10.4 Polar Coordinates: Graphs

Polar coordinates offer a unique way to describe points and graph equations in mathematics. Instead of using x and y, we use distance from the origin (r) and angle (θ). This system is particularly useful for certain types of curves that are difficult to represent in rectangular coordinates.

Graphing polar equations involves creating a table of r and θ values, plotting points, and connecting them smoothly. Common polar curves include cardioids, limaçons, and rose curves. Understanding how to convert between polar and rectangular forms helps in analyzing these graphs and their properties.

Graphing Polar Equations

Introduction to Polar Coordinates

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• The polar coordinate system uses a polar plane to represent points
• Points are described using two components:
  • Radial distance (r): distance from the origin (pole)
  • Angular coordinate (θ): angle from the positive x-axis

Graphing polar equations

• Create a table of $\theta$ and $r$ values to graph a polar equation $r=f(\theta)$
  • Select $\theta$ values from $0$ to $2\pi$, usually in increments of $\frac{\pi}{6}$ or $\frac{\pi}{4}$ ($\frac{\pi}{4}$ represents 45°)
  • Use the given equation to calculate the corresponding $r$ values for each $\theta$ value
• Plot the points $(r,\theta)$ on the polar grid
  • $\theta$ determines the direction from the pole (origin) (0° points directly to the right)
  • $r$ determines the distance from the pole (2 units from the pole at a 30° angle)
• Smoothly connect the plotted points to create the graph (polar curve)
• Use symmetry properties to graph the curve more efficiently
  • The graph is symmetric about the polar axis if $f(-\theta)=f(\theta)$
  • The graph is symmetric about the pole if $f(\theta+\pi)=f(\theta)$
  • The graph is symmetric about the $\theta=\frac{\pi}{2}$ line if $f(-\theta)=-f(\theta)$ ($\theta=\frac{\pi}{2}$ represents the vertical line)

Common polar curve types

• Cardioids have the general form $r=a(1\pm\cos\theta)$ or $r=a(1\pm\sin\theta)$
  • The graph resembles a heart shape with a dimple or indentation
  • The orientation of the dimple depends on the sign inside the parentheses (plus sign dimple faces the pole)
• Limaçons have the general form $r=a\pm [b](https://www.fiveableKeyTerm:b)\cos\theta$ or $r=a\pm b\sin\theta$, where $a,b>0$
  • The graph is a single loop with an inner loop if $a>b$ (inner loop is larger when $a$ is much greater than $b$)
  • The graph is a cardioid if $a=b$
  • The graph is a single loop with a dimple if $a<b$ (dimple is more pronounced when $b$ is much greater than $a$)
• Rose curves have the general form $r=a\cos(n\theta)$ or $r=a\sin(n\theta)$, where $n$ is a positive integer
  • The graph has $n$ petals if $n$ is odd (3 petals for $n=3$)
  • The graph has $2n$ petals if $n$ is even (8 petals for $n=4$)
  • $|a|$ determines the maximum radius of the petals (petals extend 5 units from the pole if $|a|=5$)

Polar to rectangular conversion

• Convert from polar to rectangular form:
  1. Replace $r\cos\theta$ with $x$ and $r\sin\theta$ with $y$
  2. Simplify the equation after substitution
• Convert from rectangular to polar form:
  1. Replace $x^2+y^2$ with $r^2$ and $\frac{y}{x}$ with $\tan\theta$ (or use $\theta=\arctan(\frac{y}{x})$)
  2. Simplify the equation after substitution
• Converting between forms helps identify key graph features
  • Rectangular form equations show symmetry about the $x$-axis, $y$-axis, or origin
  • Polar form equations show symmetry about the polar axis, the pole, or the $\theta=\frac{\pi}{2}$ line
• Analyzing the domain and range in both forms provides a comprehensive understanding of the graph's behavior
• Parametric equations can be used to represent polar curves, with r and θ as parameters