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Calculus 3 (Math 221) Notes and Study Guide

Brenton LeMesurier

Section 1.3 Polar Coordinates

References.

Defining Polar Coordinates.

Polar coordinates describe the location of a point \(P\) in the plane in terms of
  • the polar distance \(r\) from a reference point \(O\text{,}\) the pole, and
  • the polar angle \(\theta\text{,}\) describing the direction of motion from \(O\) to \(P\) relative to a direction considered horizontal.
In comparison to standard cartesian coordinates \((x,y)\) for \(P\text{,}\) using the pole as the origin and with the positive \(x\)-axis as the horizontal direction, the polar distance is easily described:
\begin{equation} r = |OP| = \sqrt{x^2+y^2}.\tag{1.3.1} \end{equation}
The angle is measured from the positive \(x\)-axis to the ray \(\overrightarrow{OP}\text{,}\) going in the direction towards the positive y-axis (so “anti-clockwise”).
It is not so simple to give a formula for it in terms of \(x\) and \(y\text{,}\) so it helps to go the other way first:

Cartesian Coordinate Values from Polar Coordinate Values.

The point \(P\) with cartesian coordinates \((x,y)\) lines an the circle of radius \(r\text{,}\) center \((0,0)\text{,}\) so the angle \(\theta\) determines its coordinates to be
\begin{equation} x = r \cos \theta, \; y = r \sin \theta.\tag{1.3.2} \end{equation}

Polar Coordinate Values from Cartesian Coordinate Values.

Getting \(r\) from \(x\) and \(y\) was easy. To get \(\theta\text{,}\) first divide, getting \(\ds\tan \theta = \frac{y}{x}, \; \text{ if } x \neq 0\)
There are two problems: the case \(x=0\) and the multiple angles with the same tangent.
We can of course restrict the allowed \(\theta\) values to “one rotation”; two favorite choices are
  • \(-\pi < \theta \leq \pi\) to keep the size of \(\theta\) small (with a bias to positive values) and
  • \(0 \leq \theta < 2\pi\) to keep \(\theta\) positive and still as small as possible.
However that still leaves two possible values for \(\theta\text{,}\) differing by \(\pi\text{,}\) and using \(\theta = \arctan(y/x)\) does not always give the correct value: it always gives an angle \(-\pi/2 < \theta < \pi/2\) and so a point in the right half-plane.

Polar Coordinate Values from Cartesian Coordinate Values: A Solution.

Often we prefer the smallest magnitude for \(\theta\text{,}\) and use the value in \((-\pi,\pi]\text{.}\) (Excluding \(\theta=-\pi\) as it would be redundant.) Here is one way to do that:
\begin{equation} \theta = \left\{ \begin{array}{rl} \arctan(y/x) &\text{ if }x > 0\text{, so }-\pi/2 < \theta < \pi/2 \\ \arctan(y/x) - \pi &\text{ if }x < 0\text{ and }y < 0\text{, so }-\pi < \theta < -\pi/2 \\ \arctan(y/x) + \pi & \text{ if }x < 0\text{ and }y \geq 0\text{, so }\pi/2 < \theta \leq \pi \\ \pi/2 &\text{ if }x = 0\text{ and }y > 0 \\ -\pi/2 &\text{ if }x = 0\text{ and }y < 0 \end{array} \right.\tag{1.3.3} \end{equation}
This still omits the case of the pole, where \(x=y=0\) (so \(r=0\)): there the polar angle is ill-defined, but the good news is that any value of \(\theta\) is acceptable, in that the equations (1.3.2) give the correct cartesian coordinates.

Simpler Equations for Getting from Cartesian to Polar Coordinates.

