Problem 35 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation. $$r=\sqrt{3}+\cos \theta$$

Step-by-Step Solution

Answer

The graph is a limacon with an inner loop, reflecting its shape on both sides of the x-axis.

Step 1: STEP 1: Understand the Polar Equation

The polar equation given is \( r = \sqrt{3} + \cos \theta \). In polar coordinates, \( r \) represents the radial distance from the origin, and \( \theta \) is the angle. The formula indicates that the distance, \( r \), varies with the cosine of the angle \( \theta \).

Step 2: STEP 2: Determine the Shape

This equation is a type commonly known as a limacon. The standard limacon form is \( r = a + b \cos \theta \). Comparing with our equation, we have \( a = \sqrt{3} \) and \( b = 1 \) indicating it causes a loop since \( \, a > b \,\).

Step 3: STEP 3: Find Critical Points

Evaluate the function's output at key points: - At \( \theta = 0 \), \( r = \sqrt{3} + 1 = \sqrt{3} + 1 \).- At \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \), \( r = \sqrt{3} \).- At \( \theta = \pi \), \( r = \sqrt{3} - 1 \).

Step 4: STEP 4: Plot the Points and Curve

Plot the points found in Step 3. Connect these points, smoothly sketch the full curve using the symmetry about the x-axis due to the cosine function in the equation. Also, note that the curve will have a loop inside as \( a > b \).

Step 5: STEP 5: Finalize the Sketch

Make sure the sketch visually represents the loop at \( \theta = \pi \), showing it inside the main part of the curve. The loop occurs because at some angles, \( r \) becomes less than zero, meaning those parts of the curve reflect about the origin.

Key Concepts

Limacon

A limacon is a special type of curve found in polar coordinates. It can appear in various forms, including limacons with loops, dimples, or without any noticeable inward features, depending on the values in its equation. A typical limacon can be expressed in the form \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). The key is the relationship between \( a \) and \( b \):

• If \( a > b \), the limacon will have a loop.
• If \( a = b \), the curve becomes a cardioid, which looks like a heart shape.
• If \( a < b \), the limacon will not have a loop, but might have a dimple or be convex.

In our exercise, the equation \( r = \sqrt{3} + \cos \theta \) is a limacon with a loop. This is because \( a = \sqrt{3} \) and \( b = 1 \), and we observe \( a > b \). Understanding the limacon's form helps in identifying the curve's shape and its characteristics when plotted.

Graphing Polar Equations

Graphing polar equations involves plotting points based on how the radial distance \( r \) changes with the angle \( \theta \). To graph these equations, one follows a set of logical steps:

• Start by understanding the given polar equation. Identify the types, like circles, limacons, or roses.
• Determine the key characteristics, such as symmetry, maximum and minimum values of \( r \), and intersections with the axes.
• Choose important angles, typically at intervals like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), and calculate the corresponding \( r \) values.
• Plot these \( (r, \theta) \) points in the polar coordinate system.
• Connect these points smoothly, taking into account any symmetry and distinct features of the curve.

The exercise solution shows these steps in action for the limacon \( r = \sqrt{3} + \cos \theta \). It helps to plot points and note areas where \( r \) is less than zero, causing parts of the curve to reflect across the origin. This attention to detail ensures an accurate representation of the curve.

Trigonometric Functions

Trigonometric functions like \( \cos \theta \) and \( \sin \theta \) are fundamental in graphing polar equations. They dictate how the radial distance \( r \) varies with the angle \( \theta \). Here are some key points:

• Trigonometric functions are periodic, with specific ranges and symmetries that affect the curve's shape.
• \( \cos \theta \) is symmetric about the x-axis, meaning the limacon will be symmetric as well.
• Critical angles, where \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), help in identifying maximum, minimum, and key symmetry points.
• Values of \( a \) and \( b \) in equations like \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \) determine the existence and size of loops and dimples.

In the given exercise, the trigonometric function \( \cos \theta \) plays a crucial role by modifying the radial distance and introducing a loop where deviations like \( \frac{\pi}{2} \) reflect unique values for \( r \), essential to the limacon's shape.