Introduction to Polar Coordinates, precalculus lesson

A radar screen has no grid of streets. A beam sweeps around a center, and each blip is reported by how far away it is and in which direction. Those two numbers are enough to find any point in the plane. Polar coordinates turn that habit into a coordinate system, and for anything that turns around a center, such as circles, spirals and orbits, they are often simpler than $x$ and $y$ .

Key terms

Pole

Definition The pole is a fixed point $O$ of the plane chosen as the center of a polar coordinate system. Every distance in the system is measured from $O$ , and every angle has its vertex at $O$ . When polar and rectangular coordinates are drawn on the same plane, the pole is placed at the origin $(0, 0)$ .

In plain words The pole is the center of the radar screen. Distances are measured from it and directions turn around it.

Example The points at distance $2$ from the pole form the circle of radius $2$ centered at $O$ . A polar grid is drawn from circles like this one, centered at the pole and spaced one unit apart, together with rays leaving the pole at equal angles.

Polar axis

Definition The polar axis is a fixed ray that starts at the pole. It is the direction of angle $0$ . In the standard setup it is the positive $x$ -axis.

In plain words It is where the radar beam starts before it begins to turn.

Example The point on the polar axis at distance $5$ from the pole has polar coordinates $(5, 0)$ , and in the standard setup it is also the rectangular point $(5, 0)$ . The ray pointing the opposite way, the negative $x$ -axis, is the direction of angle $\pi$ .

Angle

Definition In polar coordinates the angle $\theta$ is a directed angle with its vertex at the pole and its initial side on the polar axis. A positive $\theta$ is a counterclockwise rotation and a negative $\theta$ is a clockwise one, and any real number is allowed. Angles are in radians unless a degree sign is shown. Two angles whose difference is an integer multiple of $2\pi$ , or of $360^\circ$ , are coterminal, which means they have the same terminal side.

In plain words The angle tells you which way to face. Start facing along the polar axis and turn by $\theta$ , to the left for positive angles and to the right for negative ones.

Example The angle $\frac{3\pi}{4}$ , which is $135^\circ$ , points up and to the left, halfway between the positive $y$ -axis and the negative $x$ -axis. The angles $-\frac{\pi}{2}$ and $\frac{3\pi}{2}$ are coterminal, and both point straight down.

Radius

Definition Let $r > 0$ and let $\theta$ be a real number. The point with polar coordinates $(r, \theta)$ is the point on the terminal side of $\theta$ at distance $r$ from the pole. The first coordinate $r$ is the radius, or radial coordinate, and the second coordinate $\theta$ is the angle, or angular coordinate. The order is always distance first and angle second.

In plain words Face the direction $\theta$ , then walk $r$ units straight out from the pole.

Example The point $(3, \frac{\pi}{6})$ lies $3$ units from the pole on the ray at $30^\circ$ above the polar axis. Every point $(3, \theta)$ lies on the circle of radius $3$ around the pole, and changing $\theta$ only slides the point around that circle.

Negative radius

Definition For $r < 0$ , the point $(r, \theta)$ is the point at distance $\lvert r \rvert$ from the pole on the ray opposite the terminal side of $\theta$ . That opposite ray is the terminal side of $\theta + \pi$ , so for all real $r$ and $\theta$ the pairs

(r, \theta) \quad \text{and} \quad (-r, \theta + \pi)

name the same point.

In plain words A negative radius means face the direction $\theta$ and then walk backwards $\lvert r \rvert$ units, through the pole and out the other side.

Example To plot $(-3, \frac{\pi}{6})$ , face the ray at $30^\circ$ and walk $3$ units the opposite way. You land on the ray at $30^\circ + 180^\circ = 210^\circ$ , in the third quadrant. The same point is $(3, \frac{7\pi}{6})$ .

The origin

Definition A radius of $0$ gives the pole, whatever the angle. So $(0, \theta)$ names the origin for every real number $\theta$ . The origin is the only point that has a polar name with every possible angle.

