From a Polar Point to Its Coordinates
A polar point describes a location using a signed distance from the origin and an angle. Cartesian coordinates describe that same location by its horizontal and vertical displacements. Converting from polar to Cartesian coordinates means finding those two displacements.
The conversion uses trigonometry to resolve the radius into horizontal and vertical components. A negative radius is allowed: it places the point in the direction opposite the one specified by the angle, so the sign of \(r\) must be included in both coordinates.
Why the Formulas Work
First consider a point whose radius \(r\) is positive. The radius and the coordinate axes form a right triangle: for an acute angle \(\theta\) in the first quadrant, the horizontal leg has length \(x\) and the vertical leg has length \(y\). Relative to the angle \(\theta\), the horizontal leg is adjacent and the vertical leg is opposite. Therefore, by the definitions of cosine and sine,
Multiplying each equation by \(r\) gives \(x=r\cos\theta\) and \(y=r\sin\theta\). For angles outside the first quadrant, the triangle gives only the magnitudes \(|x|\) and \(|y|\); the signs of the coordinates are determined by the quadrant. The same formulas hold with the signs built in, because the unit-circle definition of \((\cos\theta,\sin\theta)\) makes cosine and sine positive or negative according to the angle’s direction.
Now let \(r\) be negative. The point lies opposite the direction indicated by \(\theta\), so both components are multiplied by a negative number. The same equations still apply without modification. If \(r=0\), both coordinates are zero regardless of \(\theta\).
Using the Conversion Formulas
To convert a polar point, evaluate the cosine and sine of its angle, multiply each value by the signed radius, and write the resulting ordered pair as \((x,y)\). Keep the coordinate order straight: cosine gives \(x\), while sine gives \(y\).
Exact trigonometric values are often available for standard angles such as \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\). If an angle is given in degrees, use degree-mode values; if it is given in radians, use radian-mode values. The formulas work in either unit, provided the angle is interpreted consistently.
Worked Example: A Positive Radius in the First Quadrant
Convert \((4,\frac{\pi}{3})\) to Cartesian coordinates.
Use \(x=r\cos\theta\) and \(y=r\sin\theta\), with \(r=4\) and \(\theta=\frac{\pi}{3}\). The exact trigonometric values are \(\cos\frac{\pi}{3}=\frac12\) and \(\sin\frac{\pi}{3}=\frac{\sqrt3}{2}\). Thus
The Cartesian point is \((2,2\sqrt3)\). Both coordinates are positive, as expected for an angle of \(\frac{\pi}{3}\) and a positive radius. Check the distance from the origin: \(x^2+y^2=2^2+(2\sqrt3)^2=4+12=16=4^2\), as required for a radius of \(4\). Substitution into the original conversion formulas also gives \(4\cos\frac{\pi}{3}=2\) and \(4\sin\frac{\pi}{3}=2\sqrt3\).
Negative Radii
A negative radius does not mean that the point has negative distance from the origin. It means that the point is located the stated distance in the direction opposite the angle. The conversion formulas handle this by carrying the negative sign into both products.
For instance, the polar representations \((r,\theta)\) and \((-r,\theta+\pi)\) describe the same point when the angles are interpreted modulo a full turn. This is useful for understanding a negative radius, but it does not change the conversion procedure: use the \(r\) and \(\theta\) that were actually given.
Worked Example: A Negative Radius with a Second-Quadrant Angle
Convert \((-6,\frac{5\pi}{6})\) to Cartesian coordinates.
The angle \(\frac{5\pi}{6}\) has \(\cos\frac{5\pi}{6}=-\frac{\sqrt3}{2}\) and \(\sin\frac{5\pi}{6}=\frac12\). Multiplying each value by \(r=-6\) gives
The Cartesian point is \((3\sqrt3,-3)\), which lies in the fourth quadrant. This may initially seem surprising because the angle \(\frac{5\pi}{6}\) points into the second quadrant. The negative radius reverses that direction. Check the distance: \((3\sqrt3)^2+(-3)^2=27+9=36=(-6)^2\). The original formulas also reproduce the coordinates: \((-6)(-\frac{\sqrt3}{2})=3\sqrt3\) and \((-6)(\frac12)=-3\).
Angles, Signs, and Location
The signs of \(\cos\theta\) and \(\sin\theta\) indicate the direction for a positive radius. A negative radius reverses both signs. It is therefore helpful to consider the signs of the two products, rather than deciding the quadrant from the angle alone.
| Angle’s quadrant (positive radius) | Sign of \(x\) | Sign of \(y\) |
|---|---|---|
| First | Positive | Positive |
| Second | Negative | Positive |
| Third | Negative | Negative |
| Fourth | Positive | Negative |
For a negative radius, reverse both signs in the relevant row. If an angle lies on an axis, one coordinate is zero; the same multiplication rules still determine the other coordinate. If \(r=0\), the point is the origin for every angle.
Worked Example: A Negative Radius and an Angle on an Axis
Convert \((-5,\pi)\) to Cartesian coordinates.
At \(\theta=\pi\), \(\cos\pi=-1\) and \(\sin\pi=0\). Therefore,
The Cartesian point is \((5,0)\). Although the angle \(\pi\) points left, the negative radius reverses that direction, placing the point on the positive \(x\)-axis. Checking by substitution, \(x=(-5)(-1)=5\) and \(y=(-5)(0)=0\). Its distance from the origin is \(\sqrt{5^2+0^2}=5=|-5|\).
A Full Conversion Process
The following process keeps the calculation organized, especially when a negative radius or an angle on an axis is involved.
Identify the signed radius \(r\) and angle \(\theta\). Note whether the angle is in degrees or radians.
Evaluate \(\cos\theta\) and \(\sin\theta\), including their signs. Use exact values when available.
Calculate \(x=r\cos\theta\) and \(y=r\sin\theta\). Do not replace a negative radius with its absolute value.
Confirm that the signs and axis location make sense, then verify that \(x^2+y^2=r^2\).
Common Mistake: Dropping the Negative Sign
A frequent error is to treat \(r\) as a distance that must be positive, even when the given polar coordinate has a negative radius. That changes the point by reversing its direction a second time—or by failing to reverse it at all.
The correction is to use the signed value \(r=-4\) in both formulas:
The correct point is \((-2,-2\sqrt3)\), in the third quadrant. Its distance check is \((-2)^2+(-2\sqrt3)^2=4+12=16=(-4)^2\). The key is not to change the formulas for negative radii; it is to preserve the sign when multiplying.
Why This Conversion Matters
Polar coordinates describe a point by direction and signed distance, while Cartesian coordinates describe its horizontal and vertical position. Converting between these descriptions lets you interpret the same point in whichever coordinate system is useful for a particular calculation or graph.
For coordinate conversion, the essential information is local and direct: evaluate two trigonometric functions and multiply by the radius. No equation-solving is required. Before moving on, make a habit of checking both the arithmetic and the geometry: a positive radius follows the angle’s direction, a negative radius reverses it, and the distance from the origin is \(|r|\).
Check Your Understanding
Use the conversion formulas to answer each question. Keep negative radii signed throughout your calculations.
- Convert \((2,\frac{\pi}{6})\) to Cartesian coordinates.
- Convert \((-4,\frac{3\pi}{2})\) to Cartesian coordinates.
- For \((3,\frac{5\pi}{4})\), what are the signs of \(x\) and \(y\), and what are the exact coordinates?
- Explain why a negative radius reverses the direction indicated by the angle.
- What Cartesian point results when \(r=0\), and does the angle affect the answer?
