8.4 Polar Coordinates: Graphs

Polar coordinates offer a unique way to describe points and curves using distance and angle. They're especially useful for circular or spiral shapes that are tricky to graph in rectangular coordinates. Understanding symmetry in polar equations helps simplify graphing.

Graphing polar equations involves evaluating r for key θ values and plotting points. Recognizing classic polar curves like cardioids, limaçons, and roses is crucial. These curves pop up in various fields, from mathematics to engineering, making them important to master.

Polar Coordinate System

Symmetry in polar equations

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• Symmetry with respect to the polar axis ($\theta=0$)
  • Equation satisfies $r(\theta) = r(-\theta)$
  • Graph is a mirror image across the polar axis ($\theta=0$ or the positive x-axis)
  • Example: $r = 2 \cos \theta$
• Symmetry with respect to the pole (origin)
  • Equation satisfies $r(\theta) = r(\theta + \pi)$
  • Graph is symmetric about the origin
  • Rotating the graph by $\pi$ radians ($180^\circ$) about the origin results in the same graph
  • Example: $r = 1 + \cos \theta$
• Symmetry with respect to the vertical line $\theta=\frac{\pi}{2}$
  • Equation satisfies $r(\frac{\pi}{2} - \theta) = r(\frac{\pi}{2} + \theta)$
  • Graph is a mirror image across the vertical line $\theta=\frac{\pi}{2}$ (positive y-axis)
  • Example: $r = \sin(2\theta)$
• Symmetry with respect to the horizontal line $\theta=\pi$
  • Equation satisfies $r(\pi - \theta) = r(\pi + \theta)$
  • Graph is a mirror image across the horizontal line $\theta=\pi$ (negative x-axis)
  • Example: $r = 2 \sin \theta$

Graphing techniques for polar equations

• Determine the domain of the polar equation
  • Usually $0 \leq \theta \leq 2\pi$, but may be restricted based on the equation
• Evaluate $r$ for key values of $\theta$
  • Common angles: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2},$ and $2\pi$
  • Substitute these angles into the equation to find corresponding $r$ values
• Plot the points $(r, \theta)$ in the polar coordinate system
  1. Convert $(r, \theta)$ to rectangular coordinates (cartesian coordinates)
     • $x = r \cos \theta$
     • $y = r \sin \theta$
  2. Plot the point $(x, y)$ in the rectangular coordinate system
• Use symmetry properties to complete the graph
  • Reflect plotted points across the axes or lines of symmetry identified earlier
  • Helps to minimize the number of calculations needed

Classic polar curve identification

• Cardioids: $r = a(1 \pm \cos \theta)$
  • Heart-shaped curve
  • Symmetric about the polar axis
  • Example: $r = 2(1 + \cos \theta)$
• Limaçons: $r = a \pm b \cos \theta$ or $r = a \pm b \sin \theta$
  • Inner loop appears when $|b| < |a|$
    • Example: $r = 2 + \cos \theta$
  • Cardioid-like curve appears when $|b| = |a|$
    • Example: $r = 1 + \sin \theta$
  • Dimpled curve appears when $|b| > |a|$
    • Example: $r = 1 + 2\cos \theta$
• Rose curves: $r = a \cos(n\theta)$ or $r = a \sin(n\theta)$
  • $n$ petals if $n$ is odd
    • Example: $r = \cos(3\theta)$ has 3 petals
  • $2n$ petals if $n$ is even
    • Example: $r = \sin(4\theta)$ has 8 petals
  • Symmetric about the polar axis when $n$ is odd
  • Symmetric about the pole when $n$ is even

Additional Concepts in Polar Coordinates

• Polar form and complex plane representation
  • Polar form expresses complex numbers using radial distance and angle
  • Useful for visualizing complex numbers in the complex plane
• Periodic functions in polar coordinates
  • Many polar equations represent periodic functions
  • The period depends on the equation and affects the shape of the graph