Polar Coordinates
A radar screen does not report a plane as “three miles east and four miles north.” It reports a distance from the antenna and an angle off due north. That is a polar address: how far out, and which way. Polar coordinates describe a point by a directed distance r from the origin and an angle theta from the positive x-axis, instead of by horizontal and vertical offsets.
By the end of this page you can:
- Plot a point given as a polar pair (r, theta).
- Convert a polar point to rectangular form using x = r cos theta and y = r sin theta.
- Convert a rectangular point to polar form using r = sqrt(x^2 + y^2) and tan theta = y/x, choosing theta in the correct quadrant.
- Explain why a single point has infinitely many polar names.
The floor. Carry away the two conversion pairs: x = r cos theta with y = r sin theta going one way, and r^2 = x^2 + y^2 with tan theta = y/x going back. A point in the plane has one rectangular name but many polar names.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 8 |
| Section | 8.1 Polar Coordinates |
| Open alternate | OpenStax Precalculus 2e, 10.3 Polar Coordinates |
| Difficulty | Core |
| Time | About 30 minutes |
Before You Start
The Idea
A polar pair (r, theta) places a point by walking out a distance r along the ray at angle theta from the positive x-axis. The link to rectangular coordinates comes straight from right-triangle trigonometry: the horizontal leg is r cos theta and the vertical leg is r sin theta. Going the other direction, the distance from the origin is r = sqrt(x^2 + y^2), and the angle satisfies tan theta = y / x.
One feature has no rectangular analogue: a point has many polar names. Adding any full turn of 2 pi to theta lands on the same spot, so (r, theta) and (r, theta + 2 pi) name the same point. A negative r flips the ray to the opposite direction, so (-r, theta) equals (r, theta + pi). The origin itself is (0, theta) for every theta.
Key Formulas
Polar to rectangular:
\[ x = r \cos\theta, \qquad y = r \sin\theta \]
Rectangular to polar:
\[ r^2 = x^2 + y^2, \qquad \tan\theta = \frac{y}{x} \quad (x \neq 0) \]
Source: Stewart 7e, Section 8.1 Polar Coordinates.
When converting back, the equation tan theta = y / x does not by itself fix theta, because two angles a half-turn apart share the same tangent. Pick theta in the quadrant where (x, y) actually lies. Dropping that step is the common error: arctan on a calculator always returns an angle in the right half-plane, so a point in the second or third quadrant needs pi added to the calculator value.
Worked Example
Convert the polar point (4, 2 pi / 3) to rectangular coordinates.
Apply x = r cos theta and y = r sin theta with r = 4 and theta = 2 pi / 3.
Show the answer
The angle 2 pi / 3 is in the second quadrant, where cosine is negative and sine is positive:
\[ \cos\frac{2\pi}{3} = -\frac{1}{2}, \qquad \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \]
So:
\[ x = 4 \cos\frac{2\pi}{3} = 4 \left(-\frac{1}{2}\right) = -2 \]
\[ y = 4 \sin\frac{2\pi}{3} = 4 \left(\frac{\sqrt{3}}{2}\right) = 2\sqrt{3} \]
The rectangular point is (-2, 2 sqrt 3). The negative x and positive y agree with a second-quadrant location, which is a quick sanity check on the signs.
Connections
This builds on reading sine and cosine off the unit circle, since the conversion formulas are the unit-circle coordinates scaled by r, and on the distance formula, which gives r as the distance from the origin. It leads into graphing polar equations, where r is written as a function of theta to trace curves like cardioids and roses, and into the trigonometric form of complex numbers, where a complex number is plotted in the plane and described by its modulus and argument exactly the way a point is described by r and theta.
Resources
- Primary: Stewart, Redlin, Watson, Precalculus 7e, Section 8.1 Polar Coordinates.
- Alternate: OpenStax Precalculus 2e, Section 10.3 Polar Coordinates.
Last updated: 2026-06-24