Problem 42 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=5+4 \sin \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \[r=5+4 \sin \theta\] results in a Limaçon, a polar curve. When \(r\) and \(\theta\) spans between \(0\) and \(2\pi\), the graph forms a complete loop showing the symmetrical nature of the \(\sin\) function in the equation.

Step 1: Identify the nature of polar equations

Understand that the graph of a polar equation is a curve that consists of points that have the polar coordinates satisfying the equation. The equation given, \[r=5+4 \sin \theta\], is in the form of a limaçon, a polar curve that is defined by the equation in the form \[r = a + b \sin \theta\] or \[r = a + b \cos \theta\].

Step 2: Construct a table of values

Prepare a table of values that includes various angles \(\theta\) and the corresponding radius \(r\). Start by selecting values of \(\theta\) from \[0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{5\pi}{4}, \frac{4\pi}{3}, \frac{3\pi}{2}, \frac{5\pi}{3}, \frac{7\pi}{4}, \frac{11\pi}{6} \] and \[2\pi\]. These angles give a good spread across all four quadrants. Compute the radius \(r\) for each of these angles using the given equation.

Step 3: Plot the points

Using the table of values, plot the points on a polar graph for each pair of polar coordinates \((r, \theta)\). Make sure to understand that \(\theta\) is the angle made with the positive part of the x-axis and \(r\) is the distance from the origin.

Step 4: Sketch the graph

After all points are plotted, join the points together starting from \(\theta = 0\) and working way up to \(\theta = 2\pi\). The line should loop around before ending at the point you started, forming a complete graph of the polar equation.

Key Concepts

Sketching Polar Graphs

When sketching polar graphs, the process begins with understanding the polar equation that provides the relationship between the radius r and the angle θ. Every point on the graph is denoted by the polar coordinates (r, θ) which represent the point's distance from the origin (the radius r) and its angle with the positive x-axis (the angle θ).

To sketch the graph of a polar equation, follow these steps:

• Identify the type of polar equation and its characteristics, which will inform you about the general shape of the curve. For instance, certain equations might represent spirals, circles, or limaçon curves.
• Construct a table of values by choosing a range of angles θ and calculating the corresponding radius r for each angle from the polar equation.
• Plot points using the radius and angle values from your table, mapping each point on polar graph paper, which has concentric circles and radial lines to help locate points.
• Finally, join the plotted points to visualize the shape of the graph. It's important to consider symmetry and commonality between quadrants to help make accurate sketches.

With practice, you'll gain intuition for how different types of polar equations look when graphed, and you'll be able to sketch increasingly complex polar graphs with ease.

Limaçon Curve

The limaçon curve is a fascinating polar graph with a distinctive snail-like or teardrop shape. It's one of the more recognizable figures in polar graphing, and it's defined by equations of the form r = a + b sin θ or r = a + b cos θ where a and b are constants.

The shape of a limaçon curve is heavily influenced by the relationship between constants a and b. When |b| > |a|, the limaçon has an inner loop. If |b| = |a|, it has a cardioid shape. Without the inner loop, it represents a dimpled or convex limaçon depending on whether |b| is less than or greater than |a|, but still close in value. To sketch a limaçon successfully, key angle points should be selected to capture the cusps and loops precisely.

For the given example r = 5 + 4 sin θ, you would plot points for angles that highlight the characteristic features of the limaçon, such as at θ = π/2 where the inner loop might occur. By joining these points smoothly, keeping in mind the general limaçon properties, you could sketch its unique shape accurately.

Polar Coordinates

The polar coordinate system is an alternative to the Cartesian coordinate system. Instead of using x and y coordinates to define a location on a plane, polar coordinates use a distance from a central point (the origin) and an angle from a reference direction, typically the positive x-axis.

The polar coordinates (r, θ) consist of:

• The radial coordinate r, which measures how far away the point is from the origin.
• The angular coordinate θ, representing the counterclockwise angle from the positive x-axis to the point.

Polar coordinates are especially useful when dealing with problems involving circular or rotational symmetry. It's crucial to remember that a single point can be represented by multiple polar coordinates due to the periodic nature of angles and the fact that negative radii can place points in the opposite direction.

Converting back and forth between Cartesian and polar coordinates requires trigonometry. To convert polar coordinates to Cartesian, use x = r cos θ and y = r sin θ. For the reverse, the formulas are r = √(x² + y²) and θ = atan2(y, x), where atan2 is the two-argument arctan function that considers the signs of both variables to determine the correct quadrant of the angle.