Problem 24 - Algebra and Trigonometry

Question

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{8}{1+\cos \theta} $$

Step-by-Step Solution

Answer

The viewing window that both captures the entire graph of the polar equation \(r=\frac{8}{1+\cos \theta}\) and minimizes the unused portion of the screen would have theta running from -30 to 30 (with a step of 3) and r from -8 to 4 (with a step of 1).

Step 1: Understand the function's characteristics

First, understand the characteristics of the given polar equation \(r=\frac{8}{1+\cos \theta}\). This is a Limacon with a loop because the ratio of coefficients (8/1) is greater than 1 and less than 2. The pole, graph's center, is at origin and the maximum distance from the pole, which indicates the size of the Limacon, is the coefficient of the cosine term +1, which is 2 in this case.

Step 2: Determine the range of theta

Next, determine the range of theta. Because the Limacon completes a loop, theta should run through all possible angles from 0 to 2pi to capture the full graph.

Step 3: Determine the range of r

Then, determine the range of r. By replacing \(\theta\) in the equation by values between 0 and 2pi and checking the maximum and minimum 'r' outputs, we know that the r values lie between 0 and 8, because the maximum r value is when \(\cos \theta = -1\), which gives \(r = 8\).

Step 4: Establish the viewing rectangle

Now establish the viewing rectangle (window) that shows a complete graph and minimizes unused portions of the screen. Consider all the information above, the appropriate window values would be for theta :[-30, 30, 3] and for r: [-8, 4, 1]. This window will capture the full graph of the limacon and minimize the unused portion of the screen.

Key Concepts

Limacon

The Limacon is an interesting shape in polar coordinates described by equations like \( r = a + b\cos\theta \) or \( r = a + b\sin\theta \). It is named after the French scientist "Limaçon," meaning snail. The shape varies depending on the values of \( a \) and \( b \).

There are four main forms of Limacons:

• Limacon with an inner loop: occurs when \( \frac{b}{a} > 1 \). It looks like a looping snail shell.
• Cardioid: occurs when \( \frac{b}{a} = 1 \). This heart-like shape is popular in trigonometry.
• Limacon dimpled: occurs when \( 1 > \frac{b}{a} > 0.5 \). It has a slight indentation.
• Convex Limacon: occurs when \( \frac{b}{a} < 0.5 \). This shape is more circular and smooth.

In our case, the equation \( r = \frac{8}{1+\cos\theta} \) describes a Limacon with a loop since \( \frac{b}{a} = 8 \), which fits within the range for a looping Limacon.

viewing rectangle

A viewing rectangle is the range of coordinates selected on a graphing tool to display a complete and clear graph with minimal empty space on the screen.

To determine the viewing rectangle for a polar graph like a Limacon, we look at both \( r \) and \( \theta \):

• For \( \theta \): The range typically covers from 0 to \( 2\pi \), capturing all possible angles to form a complete graph.
• For \( r \): Analyze the minimum and maximum values of the radius; this shapes the outer boundary of the graph.

For the given exercise, the suggested viewing rectangle is specified as [-30, 30, 3] for \( \theta \) and [-8, 4, 1] for \( r \). This selection ensures that the entire shape of the Limacon is captured, and extraneous empty space is minimized on the display.

polar equation

Polar equations express points in a plane using radial and angular coordinates, \( (r, \theta) \). Unlike Cartesian equations \((x, y)\), polar equations revolve around the center or pole and rely on angles and distances.

Common polar equations include:

• Circular equations: such as \( r = a \), form perfect circles at any given radius.
• Rose curves: with equations like \( r = a \sin(k\theta) \), creating petal-like patterns.
• Spirals: like the Archimedean spiral, formulated as \( r = a + b\theta \).
• Limacons: as in this exercise, formulated as \( r = \frac{8}{1 + \cos\theta} \). These create distinct shapes based on ratios of angle coefficients.

Polar equations are crucial for visualizing phenomena like waves, orbits, and circular patterns in mathematics and physics.

r-theta graph

The r-theta graph, or polar graph, is a visual representation of data in polar coordinates \((r, \theta)\). Unlike traditional Cartesian graphs, where points are mapped by x and y coordinates, in polar graphs:

• The radius \( r \): indicates the distance from the origin.
• The angle \( \theta \): specifies the direction from the polar axis (usually the positive x-axis).

In r-theta graphs:

• Curves are plotted around the center point, creating circular and spiral patterns commonly used in engineering and navigation.
• The complete graph depends on the adequate range for \( \theta \), typically from 0 to \( 2\pi \). Adjusting this range ensures the entire shape, such as the Limacon, is visualized on the graph.
• Analysis of r-theta graphs is valuable for understanding periodic functions and wave-like behaviors due to their circular structure.

R-theta graphs highlight the beauty of symmetry and periodic repetition, offering unique insights that Cartesian graphs might not.