Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given cylindrical or spherical equation. $$ r=3 \cos \theta $$

Step-by-Step Solution

Answer

The graph is a circle centered at \((\frac{3}{2}, 0)\) with radius \(\frac{3}{2}\).

Step 1: Understanding the Equation

The given equation is in polar coordinates: \( r = 3 \cos \theta \). This equation represents a conic section, specifically a circle or an ellipse.

Step 2: Identify Type of Curve

The equation \( r = 3 \cos \theta \) is a form of a limaçon, which simplifies to a circle when the equation takes the form \( r = a + a \cos \theta \). Here, with \( a = 3 \), the equation simplifies to a circle centered at \((\frac{3}{2}, 0)\) with radius \(\frac{3}{2}\).

Step 3: Convert to Cartesian Coordinates

To visualize the graph, convert the polar equation to Cartesian coordinates using the relations \( r = \sqrt{x^2 + y^2} \) and \( x = r \cos \theta \). Applying these gives: \( x = 3 \cos \theta \cdot \cos \theta = 3 - y^2 \). Therefore, the Cartesian form is \((x - \frac{3}{2})^2 + y^2 = (\frac{3}{2})^2\).

Step 4: Plot the Graph

Using the Cartesian equation \((x - \frac{3}{2})^2 + y^2 = (\frac{3}{2})^2\), sketch the circle with center at \((\frac{3}{2}, 0)\) and radius \(\frac{3}{2}\). The circle should be symmetrical around the x-axis and complete.

Step 5: Verify the Plot

Ensure that the graph is correct by checking a few points. For example, substitute \( \theta = 0 \) and \( \theta = \pi \): at \( \theta = 0 \), \( r = 3 \), thus point \((3, 0)\) is valid. At \( \theta = \pi \), \( r = -3 \), which wraps around the origin to the same circle.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They play an essential role in the study of geometry and trigonometry. The primary types of conic sections include:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

These sections are distinguished by the angle and position of the intersecting plane in relation to the cone.
When dealing with equations such as \( r = 3 \cos \theta \), the graph can be a circle or an ellipse in polar coordinates. In this case, it simplifies to a circle, as indicated by its form \( r = a + a \cos \theta \), where \( a = 3 \).
Understanding conic sections aids in recognizing and predicting the shape of graphs based on their equations and parameters, providing insight into their symmetry and orientation.

Cartesian Coordinates

The Cartesian coordinate system allows us to describe the position of points in a plane using two numbers: one for the horizontal axis (x-axis) and one for the vertical axis (y-axis). This method provides a straightforward way to visualize and plot equations that describe geometric shapes.
To convert from polar coordinates, such as \( (r, \theta) \), to Cartesian coordinates, we use specific relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

The conversion of the given polar equation \( r = 3 \cos \theta \) into Cartesian form is an excellent exercise to visualize the problem in another coordinate system. This transformation helps verify and accurately graph the represented shape on familiar axes.

Limaçon

The limaçon is a type of polar graph that appears as a distorted circle. It's a fascinating figure that depends heavily on its equation form, generally written as \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Depending on the coefficients \( a \) and \( b \), a limaçon can:

• Have a loop
• Be heart-shaped (a cardioid)
• Look like a circle
• Resemble a dimpled structure

In the exercise, the equation \( r = 3 \cos \theta \) simplifies to a circle, where both coefficients are equal \( a = b = 3 \). This represents a unique case of the limaçon where the shape becomes perfectly symmetrical, forming a circle centered at \( (\frac{3}{2}, 0) \). Understanding the properties of limaçons is crucial for interpreting more complex polar curves.

Graphing Polar Equations

Graphing polar equations involves plotting points on a plane using a radius and an angle, rather than x and y coordinates. This can provide unique insights and different visual representations of shapes.
To graph an equation like \( r = 3 \cos \theta \), which forms a circle:

• Recognize the symmetry of the equation, in this case about the x-axis.
• Plot critical points such as when \( \theta = 0 \) and \( \theta = \pi \).
• Understand that negative values of \( r \), like \( \theta = \pi \), effectively trace the same distance on the opposite side of the origin.

Ensuring an accurate graph offers concrete visual confirmation, aiding comprehension and providing an opportunity to validate theoretical predictions. Graphing in polar coordinates can seem daunting, but by mastering simple transformations, several complex shapes become easier to handle and interpret.