The Polar Coordinate System (College Board AP® Precalculus): Revision Note

Polar coordinates

What are polar coordinates?

• The polar coordinate system is an alternative to the rectangular (Cartesian) coordinate system for locating points in the plane

• It is built on a grid of

  • circles centered at the origin

  • and lines passing through the origin

• The positive $x$-axis is called the polar axis

• A point in the polar coordinate system is described by an ordered pair $\mathbf{(}\mathit{r}\mathbf{,}\mathbf{ }\mathit{\theta }\mathbf{)}$

  • $r$ is the radius

    • This is a signed value related to the distance of the point from the origin

      • It can be positive or negative (see the next section below for details)

    • The numbers on the above grid (1, 2, 3, 4, 5) correspond to distances $r$ from the origin

  • $\theta$ is the angle in standard position whose terminal ray includes the point

    • Remember that angles are usually measured in radians in AP® Precalculus

• Values of $\mathit{\theta }$ for the radial lines on the grid can vary just as the same angle in standard position can be represented by different values

  • The radial lines on the grid above divide the circle into 24 parts

    • so each line is $\frac{2\pi }{24}=\frac{\pi }{12}$ radians away from the ones on either side

  • Starting from zero at the polar axis, the $\theta$ values

    • increase counter-clockwise

      • $0,\frac{\pi }{12},\frac{2\pi }{12}=\frac{\pi }{6},\frac{3\pi }{12}=\frac{\pi }{4},$ etc.

      • up to $2\pi$ (which is in the same position as 0)

    • and decrease clockwise

      • $0,-\frac{\pi }{12},-\frac{2\pi }{12}=-\frac{\pi }{6},-\frac{3\pi }{12}=-\frac{\pi }{4},$ etc.

      • down to $-2\pi$ (which is in the same position as 0)

• These angular values can also go beyond $2\pi$ or $-2\pi$

  • E.g. continuing to go counter-clockwise from $2\pi$

    • the $\frac{\pi }{12}$ line becomes $\frac{\pi }{12}+2\pi =\frac{25\pi }{12}$

    • the $\frac{\pi }{6}$ line becomes $\frac{\pi }{6}+2\pi =\frac{13\pi }{6}$

    • etc.

  • Or continuing to go clockwise from $-2\pi$

    • the $-\frac{\pi }{12}$ line becomes $-\frac{\pi }{12}-2\pi =-\frac{25\pi }{12}$

    • the $-\frac{\pi }{6}$ line becomes $-\frac{\pi }{6}-2\pi =-\frac{13\pi }{6}$

    • etc.

How do I interpret the radius value r?

• For a point with polar coordinates $\mathbf{(}\mathit{r}\mathbf{,}\mathbf{ }\mathit{\theta }\mathbf{)}$

  • Draw the angle $\mathit{\theta }$ in standard position

    • This gives the terminal ray

  • If $\mathit{r}\mathbf{>}\mathbf{0}$, the point lies on the terminal ray

    • at distance $r$ from the origin

  • If $\mathit{r}\mathbf{<}\mathbf{0}$, the point lies on the opposite ray (the ray pointing in the reverse direction)

    • at distance $\left|r\right|$ from the origin

  • If $\mathit{r}\mathbf{=}\mathbf{0}$, the point is at the origin, regardless of the value of $\theta$

• Because of its behavior, $r$ is called a signed radius value

  • It behaves like a coordinate on a number line laid along the terminal ray

  • where the positive direction is the direction of the terminal ray itself

Can the same point have more than one set of polar coordinates?

• Yes, unlike with rectangular coordinates, a single point in the plane can be represented by many different polar coordinate pairs

• This happens for two reasons

  • Adding full revolutions to the angle

    • the coordinates $(r, \theta )$ and $(r, \theta +2\pi )$ represent the same point

    • More generally, $(r, \theta +2\pi k)$ for any integer $k$ gives the same point

  • Using a negative radius

    • the coordinates $(r, \theta )$ and $(-r, \theta +\pi )$ represent the same point

    • Rotating the terminal ray by an extra $\pi$ reverses its direction

      • and making $r$ negative reverses it back

• E.g. the point with polar coordinates $\left(2, \frac{\pi }{3}\right)$ can also be written as

  • $\left(2, \frac{\pi }{3}+2\pi \right)=\left(2,\frac{7\pi }{3}\right)$

  • or as $\left(-2, \frac{\pi }{3}+\pi \right)=\left(-2, \frac{4\pi }{3}\right)$

  • or in infinitely many other ways

How do I convert from polar coordinates to rectangular coordinates?

• Given a point with polar coordinates $(r, \theta )$, the rectangular coordinates $(x, y)$ can be found using

  • $x=r\cos \theta$

  • $y=r\sin \theta$

• These formulas come directly from the right-triangle definitions of sine and cosine

  • and they work for all values of $r$ and $\theta$ (including negative $r$)

• E.g. to convert $\left(4, \frac{2\pi }{3}\right)$ to rectangular coordinates

  • $x=4\cos \frac{2\pi }{3}=4·\left(-\frac{1}{2}\right)=-2$

  • $y=4\sin \frac{2\pi }{3}=4·\frac{\sqrt{3}}{2}=2\sqrt{3}$

  • So the rectangular coordinates are $(-2, 2\sqrt{3})$

How do I convert from rectangular coordinates to polar coordinates?

• Given a point with rectangular coordinates $(x, y)$, the polar coordinates $(r, \theta )$ can be found using:

  • $r=\sqrt{x^2+y^2}$

  • $\theta =\arctan \left(\frac{y}{x}\right)$ for $x>0$

  • $\theta =\arctan \left(\frac{y}{x}\right)+\pi$ for $x<0$

    • The adjustment by $\pi$ when $x<0$ is needed because the $\arctan$ function only returns angles in the interval $\left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$

      • This range corresponds to terminal rays in the right half-plane (Quadrants I and IV)

      • For points in the left half-plane (Quadrants II and III), adding $\pi$ rotates the terminal ray to the correct side

• E.g. to convert $(-3, 3)$ to polar coordinates:

  • $r=\sqrt{(-3{)}^2+3^2}=\sqrt{18}=3\sqrt{2}$

  • Since $x<0$, $\theta =\arctan \left(\frac{3}{-3}\right)+\pi =\arctan (-1)+\pi =-\frac{\pi }{4}+\pi =\frac{3\pi }{4}$

  • So one set of polar coordinates is $\left(3\sqrt{2}, \frac{3\pi }{4}\right)$