Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2-\cos \theta $$

Step-by-Step Solution

Answer

The graph is a dimpled limaçon, symmetric about the polar axis.

Step 1: Identify the Polar Equation

The given polar equation is \( r = 2 - \cos \theta \). This indicates that the relationship between \( r \) and \( \theta \) is not linear but rather a trigonometric function that can take on various shapes.

Step 2: Identify the Type of Curve

The equation \( r = 2 - \cos \theta \) resembles the equation for a limaçon with a dimple, based on its standard form \( r = a - b\cos\theta \). In this case, \( a = 2 \) and \( b = 1 \). Since \( a > b \), we expect a limaçon with an inner dimple.

Step 3: Determine Key Points

Evaluate the equation at key angles: - For \( \theta = 0 \), \( r = 2 - \cos(0) = 1 \).- For \( \theta = \frac{\pi}{2} \), \( r = 2 - \cos(\frac{\pi}{2}) = 2 \).- For \( \theta = \pi \), \( r = 2 - \cos(\pi) = 3 \).- For \( \theta = \frac{3\pi}{2} \), \( r = 2 - \cos(\frac{3\pi}{2}) = 2 \).- For \( \theta = 2\pi \), \( r = 2 - \cos(2\pi) = 1 \).

Step 4: Sketch the Graph Using Key Points

Plot the points obtained from the key angles:- Start by plotting \( (1, 0) \), \( (2, \frac{\pi}{2}) \), \( (3, \pi) \), \( (2, \frac{3\pi}{2}) \), and \( (1, 2\pi) \).- The graph will pass through these points, creating a dimpled limaçon shape. The dimples occur because as \( \cos \theta \) varies, \( r \) increases or decreases, particularly around \( \theta = \pi \), which is the furthest point (\( r = 3 \)).

Step 5: Confirm the Graph's Symmetry

Check for symmetry. The equation \( r = 2 - \cos \theta \) is symmetric about the polar axis (horizontal axis) because replacing \( \theta \) with \( -\theta \) does not change the equation. Additionally, the graph may exhibit rotational symmetry depending on the complete cycle around the pole.

Step 6: Final Sketch

Connect the points smoothly, ensuring that the curve displays the characteristics of a dimpled limaçon. The graph should be circular at the closed points, bending inward near \( \theta = \pi \) to form the dimple.

Key Concepts

Polar Equations

Polar equations describe points in a plane using a distance from a central point (the pole) and an angle from a reference direction (usually the positive x-axis). In contrast to Cartesian coordinates, which use x and y values, polar coordinates use the radius \( r \) and the angle \( \theta \). Here are some key points about polar equations:

• The radius \( r \) is the length from the origin to the point.
• The angle \( \theta \) is measured counterclockwise from the positive x-axis.

This coordinate system is particularly useful for equations involving circular shapes, as it can simplify the description and graphing of certain curves. For example, the polar equation \( r = 2 - \cos \theta \) describes a specific type of curve called a limaçon. Understanding how these equations are structured helps in sketching their associated graphs.

Limaçon

A limaçon is a type of polar curve represented by equations of the form \( r = a \pm b \cos \theta \) or \( r = a \pm b \sin \theta \). These curves often have intriguing shapes that can appear as rounded cardiods or with inward dimples, depending on the values of \( a \) and \( b \).For the equation \( r = 2 - \cos \theta \):

• Here, \( a = 2 \) and \( b = 1 \), resulting in a limaçon with a dimple.
• Since \( a > b \), the curve has a characteristic dimple, unlike when \( a = b \), which produces a cardioid.

The limaçon is a rich family of polar curves, and understanding their formation can help in visualizing and sketching these complex shapes.

Trigonometric Functions

Trigonometric functions like sine and cosine play a major role in polar equations. They define how the radius \( r \) changes with the angle \( \theta \). In the equation \( r = 2 - \cos \theta \), the cosine function dictates the shape by altering \( r \) as \( \theta \) varies.This is important for:

• Determining key points on the graph. For example, when \( \theta = 0 \), \( r = 1 \), and for \( \theta = \pi \), \( r = 3 \).
• Identifying symmetry and periodic properties in the graph.

Understanding how trigonometric functions affect polar graphs is crucial because the nature of sine and cosine leads to the periodic oscillations characteristic of these curves.

Graph Symmetry

Graph symmetry in polar coordinates helps predict the behavior and appearance of curves without plotting every point. It involves recognizing patterns in the equation when changing the angle or radius.For the polar equation \( r = 2 - \cos \theta \):

• The equation remains unchanged when \( \theta \) is replaced with \( -\theta \), indicating symmetry about the horizontal axis.
• This kind of symmetry helps confirm the limaçon is consistent above and below this axis, simplifying graphing.

Identifying symmetry in graphs is crucial when sketching them quickly and accurately, showing the power of analyzing equations beyond just plotting points.