Calculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

“As the crow flies” is an idiom used to describe the most direct path between two points. The polar coordinate system is a useful parametrization of the plane that, rather than describing horizontal and vertical position relative to the origin in the usual way, describes a point in terms of distance from the origin and direction. The origin is also known as the pole (hence polar coordinates).

Let $\overline{OP}$ be a line segment from the origin to a given point $P$ in the plane. The length of $\overline{OP}$ is the distance (or radius) $r$ from the origin to $P\text{.}$ The polar axis is a ray starting at the origin.

To define the "direction" of $P\text{,}$ we form an angle $\theta$ by letting the polar axis serve as the initial ray and $\overrightarrow{OP}$ as the terminal ray. We will set the positive $x$-axis as the polar axis and assume the movement in the positive direction is counter-clockwise (as in trigonometry). Notice that, unlike in the rectangular (or Cartesian) coordinate system, the polar coordinates $(r,\theta )$ for a point are not unique, as we could turn either way to face a given point (or even spin around a number of times before facing that direction).

Furthermore, by allowing $r$ to be negative, we can also "walk backwards" to get to a point by facing in the opposite direction. Rather than the grid lines defined by specific values for $x$ and $y$ in the rectangular coordinate system, specific values of $r$ correspond to circles of radius $r$ centered about the origin, and specific values of $\theta$ correspond to lines going through the pole (called radial lines).

If a polar graph is symmetric about the $x$-axis, then if the point $(r,\theta )$ lies on the graph, then the point $(r,-\theta )$ or $(-r,\pi -\theta )$ also lies on the graph.

If a polar graph is symmetric about the $y$-axis, then if the point $(r,\theta )$ lies on the graph, then the point $(r,\pi -\theta )$ or $(-r,-\theta )$ also lies on the graph.

If a polar graph is rotationally symmetric about the origin, then if the point $(r,\theta )$ lies on the graph, then the point $(-r,\theta )$ or $(r,\pi +\theta )$ also lies on the graph.