Problem 43 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the polar equation. Choose the domain of \(\theta\) to make sure you produce the entire graph. $$r=\cos (\theta / 2)$$

Step-by-Step Solution

Answer

Graph \(r = \cos(\theta/2)\) from \(0\) to \(4\pi\) to capture the complete limacon with an inner loop.

Step 1: Understand the Polar Equation

The given polar equation is \(r = \cos(\theta/2)\). Polar equations relate the radius \(r\) to the angle \(\theta\). In this case, \(r\) is determined by the cosine of half the angle, \(\theta/2\). This equation suggests a symmetry because the cosine function is periodic and can produce complete cycles within specific intervals.

Step 2: Determine the Domain for \(\theta\)

The cosine function has a period of \(2\pi\). However, since we have \(\theta/2\), the period of \(r = \cos(\theta/2)\) becomes \(4\pi\). Thus, to observe a complete graph, \(\theta\) should range from \(0\) to \(4\pi\). This ensures that we capture all possible values of \(r\) one full cycle of the modified cosine function.

Step 3: Graph the Polar Equation

Use a graphing device capable of plotting polar coordinates. Input the equation \(r = \cos(\theta/2)\) with \(\theta\) set to range from \(0\) to \(4\pi\). The graph is typically known as a "limacon" with inner loop, exhibiting symmetry about the polar axis.

Step 4: Analyze the Graph

Once the equation is graphed, you can observe the nature of the limacon. Notice how it loops inward, crossing the pole of the graph. This is typical behavior of polar graphs involving cosine where the angle is divided by a factor.

Key Concepts

Graphing Polar Coordinates

When graphing in polar coordinates, we switch from the conventional Cartesian coordinate system, which is based on \((x, y)\), to focusing on points defined by a radius and an angle - \((r, \theta)\). This method utilizes a circle-like approach where the location of a point is determined by its distance from the origin (the pole) and the angle from the positive x-axis (polar axis).

• Each point in polar coordinates is expressed with the format \((r, \theta)\).
• \(r\) represents the radius, the distance from the pole.
• \(\theta\) represents the angle, measured in radians from the polar axis.

Graphing polar equations requires understanding of how to convert between polar and Cartesian coordinates, if needed. In visualization, using tools capable of plotting polar graphs helps in drawing accurate representations without manual calculations.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent, are fundamental in polar graphing. Their periodic nature plays an important role in determining the shape and symmetry of polar graphs.For the equation \(r = \cos(\theta / 2)\), cosine is the trigonometric function at play.

• The cosine function oscillates between -1 and 1, impacting how \(r\) varies as \(\theta\) changes.
• In this polar equation, the cosine influences the size of \(r\) while maintaining a periodicity defined by the changes in \(\theta\).

As \(\theta\) varies, the cosine function transforms the graph's radius periodically, creating interesting geometric shapes like limacons, circles, or even roses.Visualizing these shapes often requires graphing devices, as the angles and the corresponding radius can form intricate patterns across a full cycle.

Symmetry in Graphs

Symmetry is a crucial aspect in understanding and simplifying the graphing of polar coordinates.Graphs may exhibit symmetry such as symmetry about the polar axis, the pole, or even rotational symmetry.

• Polar Axis Symmetry: If replacing \(\theta\) with \(-\theta\) yields an equivalent equation, then the graph is symmetric about the polar axis.
• Pole Symmetry: If replacing \(r\) with \(-r\) keeps the equation consistent, the graph is symmetric about the pole (origin).
• Rotational Symmetry: A graph can repeat its structure over particular angle increments.

For example, the polar equation \(r = \cos(\theta / 2)\) presents symmetry properties that simplify its understanding and graphing.Typically, cosine-based polar equations reflect symmetry related to their orientation and periodicity across the plane.

Limacon Curve

The limacon is a unique type of polar curve characterized by a notable loop or a dimple depending on its equation configuration.For a limacon like the one described by \(r = \cos(\theta/2)\), certain patterns emerge in its graph:

• Inner Loop: Present when the graph of the function dips below zero, forming a small loop inside the main body of the curve.
• Symmetry: Often exhibits symmetry about the polar axis due to the properties of the cosine function.
• Periodicity: Demonstrates a full cycle when \(\theta\) expands over the defined domain, such as from 0 to \(4\pi\) in this case.

Graphing a limacon involves observing these features, rendering the fascinating curves and loops representative of polar geometries.Understanding the behavior of the curve through its equation enables us to anticipate its geometric form and identify distinctive traits across different configurations.