In plain words If you walk zero units, the direction you face makes no difference. You stay at the center.

Example The pairs $(0, 0)$ , $(0, \frac{\pi}{2})$ and $(0, -5)$ all name the origin. This matters later, when you test whether a polar curve passes through the origin. One angle that gives $r = 0$ is enough.

Nonunique coordinates

Definition Let $P$ be a point other than the pole and let $(r, \theta)$ be one pair of polar coordinates for it. Then the pairs that name $P$ are exactly

\begin{aligned} &(r, \theta + 2n\pi) \quad \text{and} \\[0.4em] &(-r, \theta + (2n + 1)\pi), \qquad n \in \mathbb{Z}. \end{aligned}

The pole is named by $(0, \theta)$ for every real $\theta$ . If you require $r > 0$ and $0 \le \theta < 2\pi$ , each point other than the pole gets exactly one name. The range $-\pi < \theta \le \pi$ works the same way, and books use both.

In plain words A full turn brings you back to the same ray, and so does a half turn combined with walking backwards. That is why one point has infinitely many polar names, while its rectangular name is unique.

Example The point $(2, \frac{\pi}{3})$ also has the names

\begin{gathered} \left(2, \frac{7\pi}{3}\right), \quad \left(2, -\frac{5\pi}{3}\right), \\[0.8em] \left(-2, \frac{4\pi}{3}\right), \quad \left(-2, -\frac{2\pi}{3}\right). \end{gathered}

The first two add and subtract $2\pi$ . The last two flip the sign of the radius and add or subtract $\pi$ .

Plotting points

Definition To plot the point $(r, \theta)$ , draw the terminal side of $\theta$ from the pole. If $r > 0$ , mark the point at distance $r$ along that ray. If $r < 0$ , mark the point at distance $\lvert r \rvert$ along the opposite ray. If $r = 0$ , mark the pole.

In plain words Turn first, then walk. On a polar grid, the rays give the direction and the circles give the distance.

Example Here are four points and where each one lands.

Point	Direction faced	Walk	Location
$(3, \frac{\pi}{2})$	straight up	$3$ forward	positive $y$ -axis, $3$ units up
$(2, -\frac{3\pi}{4})$	down and left, at $-135^\circ$	$2$ forward	third quadrant
$(-1, \frac{\pi}{3})$	up and right, at $60^\circ$	$1$ backward	third quadrant, on the ray at $240^\circ$
$(-2, \pi)$	left	$2$ backward	positive $x$ -axis, $2$ units right

The last point is $(2, 0)$ . A negative radius can land you on the polar axis itself.

Check yourself

Where is the point $(-4, \frac{2\pi}{3})$ ? Name its quadrant and give a polar name for it with a positive radius.

Show the answer

Face the ray at $120^\circ$ and walk $4$ units backwards. You land on the ray at $120^\circ + 180^\circ = 300^\circ$ , in the fourth quadrant. A name with a positive radius is $(4, \frac{5\pi}{3})$ .

Give two more names for the point $(5, \frac{\pi}{4})$ , one with $r > 0$ and $-2\pi \le \theta < 0$ , and one with $r < 0$ and $0 \le \theta < 2\pi$ .

Show the answer

Subtracting $2\pi$ from the angle gives $(5, -\frac{7\pi}{4})$ . Flipping the sign of the radius and adding $\pi$ gives $(-5, \frac{5\pi}{4})$ .

Do $(3, \frac{5\pi}{6})$ and $(-3, -\frac{\pi}{6})$ name the same point?

Show the answer

Yes. Flipping the sign of the radius and adding $\pi$ turns $(-3, -\frac{\pi}{6})$ into

\left(3, -\frac{\pi}{6} + \pi\right) = \left(3, \frac{5\pi}{6}\right).

Introduction to Polar Coordinates

Key terms

Pole

Polar axis

Angle

Radius

Negative radius

The origin

Nonunique coordinates

Plotting points

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus