Polar coordinates offer a different way to describe points in a plane using distance and angle. This system is especially useful for circular or spiral shapes. Understanding how to switch between polar and Cartesian coordinates is key for solving certain types of problems.

In this part of the chapter, we'll learn how to use polar coordinates and transform between systems. This knowledge sets us up for tackling double integrals in polar form, which can simplify calculations for circular regions.

Polar Coordinate System

Defining Polar Coordinates

• Polar coordinates represent points in a two-dimensional plane using a radius $r$ and an angle $\theta$
• The radius $r$ measures the distance from the origin (pole) to the point
• The angle $\theta$ measures the counterclockwise rotation from the positive x-axis to the line segment connecting the origin to the point
• Polar coordinates are denoted as $(r, \theta)$ where $r \geq 0$ and $\theta$ can take any real value (radians or degrees)

Relationship to Cartesian Coordinates

• Polar coordinates can be converted to Cartesian coordinates $(x, y)$ using the following equations:
  • $x = r \cos(\theta)$
  • $y = r \sin(\theta)$
• Conversely, Cartesian coordinates can be converted to polar coordinates using:
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}(\frac{y}{x})$ (principal value) or $\theta = \tan^{-1}(\frac{y}{x}) + \pi$ (if $x < 0$)
• The origin in polar coordinates is represented as $(0, \theta)$ for any value of $\theta$

Coordinate Transformation

Polar to Cartesian Transformation

• To transform a point from polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$, use the equations:
  • $x = r \cos(\theta)$
  • $y = r \sin(\theta)$
• Example: Transform the polar point $(3, \frac{\pi}{4})$ to Cartesian coordinates
  • $x = 3 \cos(\frac{\pi}{4}) \approx 2.12$
  • $y = 3 \sin(\frac{\pi}{4}) \approx 2.12$
  • The Cartesian coordinates are approximately $(2.12, 2.12)$

Cartesian to Polar Transformation

• To transform a point from Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$, use the equations:
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}(\frac{y}{x})$ (principal value) or $\theta = \tan^{-1}(\frac{y}{x}) + \pi$ (if $x < 0$)
• Example: Transform the Cartesian point $(-1, 1)$ to polar coordinates
  • $r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$
  • $\theta = \tan^{-1}(\frac{1}{-1}) + \pi \approx \frac{3\pi}{4}$
  • The polar coordinates are $(\sqrt{2}, \frac{3\pi}{4})$

Graphing in Polar Coordinates

Polar Equations

• A polar equation is an equation in terms of $r$ and $\theta$ that describes a curve in the polar coordinate system
• The general form of a polar equation is $r = f(\theta)$, where $f$ is a function of $\theta$
• To graph a polar equation, create a table of values for $\theta$ (usually in the interval $[0, 2\pi]$ or $[0, \pi]$) and calculate the corresponding $r$ values using the equation

Graphing Techniques

• To graph a polar equation, plot the points $(r, \theta)$ in the polar coordinate system
• Connect the plotted points with a smooth curve
• Identify any symmetries in the graph, such as rotational symmetry or symmetry about the polar axis
• Example: Graph the polar equation $r = 2 \cos(3\theta)$
  • Create a table of values for $\theta$ in the interval $[0, 2\pi]$ and calculate the corresponding $r$ values
  • Plot the points $(r, \theta)$ in the polar coordinate system
  • Connect the plotted points with a smooth curve
  • The resulting graph is a three-leaved rose curve with rotational symmetry

Special Polar Graphs

• Some common polar graphs include:
  • Cardioid: $r = a(1 + \cos(\theta))$ or $r = a(1 - \cos(\theta))$
  • Rose curves: $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$, where $n$ is an integer
  • Limaçon: $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$
  • Lemniscate: $r^2 = a^2 \cos(2\theta)$
• Recognize the equations and characteristics of these special polar graphs to quickly identify them