Learning Outcomes
- Convert points and equations between polar and Cartesian coordinates and equations.
Section 15.4 Polar Coordinates (CO4)
Subsection 15.4.1 Activities
Fact 15.22.
“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).
Activity 15.23.
(a)
Plot the Cartesian point \(P=(x,y)=(\sqrt{3},-1)\) and draw line segments connecting the origin to \(P\text{,}\) the origin to \((x,y)=(\sqrt{3},0)\text{,}\) and \(P\) to \((x,y)=(\sqrt{3},0)\text{.}\)
(b)
Solve the triangle formed by the line segments you just drew (i.e. find the lengths of all sides and the measures of each angle).
(c)
Find all polar coordinates for the Cartesian point \((x,y)=(\sqrt{3},-1)\text{.}\)
(d)
Find Cartesian coordinates for the polar point \((r,\theta)=\left(-\sqrt{2},\displaystyle\frac{3\pi}{4}\right)\text{.}\)
Activity 15.24.
Graph each of the following.
(a)
\(r=1\)
(b)
\(r=-1\)
(c)
\(\theta=\displaystyle\frac{\pi}{6}\)
(d)
\(\theta=\displaystyle\frac{7\pi}{6}\)
(e)
\(\theta=\displaystyle\frac{-5\pi}{6}\)
(f)
\(1\leq r < -1\text{,}\) \(0\leq\theta\leq\displaystyle\frac{\pi}{2}\)
(g)
\(-3\leq r \leq 2\text{,}\) \(\theta=\displaystyle\frac{\pi}{4}\)
(h)
\(r \leq 0\text{,}\) \(\theta=\displaystyle\frac{-\pi}{2}\)
(i)
\(\displaystyle\frac{2\pi}{3}\leq\theta\leq\displaystyle\frac{5\pi}{6}\)
(j)
\(r=3\sec(\theta)\)
Fact 15.25.
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.
Fact 15.26.
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.
Fact 15.27.
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.
Activity 15.28.
(a)
Find a polar form of the the Cartesian equation \(x^2+(y-3)^2=9\text{.}\)
Activity 15.29.
Find a Cartesian form of each of the given polar equations.
(a)
\(r^2=4r\cos(\theta)\)
(b)
\(r=\displaystyle\frac{4}{2\cos(\theta)-\sin(\theta)}\)
Subsection 15.4.2 Videos
Subsection 15.4.3 Exercises
Exercises available at
https://tbil.org/preview/calculus/exercises/#/bank/CO4/