Often we can avoid these complications by working with the following simpler equations whenever possible:
\begin{equation} r^2 = x^2 + y^2, \quad \tan \theta = \frac{y}{x}\text{ when }x \neq 0,\tag{1.3.4} \end{equation}
with special handling of points with \(x=0\text{:}\) these are on the \(y\)-axis, so we can use
  • \(\theta = \pi/2\) if \(y > 0\)
  • \(\theta = -\pi/2\) or \(3\pi/2\) if \(y < 0\)
  • any \(\theta\) value we want if also \(y=0\text{,}\) so we are at the origin where the polar angle makes no sense.
Also, since both these equations (1.3.4) and Equations (1.3.2) giving cartesian coordinates in terms of polar coordinates make sense for all real values of \(r\) and \(\theta\text{,}\) we sometimes do not restrict to \(r \geq 0\text{,}\) \(-\pi < \theta \leq \pi\text{.}\)
This more flexible approach will help below to produce elegant descriptions of some interesting Polar Curves.

Polar Curves.

Many curves have rotational features that make them easiest to describe in terms of polar coordinates, with equations in terms of \(r\) and \(\theta\text{,}\) like
\begin{equation*} r=f(\theta). \end{equation*}
Note well: we will graph them as curves in the cartesian plane, with axes \(x\) and \(y\text{!}\)
The cartesian coordinates are given with the help of the equations (1.3.2) as
\begin{equation} x = f(\theta) \cos\theta, \quad y = f(\theta) \sin\theta\tag{1.3.5} \end{equation}
so that they are a type of parametric curve, with polar angle \(\theta\) as the parameter.
The most basic example is the equation \(r=C\text{,}\) for \(C\) a positive constant. This is a circle of radius \(C\text{.}\) Since the equation says nothing about the angle \(\theta\text{,}\) it can take any value, and with the standard range of values \((-\pi,\pi]\text{,}\) the angle \(\theta\) becomes a convenient parameter giving a parametric description of the curve by inserting \(r=C\) into the Equations (1.3.2):
\begin{equation*} x = C \cos \theta, \; y = C \sin \theta. \end{equation*}
Another example is \(r=e^\theta\text{,}\) which is the exponential spiral
\begin{equation*} x = e^\theta \cos\theta, \quad y =e^\theta \sin\theta. \end{equation*}

Example 1.3.1. The Graph of the Polar Equation \(\theta = c\text{,}\) \(c\) a constant.

The simple equation \(\theta = c\) requires a little more care to graph, and \(r\) must be used as the parameter instead of \(\theta\text{.}\)
Equation (1.3.4) gives \(\tan\theta = \tan c = y/x\text{,}\) so \(y=(\tan c) x\text{.}\) This looks like the equation of a straight line through the origin, and even the cases where \(\tan c\) does not exist make sense: they give the vertical line \(x=0\text{.}\)
However, if we restrict to \(r \geq 0\text{,}\) the curve is actually only part of this line: it is the ray starting at the origin and going in the direction specified by the angle \(c\text{.}\)
The moral is that, as always with graphs and functions, we must specify the domain: do we want to allow all \(r\) (and get a line), or \(r \geq 0\) (and get a ray)? For example, with \(r \geq 0\text{,}\)
  • \(\theta=\pi/2\) is the positive \(y\)-axis,
  • \(\theta=-\pi/2\) is the negative \(y\)-axis.

Tangents to Polar Curves.

Since any polar curve given by an equation \(r=f(\theta)\) is a parametric curve as in Equation (1.3.5), there is nothing really new here, but it is worth noting the formulas for the tangent slope \(dy/dx\text{:}\)
\begin{equation*} \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{\ds\frac{dr}{d\theta}\sin \theta+r\cos\theta}{\ds\frac{dr}{d\theta}\cos\theta-r\sin\theta} \end{equation*}
Evaluation of this is a good case of the strategy I have been using recently, of evaluating key pieces and then gathering them is stages: the term \(dr/d\theta\) should be evaluated and simplified before inserting, to avoid duplicated effort.

Symmetry in Polar Coordinates.

(Omitted)

Study Guide.

Study Calculus Volume 3, Section 1.3
 2 
openstax.org/books/calculus-volume-3/pages/1-3-polar-coordinates
The main content relevant for us is up to Example 13 and Checkpoint 13 (we skip the topic of symmetry, but I suggest reading it).
Do one or several exercises from each of the ranges 136–141, 142–148, 154–157, and 158–160.
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