Problem 11 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=2 \sec \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 2\sec \theta\) is made up of two 'C'-shaped parts, originating from the main properties of the secant function. This graph is symmetrical about the x-axis.

Step 1: Understand the behavior of the Secant function

The secant function sec( \(\theta )\), is the reciprocal of the cosine function, or \( \sec \theta = 1/\cos \theta \). The cosine function gives an output of '1' at \(\theta = 0\) and \(\theta = 2\pi\), as well as '-1' at \(\theta = \pi\). Therefore, \(\sec \theta \) will also give an output of '1' at \(\theta = 0\) and \(\theta = 2\pi\), but '-1' at \(\theta = \pi\). When the cosine function equals zero, the secant function will be undefined. That's the case for \(\theta= \pi /2\) and \(\theta= -\pi /2\) .

Step 2: Conversion between polar coordinates and Cartesian coordinates

To sketch the graph, it might be easier to convert the plot into Cartesian coordinates. The conversion formulas from polar coordinates (\(r, \theta )\)) to Cartesian coordinates (\(x, y)\) can be used here. \(x = r\cos \theta\) and \(y = r \sin \theta\). Here, \(y = r \sin \theta = 2\sec\theta \sin \theta = 2 \sin \theta / \cos \theta = 2 \tan \theta\). Therefore, the plot in Cartesian coordinates will represent \(y = 2 \tan \theta\).

Step 3: Sketch the Graph

The graph can be sketched by plotting the main points computed in Step 1: A point at the tip of the 'petal' when \( \theta = 0\) and \( \theta = 2\pi \) i.e, at \( r = 2\), and two undefined points at \( \theta = +\pi/2 \) and \( \theta = -\pi/2 \). Sketch a curve resembling a 'C' that starts from the point \( r = 2 \) at \( \theta = 0\) and ends at \( r = 2\) at \( \theta = 2\pi\) . Mirror this 'C' to sketch the other part of the graph. So, the graph has two 'C' shaped distinct sections due to the undefined points of secant function.

Key Concepts

Secant Function and its Characteristics

The secant function, denoted as \( \sec \theta \), is closely related to the cosine function. Essentially, it is the reciprocal of cosine, which means \( \sec \theta = \frac{1}{\cos \theta} \). This relationship gives the secant function several distinct features that are important to understand when working with polar equations.

The cosine function has its

• maximum value of '1' at \( \theta = 0 \) and \( \theta = 2\pi \)
• minimum value of '-1' at \( \theta = \pi \)
• values undefined or infinite at angles where cosine is zero, such as \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).

This means that the secant function \( \sec \theta \):

• takes a value of '1' at \( \theta = 0 \) and \( \theta = 2\pi \)
• takes a value of '-1' at \( \theta = \pi \)
• becomes undefined at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \)

The key point is understanding where the function becomes undefined, which will help in graph interpretation.

When plotting polar graphs, knowing these characteristics will aid in understanding the coordinates derived from polar equations that involve \( \sec \theta \). This ensures clear graph sketching and analysis of features like asymptotes.

Converting Polar to Cartesian Coordinates

The process of converting polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\) is often necessary to make graph sketching easier, especially for equations involving trigonometric functions like secant. Here's how the conversion works:

In the given polar equation \( r = 2 \sec \theta \), you can find the Cartesian equivalent by using the conversion formulas.

• \( x = r \cos \theta = 2 \sec \theta \cos \theta = 2 \)
• \( y = r \sin \theta = 2 \sec \theta \sin \theta \)
• This simplifies further to \( y = 2 \tan \theta \)

So, the Cartesian form of the equation becomes \( y = 2 \tan \theta \).

Understanding these conversions helps in graphing the equations on Cartesian coordinates, which can sometimes be more intuitive than polar coordinates for visualizing functions like \( \tan \theta \) or \( \sec \theta \). It provides insight into linear relationships and undefined points, simplifying the interpretation of the graph.

Graph Sketching of Polar Equations

Sketching the graph of \( r = 2 \sec \theta \) requires a combination of understanding the secant function's behavior and performing polar to Cartesian conversion. Here are simple steps to follow:

First, identify the key points derived from the secant function.

• At \( \theta = 0 \) and \( \theta = 2\pi \), \( r = 2 \). The graph starts and ends at these positions.
• At \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \), the function is undefined. These are where asymptotes would be found in a Cartesian context.

Now, use these points to form the graph.

Start with a 'C' shaped curve:
- Begin at \( \theta = 0 \) with \( r = 2 \).
- The curve will move symmetrically and stop back again at \( \theta = 2\pi \) with \( r = 2 \).

Finally, mirror the curve to form two 'C' shaped sections reflecting the effect of the undefined outputs of the secant function. This symmetry highlights the limitations of the secant function and results in an elegant yet descriptive plot of the polar equation. When performed correctly, graph sketching provides a clear visual representation of the mathematical relationships within the equation.