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Chapter 6: Problem 49

Sketch the graph of each polar equation. $$ r=1+\cos \theta(\text { cardioid }) $$

Short Answer

Expert verified
Sketch a cardioid symmetric about the x-axis with key points where \(\theta = 0, \pi\/2, \pi,\) and \(3\pi\/2\).

Step by step solution

01

- Identify the Polar Equation

Recognize that the given polar equation is of the form \(r = 1 + \cos \theta\), which is a standard form of a cardioid.
02

- Plot Key Points

Determine key points by evaluating the equation at specific values of \(\theta\). For example, at \(\theta = 0\), \(r = 2\); at \(\theta = \pi\/2\), \(r = 1\); at \(\theta = \pi\), \(r = 0\); and at \(\theta = 3\pi\/2\), \(r = 1\).
03

- Determine Symmetry

Check the symmetry of the equation. This particular cardioid is symmetric about the polar axis (the x-axis) because \(\cos \theta\) is an even function.
04

- Sketch the Graph

Using the key points and the symmetry determined, sketch the curve. Start from \(\theta = 0\) and draw smoothly through the key points. The graph should exhibit the characteristic heart shape of a cardioid.
05

- Verify the Shape

As the range of \(r\) values and the smooth curve through the key points support the shape of a cardioid, verify that the overall shape is correct and they reflect across the polar axis as expected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cardioid
A cardioid is a special type of curve in polar coordinates defined by equations like \( r = a + b \, \text{cos} \, \theta \) or \( r = a + b \, \text{sin} \, \theta \). The shape resembles a heart, which is why it is named after the Greek word 'kardia', meaning heart. In our exercise, the equation is \( r = 1 + \, \text{cos} \, \theta \). This is a standard cardioid equation.
  • When \( \text{cos} \, \theta \) reaches its maximum at \( \theta = 0 \), \( r = 2 \).
  • As \( \theta \) changes, \( \text{cos} \, \theta \) varies between -1 and 1.
Thus, the value of \( r \) smoothly changes and forms the characteristic heart shape once graphed.
Polar Coordinates
Understanding polar coordinates is key to graphing the given equation. Instead of using x and y coordinates, in polar coordinates we use \( r \) and \( \theta \).
  • \( r \) represents the distance from the origin (pole).
  • \( \theta \) represents the angle from the polar axis (usually the x-axis).
This coordinate system helps to visualize the plot of complex curves like cardioids. Reading and plotting points involve finding the distance \( r \) and angle \( \theta \), and then connecting these points smoothly.
Graphing Polar Equations
Graphing polar equations involves several key steps. First, identify key points by substituting specific angles in radians into the equation. For our cardioid example, key points include:
  • At \( \theta = 0 \), \( r = 2 \).
  • At \( \theta = \frac{\text{蟺}}{2} \), \( r = 1 \).
  • At \( \theta = \text{蟺} \), \( r = 0 \).
  • At \( \theta = \frac{3\text{蟺}}{2} \), \( r = 1 \).
When connecting these points, ensure to capture the smooth transitions. This particular equation displays the characteristic 'heart' form due to these points.
Cosine Function Symmetry
The symmetry of the cosine function plays a crucial role in sketching the graph of our cardioid. Cosine functions, \( \text{cos} \, \theta \), are even functions. This implies that \( \text{cos}(-\theta) = \text{cos}(\theta) \).
  • For our cardioid, the equation \( r = 1 + \, \text{cos} \, \theta \) presents symmetry about the polar axis (x-axis).
  • Such symmetry simplifies the graphing process as we need to plot points for only half of the graph and then reflect them about the axis.
Understanding this symmetry enables easier and more accurate plotting of the polar graph, ensuring the resulting shape is correct and complete.

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