Problem 32 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=-3 \cos \theta$$

Step-by-Step Solution

Answer

The curve is a circle centered at (-1.5, 0) with a diameter of 3.

Step 1: Understanding the Polar Equation

We start by understanding the given polar equation \( r = -3 \cos \theta \). This is in polar form where \( r \) represents the radius, and \( \theta \) is the angle. A negative sign in front of \( \cos \theta \) indicates reflection in polar coordinates.

Step 2: Analyzing the Symmetry and Shape

The presence of \( \cos \theta \) in the equation suggests symmetry about the horizontal axis (polar axis). Because \( r = -3 \cos \theta \) involves \( \cos \theta \), this is typically a limaçon. For if \( r = a + b\cos(\theta) \) where \( a = 0 \) and \( b < 0 \), it is a circle with a center shifted to the left.

Step 3: Determine Key Points

Calculating key points will help sketch the curve. For \( \theta = 0 \), \( r = -3 \cos(0) = -3 \). For \( \theta = \pi \), \( r = -3 \cos(\pi) = 3 \). For other angles like \( \theta = \frac{\pi}{2} \), and \( \theta = \frac{3\pi}{2} \) we find \( r = 0 \). These values highlight the transition of the radius as \( \theta \) varies.

Step 4: Sketch the Curve

Using the points from Step 3, plot the points on the polar graph. With \( r = -3 \) at \( \theta = 0 \), the point is opposite in direction, so at \( \theta = \pi \), \( r = 3 \). Plot the point at (3, \( \pi \)) and likewise for others, forming a circle centered at (-1.5, 0) in cartesian form, resulting in diameter 3.

Key Concepts

Polar Equations

Polar equations express the relationship between the radial distance \( r \) from the origin and the angle \( \theta \) from a fixed direction, often the positive x-axis in a plane. Unlike Cartesian coordinates which use \( x \) and \( y \), polar coordinates offer a convenient way to describe curves and circles in terms of direction and distance.
Polar coordinates are especially useful when dealing with circular or spiral patterns. The equation \( r = -3 \cos \theta \) is an example of a polar equation. In this, \( r \) can be positive or negative. A negative \( r \) implies moving in the opposite direction of \( \theta \), essentially reflecting over the origin.
This form lends itself to symmetry and rotation, offering a unique way to understand the distances and angles without confinement to straight lines. In polar equations, trigonometric functions define how \( r \) changes with \( \theta \). Modeling circles becomes intuitive as values vary sinusoidally over \( \theta \):

• \( \cos \theta \) typically reflects circles symmetric across the horizontal axis.
• \( \sin \theta \) symmetry crosses the vertical axis.

Limaçon Curves

The limaçon is a particular family of polar curves and comes into play with equations of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). It's named after the French word for "snail" due to its distinctive shape.
When \( a = 0 \) and \( b eq 0 \), we typically deal with circular variations, however, the entire graph is still considered a limaçon due to its formation principles. For example, the equation we examined, \( r = -3 \cos \theta \), essentially forms a circle with a shifted center.
Limaçon curves might appear as:

• Simple limacons (dimpled configurations where one end is indented).
• Inner-loop limacons, forming a loop within themselves.
• Cardioids, when \( a = b \), resembling a heart shape.

This variety offers insight into polar geometry, vividly illustrating the interplay between linear constants and angles, affecting the graph's bulges and loops.

Symmetry in Polar Graphs

Symmetry in polar graphs is a fascinating characteristic that helps simplify complex calculations and predict how polar curves behave.
A polar graph like \( r = -3 \cos \theta \) reveals symmetry across the polar axis, which is analogous to the horizontal axis in Cartesian coordinates.
Here are some key observations:

• **Reflection over Polar Axis (Horizontal Axis):** Graphs involving functions like \( r = a \cos \theta \) generally show mirror symmetry about the polar axis.
• **Reflection over Vertical Line:** If \( \sin \theta \) is predominant, the graph is typically symmetric about the vertical line \( \theta = \frac{\pi}{2} \).
• **Origin Symmetry:** Some graphs also show origin symmetry, where \( (r, \theta) \) translates to \((-r, \theta + \pi) \).

This symmetrical property allows graphing with fewer errors by observing patterns and relationships without recalculating every point.
Understanding these symmetries reduces complexity in graphing, offering a shortcut to visualize and sketch polar curves effectively.