Problem 16 - Precalculus

Question

Graph the polar equations. $$r=1+2 \cos \theta$$

Step-by-Step Solution

Answer

The graph is a limaçon with an inner loop, symmetric about the polar axis.

Step 1: Understand the Equation

The given polar equation is \( r = 1 + 2 \cos \theta \). In polar coordinates, \( r \) represents the distance from the origin (pole) to a point, and \( \theta \) is the angle from the positive x-axis. The cosine function indicates that the graph will be symmetric about the polar axis.

Step 2: Identify the Type of Polar Graph

The equation \( r = 1 + 2 \cos \theta \) matches the format of a limaçon with an inner loop since \( a = 1 \) and \( b = 2 \), where \( a < b \). This type of limaçon will have a loop inside.

Step 3: Determine Key Values of \( \theta \)

Evaluate the equation for specific values of \( \theta \). When \( \theta = 0 \), \( r = 1 + 2 \times 1 = 3 \); when \( \theta = \pi \), \( r = 1 + 2 \times (-1) = -1 \) (representing a loop); when \( \theta = \frac{\pi}{2} \) or \( \theta = \frac{3\pi}{2} \), \( r = 1 \).

Step 4: Plot Key Points

Plot these critical points: (3, 0), (1, \( \frac{\pi}{2} \)), (1, \( \frac{3\pi}{2} \)), and (2, \( \pi \)). The negative value at \( \theta = \pi \) indicates that the point is actually on the opposite side of the pole.

Step 5: Draw the Limaçon

Connect the points thoughtfully to represent the shape of a limaçon with an inner loop. The graph should show a large loop extending to \( r = 3 \) and a small inner loop crossing the origin due to the negative value of \( r \) when \( \theta = \pi \).

Step 6: Verify Symmetry

Check the graph for symmetry about the polar axis (i.e., the x-axis). The equation is even in \( \theta \), confirming that the graph should reflect symmetrically along this axis, supporting the limaçon shape.

Key Concepts

Limaçon

A limaçon is a type of curve represented by polar equations of the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). These curves are very distinct in shape, and understanding them can be quite fascinating.

• Forms of Limaçons: A limaçon can either have an inner loop, be dimpled, a cardioid, or be convex. This depends on the relationship between \( a \) and \( b \).
  - Inner Loop: This occurs when \( a < b \). Imagine a smaller loop inside, like \( r = 1 + 2 \cos \theta \), where the small loop creates a fascinating intersection.
  - Cardioid: If \( a = b \), it forms a heart-like shape.
  - Dimpled Limaçon: If \( a > b \) and \( a < 2b \), there is a slight indentation.
  - Convex Limaçon: When \( a \geq 2b \), the limaçon looks like a flattened circle.

Remember, identifying the type of limaçon can help you sketch it accurately. Each variation tells a different story in its reflection and symmetry.

Polar Coordinates

Polar coordinates provide an alternative to the Cartesian coordinate system, capturing points based on their distance from the origin and the angle from the positive x-axis. This offers a different way to view graph-related problems.

• Components of Polar Coordinates:
  - The point \( (r, \theta) \) denotes:
  - \( r \): The radial distance from the origin (also known as the pole).
  - \( \theta \): The angular direction from the positive x-axis.

Using polar coordinates can simplify certain equations and reveal beautiful symmetries not easily seen in Cartesian forms. This is particularly evident in graphs with rotary symmetry, like curves ridding on trigonometric identities.
In essence, polar coordinates shine light on a plane's geometry that isn't always visible otherwise. It's about seeing circles, curves, and loops in a nuanced light.

Graph Symmetry

Graph symmetry in polar equations can help simplify graphing and understanding the underlying properties of the curves. Especially for equations related to trigonometric functions, symmetry can be quite revealing.

• Types of Symmetry: In polar graphs, symmetry can often be observed around specific axes.
  - Polar Axis Symmetry: A graph symmetric about the polar axis usually involves cosine functions, where reflecting over the x-axis preserves the figure.
  - Line \( \theta = \frac{\pi}{2} \): This occurs with sine functions, showing vertical reflection symmetry.
  - Origin Symmetry: Some graphs demonstrate symmetry through inversion around the pole.

In the case of the equation \( r = 1 + 2 \cos \theta \), it's symmetric about the polar axis, reflecting a hallmark of cosine-based polar equations. Recognizing this can help predict and verify graph forms more easily, ensuring accuracy in sketching and interpretation.