Problem 29 - Precalculus with Limits

Question

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=3(1 - \cos\ \theta)\)

Step-by-Step Solution

Answer

The graph of the polar equation \(r=3(1-\cos\theta)\) is a cardioid polar graph that is oriented to the right, starts at \(r=0\) for \(\theta=0\) and \(\theta=2\pi\), and reaches \(r=6\) for \(\theta=\pi\). It is symmetric with respect to the line \(\theta = \pi/2\).

Step 1: Find the zero values of \(r\)

Set the polar equation equal to 0 to find the zero points. This means solving \(3(1 - \cos\ \theta) = 0\) for \(\theta\). This will lead to \(\theta = 0\) or \(\theta = 2\pi\). So, the polar graph touches the pole at \(\theta = 0\) and \(\theta = 2\pi\).

Step 2: Find the maximum value of \(r\)

Because the cos function results in values from -1 to 1, the minimum value of the function \(1 - \cos\ \theta\) is 0 (when \(\cos\ \theta\) is 1) and the maximum is 2 (when \(\cos\ \theta\) is -1). Therefore, \(r\) will range from 0 to \(3\times2 = 6\). \(r\) is maximum when \(\theta = \pi\) and minimum when \(\theta = 0\).

Step 3: Check for symmetry

Let’s verify if the graph is symmetric with respect to the line \(\theta = \pi/2\). If \(r(-\theta)\) is equal to \(r(\theta)\), the graph is symmetric with respect to the line \(\theta = \pi/2\). Substituting \(-\theta\) in place of \(\theta\) in \(r=3(1 - \cos\ \theta)\) gives \(r=3(1 - \cos\ (-\theta))=3(1 - \cos\ \theta)\), which shows that the graph is symmetric with respect to \(\theta = \pi/2\).

Step 4: Sketch the graph

Begin by marking the points found in step 1 (the zeros) and step 2 (the maximum r-values). Sketch the graph, keeping in mind the symmetry found in step 3. The graph starts at the pole at \(\theta = 0\) and increases to a maximum radius of 6 at \(\theta = \pi\), which repeats for \(\theta = 2\pi\). The shape described results in a cardioid polar graph.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that offers an alternative to the more familiar Cartesian (rectangular) coordinates. Instead of x and y, points are described using a distance, usually denoted as 'r,' from a fixed point called the pole (similar to the origin in Cartesian coordinates), and an angle, \( \theta \), measured from a reference line known as the polar axis (comparable to the positive x-axis).

Polar coordinates are especially useful when dealing with problems involving circular or rotational symmetry, where specifying the angle and distance from a central point is more natural than using x and y coordinates. The relationship between Cartesian and polar coordinates can be defined as \( x = r \cos \theta \) and \( y = r \sin \theta \).

Cardioid Polar Graph

The cardioid is a unique and captivating shape in polar graphing characterized by its heart-like appearance. It is defined by equations of the form \( r = a \pm a \cos \theta \) or \( r = a \pm a \sin \theta \), where 'a' is a constant. The curve possesses reflective symmetry around the central axis, the line \( \theta = 0 \) or \( \theta = \pi/2 \) depending on its orientation.

When sketching a cardioid, one typically locates key points such as zeros and maximum values of 'r,' and also assesses for symmetry. The exercise provided involves a polar equation of a cardioid, \( r=3(1 - \cos\ \theta) \), which exhibits symmetry about the vertical line \( \theta = \pi/2 \) and possesses a maximum distance of 6 units from the pole.

Finding Zeros in Polar Equations

Identifying zeros in polar equations is a vital step in sketching polar graphs. Zeros correspond to the points where the radial distance 'r' is zero, which means that these points lie at the pole of the polar coordinate system. To find zeros, you set the polar equation equal to zero and solve for \( \theta \).

In the given example, \( r=3(1 - \cos \theta) = 0 \), solving for \( \theta \) yields the angles where the graph will intersect the pole. Here, \( \theta = 0 \) or \( \theta = 2\pi \), indicating the graph touches or passes through the pole at these angles. Recognizing zeros helps in establishing the starting or ending points of the graph and ascertaining its general shape.

Symmetry in Polar Graphs

Symmetry in polar graphs simplifies the process of sketching the entire graph by considering only a segment of it and then reflecting that segment as necessary. There are three types of symmetry to check for in polar graphs: symmetry about the line \( \theta = 0 \), symmetry about the line \( \theta = \pi/2 \), and symmetry with respect to the pole.

The given cardioid, expressed by \( r=3(1 - \cos\ \theta) \), displays symmetry around \( \theta = \pi/2 \) since \( r(-\theta) = r(\theta) \). After plotting critical points, like the zeros and maximum 'r' values, the symmetry allows us to reflect these points across the axis of symmetry to complete the graph. By harnessing the power of symmetry, we can ensure our graph is both accurate and aesthetically pleasing.