11.3 Polar Coordinates and Graphs

Polar coordinates offer a unique way to describe points and curves in a plane. Instead of using x and y, we use distance from a central point and an angle. This system is perfect for circular or spiral shapes.

Graphs in polar coordinates can create beautiful patterns like roses and heart-shaped curves. These shapes are hard to describe with regular x-y coordinates, but they're simple in polar form. It's a powerful tool for certain types of math and science problems.

Polar Coordinate System

Defining the Polar Coordinate System

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• Polar coordinate system represents points in a plane using a distance from the origin (pole) and an angle from the polar axis
• Consists of a pole (origin) and a polar axis (usually the positive x-axis)
• Provides an alternative way to describe points compared to the Cartesian coordinate system ($x$-$y$ plane)
• Useful for certain types of curves and problems involving rotation or periodic behavior

Polar Points and Coordinates

• Polar point represented by an ordered pair ($r$, $\theta$)
  • $r$ represents the radial coordinate or distance from the pole to the point
  • $\theta$ represents the angular coordinate or angle formed between the polar axis and the line segment from the pole to the point
• $r$ can be positive, negative, or zero
  • Positive $r$ values locate the point $r$ units from the pole in the direction of $\theta$
  • Negative $r$ values locate the point $|r|$ units from the pole in the opposite direction of $\theta$
• $\theta$ measured in radians or degrees
  • Positive angles measured counterclockwise from the polar axis
  • Negative angles measured clockwise from the polar axis

Converting Between Coordinate Systems

• Converting from polar coordinates ($r$, $\theta$) to Cartesian coordinates ($x$, $y$):
  • $x = r \cos \theta$
  • $y = r \sin \theta$
• Converting from Cartesian coordinates ($x$, $y$) to polar coordinates ($r$, $\theta$):
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}(\frac{y}{x})$, with quadrant adjustments based on the signs of $x$ and $y$
• Points with the same polar coordinates can have different Cartesian coordinates due to the periodicity of trigonometric functions
  • Example: ($1$, $0$) and ($1$, $2\pi$) have the same Cartesian coordinates ($1$, $0$)

Polar Equations and Graphs

Polar Equations

• Polar equation is an equation that describes a relationship between $r$ and $\theta$
• Defines a set of points that satisfy the equation in the polar coordinate system
• Can be written in the form $r = f(\theta)$ or $F(r, \theta) = 0$
• Examples:
  • $r = 2$ represents a circle with radius $2$ centered at the pole
  • $r = 2\cos\theta$ represents a circle with radius $1$ centered at ($1$, $0$) in Cartesian coordinates

Graphing Polar Equations

• To graph a polar equation, create a table of values for $\theta$ and calculate the corresponding $r$ values
• Plot the points ($r$, $\theta$) in the polar coordinate system
• Connect the points smoothly to form the graph
• May need to consider multiple revolutions of $\theta$ (e.g., $0 \leq \theta \leq 4\pi$) to capture the complete graph
• Some polar equations may have multiple values of $r$ for a given $\theta$, resulting in multiple curves or loops

Symmetry in Polar Graphs

• Polar graphs can exhibit symmetry with respect to the pole or the polar axis
• Symmetry with respect to the pole (origin):
  • If $r(-\theta) = r(\theta)$, the graph is symmetric about the polar axis
  • Example: $r = 2\sin\theta$ is symmetric about the polar axis
• Symmetry with respect to the polar axis:
  • If $r(\theta) = r(\pi - \theta)$, the graph is symmetric about the vertical line $\theta = \frac{\pi}{2}$
  • If $r(\theta) = r(-\theta)$, the graph is symmetric about the horizontal line $\theta = 0$
  • Example: $r = 2\cos(2\theta)$ is symmetric about both the vertical line $\theta = \frac{\pi}{4}$ and the horizontal line $\theta = 0$

Special Polar Curves

Rose Curves

• Rose curves are polar graphs of equations in the form $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$, where $a$ is a non-zero constant and $n$ is a positive integer
• If $n$ is even, the rose curve will have $2n$ petals
• If $n$ is odd, the rose curve will have $n$ petals
• The length of the petals is determined by the absolute value of $a$
• Examples:
  • $r = 2\cos(3\theta)$ is a three-petaled rose curve
  • $r = 3\sin(4\theta)$ is an eight-petaled rose curve

Limaçons

• Limaçons are polar graphs of equations in the form $r = b \pm a\cos\theta$ or $r = b \pm a\sin\theta$, where $a$ and $b$ are non-zero constants
• The shape of the limaçon depends on the relative values of $a$ and $b$:
  • If $|a| < |b|$, the graph is a single loop (limaçon with an inner loop)
  • If $|a| = |b|$, the graph is a cardioid (heart-shaped curve)
  • If $|a| > |b|$, the graph is a limaçon with an outer loop
• Examples:
  • $r = 2 + 3\cos\theta$ is a limaçon with an inner loop
  • $r = 4 - 2\sin\theta$ is a limaçon with an outer loop

Cardioids

• Cardioids are special cases of limaçons where $|a| = |b|$
• Polar equations of cardioids have the form $r = a \pm a\cos\theta$ or $r = a \pm a\sin\theta$, where $a$ is a non-zero constant
• The curve is heart-shaped with a cusp (sharp point) at the pole
• The size of the cardioid is determined by the absolute value of $a$
• Examples:
  • $r = 3 + 3\cos\theta$ is a cardioid with a cusp at the pole and a maximum diameter of $6$ units
  • $r = 2 - 2\sin\theta$ is a cardioid with a cusp at the pole and a maximum diameter of $4$ units