and quadrant check are needed.
Worked Conversions
Worked Example: A Point in Quadrant II
Convert \((-3,3\sqrt{3})\) to polar coordinates with \(r\geq 0\) and \(0\leq\theta<2\pi\).
Step 1: Find the radius. Substitute \(x=-3\) and \(y=3\sqrt{3}\):
Step 2: Identify the quadrant and reference angle. Since \(x<0\) and \(y>0\), the point is in Quadrant II. Its reference angle is
Therefore, \(\theta=\pi-\alpha=\pi-\pi/3=2\pi/3\). The polar coordinates are \((6,2\pi/3)\).
Check: Substitution into the polar equations recovers the original point:
Both coordinates match \((-3,3\sqrt{3})\).
Worked Example: A Point in Quadrant IV and Equivalent Representations
Convert \((5,-5\sqrt{3})\) to polar coordinates with a nonnegative radius and \(0\leq\theta<2\pi\). Then give another representation of the same point using a negative radius.
Step 1: Find the radius.
Step 2: Choose the angle. Here \(x>0\) and \(y<0\), so the point is in Quadrant IV. Its reference angle is
In the requested interval, \(\theta=2\pi-\alpha=2\pi-\pi/3=5\pi/3\). Thus one representation is \((10,5\pi/3)\). The coterminal angle \(-\pi/3\) gives the same point with positive radius.
Step 3: Use a negative radius. Reversing the direction of the angle by \(\pi\) and changing the sign of the radius preserves the point. A negative-radius representation is \((-10,2\pi/3)\).
Check both representations:
Each representation recovers the original Cartesian point.
Worked Example: A Point on an Axis
Convert \((0,-7)\) to polar coordinates with \(r\geq 0\), first using \(0\leq\theta<2\pi\), and then using a negative angle.
Step 1: Find the radius.
Step 2: Identify the direction. Since \(x=0\) and \(y<0\), the point lies on the negative \(y\)-axis. Its standard angle is \(3\pi/2\), so the coordinates are \((7,3\pi/2)\). The coterminal angle \(-\pi/2\) is another valid angle with the same positive radius.
Check:
This gives \((0,-7)\), as required. The quotient \(y/x\) is undefined here, which is why axis points must be handled directly.
Equivalent Polar Representations
A single point does not have only one polar representation. Adding a full turn to the angle leaves both sine and cosine unchanged. A negative radius points in the direction opposite the angle, so changing the sign of the radius while adding \(\pi\) to the angle also leaves the Cartesian point unchanged.
For example, \((10,5\pi/3)\), \((10,-\pi/3)\), and \((-10,2\pi/3)\) all represent \((5,-5\sqrt{3})\). If a question requests a particular interval for the angle or requires a nonnegative radius, satisfy that convention in the final answer. Otherwise, equivalent forms are valid.
A Reliable Conversion Process
Use \(r=\sqrt{x^2+y^2}\) for the usual nonnegative-radius form.
Use the signs of \(x\) and \(y\) to identify its quadrant, or determine whether it lies on an axis.
For a point in a quadrant, calculate the reference angle and apply the matching quadrant rule. For an axis point, use its axis direction.
Substitute into \(x=r\cos\theta\) and \(y=r\sin\theta\); confirm that both original coordinates are recovered.
Adjust to the required angle interval or give equivalent representations if requested.
The Origin and Common Errors
For the origin, \((x,y)=(0,0)\), the radius is \(r=0\). Since \(0\cos\theta=0\) and \(0\sin\theta=0\) for every angle, the origin can be represented as \((0,\theta)\) for any \(\theta\). There is no unique direction from the origin to itself, so no single angle is determined.
A frequent error is using \(\arctan(y/x)\) and accepting its output without checking the quadrant. For \((-3,3\sqrt{3})\), that calculation gives
That angle is in Quadrant IV, but \((-3,3\sqrt{3})\) is in Quadrant II. It has the right reference-angle size, but the wrong direction. The correction is to use the signs \(x<0\) and \(y>0\), then choose \(\theta=\pi-\pi/3=2\pi/3\). Checking with \(x=r\cos\theta\) and \(y=r\sin\theta\) confirms the corrected angle.
Also avoid dividing by \(x\) when \(x=0\); use the axis rules instead. Finally, do not report only the radius: \(\sqrt{x^2+y^2}\) gives the distance but not the direction. A complete conversion needs an angle consistent with the point’s location.
Check Your Understanding
For each question, use exact values where possible and check the coordinates by substituting into the polar equations.
- Convert \((4,4)\) to polar coordinates with \(r\geq 0\) and \(0\leq\theta<2\pi\).
- Convert \((-6,-6\sqrt{3})\) to polar coordinates with \(r\geq 0\) and \(0\leq\theta<2\pi\).
- Give a positive-radius polar representation of \((0,5)\) using an angle in \([-\pi,\pi]\).
- Give a negative-radius representation equivalent to \((8,\pi/6)\).
- What is the radius of the origin, and why can its angle be any real number?
